Essays on Frege's Conception of Truth. (Grazer Philosophische Studien 75) (Grazer Philosophische Studien)

Essays on Frege’s Conception of Truth

Grazer Philosophische Studien INTERNATIONALE ZEITSCHRIFT FÜR ANALYTISCHE PHILOSOPHIE

GEGRÜNDET VON Rudolf Haller HERAUSGEGEBEN VON Johannes L. Brandl Marian David Leopold Stubenberg

VOL 75 - 2007

Amsterdam - New York, NY 2007

Essays on Frege’s Conception of Truth

Edited by

Dirk Greimann

Die Herausgabe der GPS erfolgt mit Unterstützung des Instituts für Philosophie der Universität Graz, der Forschungsstelle für Österreichische Philosophie, Graz, und wird von folgenden Institutionen gefördert: Bundesministerium für Bildung, Wissenschaft und Kultur, Wien Abteilung für Wissenschaft und Forschung des Amtes der Steiermärkischen Landesregierung, Graz Kulturreferat der Stadt Graz

In memoriam Georg Henrik von Wright

The paper on which this book is printed meets the requirements of “ISO 9706:1994, Information and documentation - Paper for documents Requirements for permanence”. Lay out: Thomas Binder, Graz ISBN: 978-90-420-2156-3 ISSN: 0165-9227 © Editions Rodopi B.V., Amsterdam - New York, NY 2007 Printed in The Netherlands

TABLE OF CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part I. Truth in Frege’s Formal System Hans SLUGA: Truth and the Imperfection of Language . . . . . . . . . . .

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Richard G. HECK, Jr.: Frege and Semantics . . . . . . . . . . . . . . . . . . . . .

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Danielle MACBETH: Striving for Truth in the Practice of Mathematics: Kant and Frege . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part II. Truth and the Truth-Values Michael BEANEY: Frege’s Use of Function-Argument Analysis and his Introduction of Truth-Values as Objects . . . . . . . . . . . . . . . . . .

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Dirk GREIMANN: Did Frege Really Consider Truth as an Object? . .

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Part III. Truth and Judgment Erich H. RECK: Frege on Truth, Judgment, and Objectivity . . . . . .

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Verena MAYER: Evidence, Judgment and Truth . . . . . . . . . . . . . .

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Part IV. The Nature of the Truth-Bearers Oswaldo CHATEAUBRIAND: The Truth of Thoughts: Variations on Fregean Themes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Marco RUFFINO: Fregean Propositions, Belief Preservation and Cognitive Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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INTRODUCTION This special edition of Grazer Philosophische Studien is dedicated to the interpretation, reconstruction and critical assessment of Gottlob Frege’s conception of truth, which lies at the core of his understanding of logic. The main doctrines of this conception are: truth is objective, i.e., the truth of a thought does not depend on the psychological act of holding the thought to be true; the sense of the word ‘true’ does not make an essential contribution to the senses of the sentences in which it occurs; the word ‘true’ is apt to indicate the essence of logic; truth is a simple concept that cannot be reduced to anything more fundamental; thoughts are the primary truth-bearers; truth is a norm of science; the True and the False are objects, not properties. The main motive for Frege’s investigations into the concept of truth derived from his conviction that in order to understand the task and the nature of logic correctly, it is necessary to get clear about the sense of the word ‘true’. He thought, in particular, that the psychologistic view of logic put forward by the majority of the logicians of his time is based on a misunderstanding of the nature of truth. Although Frege’s conception of truth is a central component of his overall logical system, there have been relatively few studies that are dedicated to it; the most important ones are collected in Gottlob Frege: Critical Assessments of Leading Philosophers, edited by Michael Beaney and Erich Reck, Vol. II, Frege’s Philosophy of Logic, London: Routledge, 2005, and in Das Wahre und das Falsche. Studien zu Freges Auffassung von Wahrheit, edited by Dirk Greimann, Hildesheim: Olms, 2003. The purpose of the present volume is to make a contribution to the important task of filling this gap by bringing together nine original essays on the topic. The main issues addressed are: the role of the concept of truth in Frege’s system, the nature of the truth-values and their relationship to truth and falsity, the relationship between truth and judgment, and Frege’s conception of the truth-bearers. The volume is divided into four corresponding parts. Most of the papers collected here were presented in an earlier form at the international symposium Frege’s Conception of Truth, held in Santa Maria, Brazil, from 7 to 9 December 2005. The papers deriving directly from

this symposium are those by Michael Beaney, Oswaldo Chateaubriand, Dirk Greimann, Danielle Macbeth, Erich Reck, Marco Ruffino, and Hans Sluga. I am pleased to also include two papers that were not given at the symposium, which are the papers by Richard Heck and Verena Mayer. I am grateful to the Department of Philosophy of the Federal University of Santa Maria for financial support. Special thanks are due to all the participants of the conference who made it such a successful event and to all the contributors to this volume for their cooperation and encouragement. Last but not least, I would also like to thank the editors of Grazer Philosophische Studien for making this volume possible. November 2006

Dirk GREIMANN

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Grazer Philosophische Studien 75 (2007), 1–26.

TRUTH AND THE IMPERFECTION OF LANGUAGE Hans SLUGA University of California at Berkeley

Summary Frege subscribed neither to a correspondence theory of truth nor, as is now frequently argued, to a simple redundancy theory of truth. He did not believe, in other words, that the word “true” can be dropped from the language without loss. He argues, instead, that in a perfect language we would not require the term “true” but that we are far from possessing such a language. A perfect language would be one that is fully adequate in the sense that it would allow us to state truths and truth-connections without ambiguities and contradictions. Ordinary language and the calculi we can construct on its basis are, on the other hand, always imperfect. In seeing these imperfections, Frege takes up an important line of late nineteenth century philosophical thinking which can be illustrated also by Nietzsche’s reflections on language. Frege and Nietzsche draw, however, diametrically opposed conclusions from the thought that our language proves imperfect.

When Frege set out in 1919 to summ arize his intellectual achievements for the historian of science Ludwig Darmstaedter, he called it distinctive of his conception of logic that it gives pre-eminence to “the content of the word ‘true’” (Frege 1979, p. 362). This insight had come to him, in fact, only slowly and over the course of some forty years. It had certainly not yet been evident in his earliest and most original work on logic, the Begriffsschrift of 1879. I have described the development of Frege’s thinking on truth elsewhere and I won’t repeat what I have said on those occasions (cf. Sluga (2001) and (2003)). My goal here is, rather, to explore a paradox that appears to arise from Frege’s idea that logic gives pre-eminence to the content of the word “true” when we conjoin it to his slightly earlier statement — in the note “My Basic Logical Insights” from 1915 — that the word “seems devoid of content” (Frege 1979, p. 323). My question is: how can the content of a word that seems devoid of content be distinctive of a conception of logic?

There is, of course, no direct contradiction in Frege’s words since he writes in the 1915 text only that the word “true” seems devoid of content, not that it actually is so. But other things he says in that note only reinforce our sense of paradox. While he rejects the idea that the word “true” might have no sense — since any sentence in which it occurred would then also lack sense — he maintains that it has only a sense that “contributes nothing to the sense of the whole sentence in which it occurs as a predicate,” and thus “does not make any essential contribution to the thought” (Frege 1979, p. 323). But how could the content of a word that contributes nothing essential to a thought be at the same time distinctive of a particular conception of logic? This is, however, exactly what Frege maintains in “My Basic Logical Insights” where he writes that precisely because the word “true” contributes nothing essential to a thought, “precisely for this reason” the word “seems fitted to indicate the essence of logic” (Frege 1979, p. 323). That the concept of truth indicates the essence of logic was no incidental idea on Frege’s part. He highlighted it also in the introductory sentences of “The Thought” in 1918-19 — an essay which was meant to serve, in turn, as an introduction to a final, informal, and philosophically inspired review of his entire work. It was at this exposed and decisive point that he spoke of the laws of logic as being nothing but laws of truth (cf. Frege (1979, p. 325)). In order to appreciate the full implications of this assertion one must recall that Frege had once spoken of arithmetic as part of logic and was still saying that it relied on logic in its deductive structure and that he considered arithmetic, in turn, to be the basis for probability theory and hence for a theory of induction and thus as the methodological base for the entire edifice of empirical science. It follows then that truth was, on Frege’s view fundamental to human knowledge as a whole and thus certainly also fundamental to the edifice of his own thought. But how could Frege imagine that truth plays such a foundational role if he also thought that the word “true” was or seemed devoid of content, if it contributed nothing essential to a thought? We are, so it seems, left with a puzzle — one that forces us to go back to the question how Frege actually understood the concept of truth and how his bewildering remarks about truth can be accommodated. There was, of course, a time when it was said with confidence that Frege had held simply and straightforwardly to a correspondence conception of truth. Some interpreters treated him even as a precursor of Tarski and his theory of truth in formalized languages. The view was not wholly unrea-

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sonable since it could draw on two connected considerations. The first was that Frege’s logic seemed to account for the truth and falsity of molecular and general propositions in terms of the truth and falsity of atomic ones and the second that his sense-reference semantics seemed to speak of atomic sentences of the subject-predicate form as true if the thing named by the subject-term falls under the concept designated by the predicate and analogously for relational sentences. The view of Frege as a correspondence theorist was shaken, however, by the belated realization that in “The Thought” he had argued strongly against that theory. For a while, most readers continued to ignore or downplay that essay as being an exception in Frege’s oeuvre and perhaps even a late aberration. But such dismissals were swept aside with the appearance of Frege’s Posthumous Writings which revealed that he had repeatedly argued against the correspondence theory of truth in papers going back to at least 1897. At this point the significance of a remark from the essay “On Sense and Reference” began also to dawn on some of Frege’s interpreters according to which “closer examination shows” that the sentence “the thought that 5 is a prime number is true” says nothing more than the simple sentence “5 is a prime number” (Frege 1997, p. 158). In consequence, Frege was reclassified now as a redundancy theorist of truth who holds that the predicate “is true” can simply be dropped from the language without any substantive loss. That characterization appeared at first sight confirmed also by “My Basic Logical Insights.” The note restates, in fact, the earlier claim of “On Sense and Reference” varying in essence only the example. The sentence “It is true that sea water is salty” Frege writes in 1915 says the same (no more and no less) than the sentence “Sea water is salty” (Frege 1979, p. 322). But a more careful reading of the 1915 note should have alerted readers that the characterization of Frege as a redundancy theorist was nevertheless unsatisfactory and just as much so as his earlier identification with the correspondence theory of truth. The redundancy theory would imply that the word “true” can be dropped from the language wherever it occurs as a predicate attached to a sentence. But Frege maintains, in fact, in “My Basic Logical Insights” that this is not always possible, that the word “true” cannot always be dispensed with. He asks himself accordingly: “How is it then that this word ‘truth’ though it seems devoid of content, cannot be dispensed with?” And he answers this question by claiming boldly: “That we cannot do so is due to the imperfection of language” (Frege 1997, p. 323). Here we encounter one more of those puzzling remarks that are so characteristic of Frege’s discussion of the concept of truth. The assertion

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that we cannot dispense with the word “true” because of the imperfection of languages calls certainly for explication and my discussion here is concerned with precisely that. This discussion will take us beyond the ascription of a redundancy view to Frege. It will lead us to conclude, instead, that he held a view more closely related to the old correspondence theory than might be suspected. While Frege rejected the idea that the truth of a sentence consists in its correspondence to a part of reality he held, instead, that in trying to express truth we must find a language that is adequate to the matters of which it speaks. I conclude therefore that in place of the idea of a theory of truth Frege adopted, in the end, what we may call an adequacy view of language. Off-hand, one might consider the issues thus raised to belong to the specialized field of Frege studies or somewhat more broadly as belonging to the technical examination of the concept of truth. I want to look at those issues here, though, in a more encompassing fashion. I begin with a review of what Frege actually said about truth specifically in his later writings and I end by placing his remarks and the issues they raise in a broader historical context. In looking at logical matters in this fashion I model my exposition on Jean van Heijenoort’s seminal essay “Logic as Language and Logic as Calculus”. I am aware that just like van Heijenoort’s piece my conclusions will have a somewhat speculative character. Frege’s critique of the correspondence theory I begin my discussion by considering first what Frege writes about truth in “The Thought”. He argues there that the attempt to define truth as correspondence breaks down since the concept is implicitly presupposed in the definition. When we try to determine whether an idea (or, for that matter, a sentence) is true, the correspondence view invites us to consider whether the idea (or the sentence) corresponds to reality. But this, Frege holds, comes to asking whether it is true that the idea or the sentence corresponds to reality. This kind of counterargument can be generalized to any other definition of truth. Any definition would have to say that a sentence is true if and only if it has a certain property T. But then in order to establish that the sentence is actually true, we would have to determine that it has the property T and this means for Frege we would have to determine that it is true that the sentence has property T. Frege draws from this the radical conclusion that not only the correspondence

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theory but also “any other attempt to define truth also breaks down”. In any such definition a characteristic would have to be specified that makes an idea or sentence true. But whenever we try to apply such a definition “the question would arise whether it were true that the characteristic were present.” It becomes apparent then “that the content of the word ‘true’ is sui generis and indefinable” (Frege 1997, p. 327). One is tempted to compare this argumentation to G. E. Moore’s proof in Principia Ethica that “good” is simple and indefinable. Any such projected definition, Moore holds, would have to be of the form “x is good if and only if x is P” where P is some none-evaluative or natural property; but any such definition would implicitly presuppose an independent grasp of the concept of good we are trying to define. For whenever we try to find out of some x whether it is good, we will have to determine not only that x actually has property P but also that having that property is really good (cf. Moore (1960, pp. 6–17)). Behind these considerations lay Moore’s commitment to a conceptual atomism according to which there are certain absolutely simple and, hence, indefinable concepts. In the essay “On the Nature of Judgment” of 1899 that served as a springboard for his and Russell’s defection from monistic idealism, Moore writes explicitly of truth as one such simple and indefinable concept and on that ground attacks Bradley’s relational conception of truth. Instead, he holds that “truth and falsehood are not dependent on the relation of our ideas to reality” (Moore 1899, p. 177). A proposition is, rather, constituted by a number of concepts together with a relation between them and “according to the nature of this relation the proposition may be either true or false”. However, this “cannot be defined, but must be immediately recognized” (Moore 1899, p. 180). While the formal parallels between Frege’s and Moore’s argument are striking we must not overlook that their conclusions are markedly different. For Moore, good is a property that things may possess but it is a non-natural property since it cannot be defined in natural terms. Frege, on the other hand, takes the indefinability of truth to show that it is no property at all. That is why the addition of the predicate “is true” to a sentence does not make an essential contribution to the thought expressed. We can compare Frege’s conclusion here to the one that R. M. Hare drew from Moore’s “naturalistic fallacy” argument. According to him the argument establishes not that good is a non-natural property but that the word “good” has a performative rather than a descriptive meaning. When we say “x is good” we are saying as much as “I like x, do so as well”; our utterance is thus, in other words, expressive and prescriptive in character (cf. Hare (1952)).

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In “The Thought” Frege speaks like Moore of “truth” in conjunction with evaluative terms like “good” and “beautiful”. According to the very first sentence of the essay “true” points the way for logic just as ‘beautiful’ does for aesthetics and ‘good’ for ethics (cf. Frege (1997, p. 325)). Since he does not elaborate on this comparison in the course of his essay, it is not easy to grasp its full implications. In order to determine those we must turn to an earlier, incomplete, and only posthumously published piece called “Logic” from 1897 which appears to have served as the immediate model for “The Thought” and certainly overlaps with it in both conceptions and formulations. As in “The Thought”, Frege rejects there the definition of truth as correspondence because “one would have to presuppose what is to be defined”. And he goes on to say that the same would hold for any explanation of the form: “A is true if it has such and such a property or if it stands in this or that relation to this or that” (Frege 1997, p. 228). Just as later on, he concludes that “truth is obviously something so primordial and simple that a reduction to something even simpler is impossible”. But in contrast to the later discussion he also suggests here that we are in consequence forced to illuminate what is unique in the predicate “true” by comparing and contrasting it to other predicates (cf. Frege (1997, p. 228)). Frege goes on to say that we may want to compare true specifically with the predicate beautiful (cf. Frege (1997, p. 231)). That comparison leads him to suggest that “logic can also be called a normative science” (Frege 1997, p. 228). But this is not quite his final word on the matter. He argues, instead, that there is an important difference between “true” and “beautiful”. While we are making an objective claim when we assert that something is true the same cannot be said when we call a thing beautiful. It is on similar grounds that he contrasts logical laws, i.e., laws of truth, in “The Thought” to moral and civil ones. When we call something beautiful or good our utterances are essentially normative in character, whereas the logical laws concern, he writes, first and foremost “what is”. As such they may, of course, in turn generate “prescriptions about asserting, thinking, judging, inferring”, but these are derivative in character and to think otherwise would court the danger of confusing different things (cf. Frege (1997, p. 325)). In “My Basic Logical Insights” he concludes therefore also that “beautiful” may, indeed, indicate the essence of aesthetics and “good” that of ethics, but that the word “true” “only makes an abortive attempt to indicate the essence of logic” (Frege 1997, p. 323). The considerations that led Frege to insist on the indefinability of truth in both “Logic” and “The Thought” are supplemented by some others

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he presented to his students at Jena in the Winter of 1910. According to Rudolf Carnap’s notes, Frege devoted almost all his attention that semester to the formal machinery of the Begriffsschrift. But he would allow himself also occasional asides of a more general, philosophical nature. On one of those occasions he said, according to Carnap: “Truth cannot be defined as ‘correspondence of an idea with reality’; for something objective cannot be compared to something subjective. Truth cannot be defined, analyzed, or reduced [to anything else]. It is something simple, primordial” (Frege 1996, p. 15). The argument is clearly elliptic but offers in passing at least a partial justification of the thesis that truth is simple and primordial. Frege appears to take it for granted that the correspondence in question would, in the first instance, have to be thought of as holding between our ideas or representations, on the one hand, and reality, on the other. But we know from his other writings that he considered such ideas or representations to be strictly speaking subjective and incommunicable. Two people are not prevented from grasping the same sense, he writes in “On Sense and Reference”, “but they cannot have the same idea … It is sometimes possible to establish differences in the ideas, or even in the sensations, of different men; but an exact comparison is not possible” (Frege 1997, p. 155). In his 1910 lectures he argues accordingly that such subjective ideas cannot be compared to objective reality. To assume that they could, would lend them an objectivity they cannot possess. If it were possible for me to determine that my idea I corresponds to a bit of reality R then I could communicate it to you by pointing to R and telling you that I now have the idea that corresponds to R. But if ideas are really subjective and incommunicable, it follows by contraposition that they cannot be said to correspond (or not correspond) to reality. Carnap’s notes do not mention whether Frege went on to consider the possibility of truth being a correspondence between something objective and something else equally objective. Could we not imagine truth to be a correspondence, for instance, between a sentence and a piece of reality? Sentences are certainly objective things in the world. But the same sentence, the same inscription, can, as Frege points out, have different meanings and can hence be considered both truth and false. Frege takes it to be evident for this reason that we cannot speak strictly of the truth or falsity of sentences but only of the truth and falsity of their senses. These senses are, of course, according to him to be conceived as fully objective. Why then should we not think of truth as a correspondence between the sense of a sentence and a bit of reality? Since Frege speaks of the senses of sentences as thoughts, we might

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also ask: why can’t we think of the truth of a thought as consisting in its correspondence to a piece of reality? Such questions force us to consider more precisely how Frege conceived of thoughts as senses of sentences. He did not, in fact, assume that besides the objectively real thoughts there exists also an ontologically independent realm of facts to which these thoughts may or may not correspond. In “The Thought” Frege declares, instead forthrightly that “a fact is a true thought” (Frege 1997, p. 342) and this remark recalls what he had said twenty years earlier: “Examples of thoughts are laws of nature, mathematical laws, historical facts” (Frege 1997, pp. 230–231). Facts are, on this view, evidently, not what thoughts are about but are themselves thoughts. While we tend to speak of thoughts as correlates of possible facts such correlates would be, in Frege’s terms, at best ideas or representations. But these are, as we have seen, unsuitable as truth-bearers not only because they are subjective but also because they are strictly speaking incommunicable. Fregean thoughts, on the other hand, constitute the world instead of being its representation. This idea is clearly not without its attractions. For it is plausible to assume that the identity criteria of facts must be intensional. The fact that Venus is the morning star is certainly different from the fact that Venus is the evening star. Facts can therefore, on Frege’s scheme, not be located at the level of reference where identity criteria are unfailingly extensional and the truth and falsity of a thought cannot be explained through its correlation or lack of such to a fact. Frege’s recognition that there are objectively true thoughts which no one has grasped comes thus to the assertion that there are facts unknown to us and this is surely something we want to grant. It follows, in any case, that Frege does not engage in a duplication of facts and correlated propositions in themselves. This kind of considerations leads naturally to the conclusions which Frege formulated so precisely for Ludwig Darmstaedter: What is distinctive about my conception of logic is that I give primacy 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 judgment; I come to the parts of a thought by analyzing the thought. (Frege 1997, p. 362)

The words might be taken to mean that truth must be considered a semantically primitive term and that all other semantic notions may be explained by means of it but that they cannot, in turn, be used to explain the notion of truth since they all presuppose it. The notes for Darmstaedter appear,

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indeed, to show how by starting with the concept of truth as basic one can come to the notions of sense and reference, the distinction of functions and objects, and to all the other fundamental notions of Fregean logic. But Frege’s formulations are not as sharp here as one would like them to be. If we follow “My Basic Logical Insights”, we should say that according to Frege truth is not some kind of property, whether natural or non-natural, whether semantic or other. He maintains, indeed, in that text that the essence of logic is really not to be found in the content of the word “true” at all but in “the assertoric force with which a sentence is uttered” (Frege 1997, p. 324). No word corresponds (or can strictly correspond) to this force. The paradoxical thing about “true” is that it seems to transform the assertoric force into a contribution to the thought. It thus “seems to make the impossible possible” (Frege 1997, p. 323). But, of course, it only seems to achieve this feat. The impression that the word “true” must have a sense is therefore ultimately misleading. “Truth” is not really a semantically simple notion. It is simple rather in the sense that the assertoric force is simple. Frege vs. Bolzano It helps at this point to contrast Frege’s treatment of truth to Bolzano’s. Their readers have occasionally noticed that the two shared certain convictions. They were both, for instance, determinedly anti-psychologistic in outlook. They both considered logic, in other words, to be concerned with something objective and not with mental processes. Frege’s objective thoughts appear to have their precise counterpart in Bolzano’s equally objective propositions in themselves. Such affinities have led some interpreters to postulate even that Frege derived some of his most distinctive ideas from Bolzano. The aficionados of “Austrian philosophy” — i.e., of the claim that there is a distinctive, influential, and separate Austrian tradition in philosophy — have been particularly insistent on this point despite the lack of any positive support for it. All we can reliably say is, rather, that Frege and Bolzano were both familiar with the philosophy of Herbart and that they may both have derived their anti-psychologism from him. (Lotze was for Frege no doubt also a source in this respect.) There are, on closer view, in any case also significant disagreements between the two and those make it rather pointless to bracket them closely together. Frege, for instance, rejected the correspondence theory of truth, as we

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have seen, whereas Bolzano was one of its determined defenders. Frege was, moreover, preoccupied with the project of constructing a logical calculus — an undertaking totally alien to Bolzano who developed his logic in the traditional manner with a view to ordinary language. These two differences are, furthermore, connected, as I will argue. Frege’s reflections on truth are, as we shall see, deeply linked to the differences he perceived between ordinary language and a formal notation. This justifies us, in fact, in saying finally that Bolzano and Frege stood on two different sides in the divide that separates classical from contemporary logic and with that also on different sides of the division between traditional philosophy which built on the assumptions and concepts of classical logic and a new kind of philosophizing that emerged in the late nineteenth century and that relied on the logic that Frege and his followers (i.e., specifically, Russell, Wittgenstein, and Carnap) set out to design. There is certainly no doubt that Bolzano fully accepted the traditional Aristotelian conception of truth and that in doing so he accepted at the same time quite uncritically also its broad philosophical implications. He did so, moreover, knowing also that some ancient and some modern philosophers had formulated alternatives to the Aristotelian doctrine of truth. He knew, for instance, that Sextus Empiricus had interpreted the Greek word “aletheia” to mean the unhidden, “to me lethon” (cf. Bolzano (1929, p. 111)) — thus anticipating Martin Heidegger’s controversial interpretation of truth as unhiddenness. He also knew of the modern principle omne ens est verum asserted by Locke, Wolff, and Baumgarten among others which sought to make “metaphysical truth” into a property of things themselves. But he dismissed such alternatives, offhand, as “utterly useless and devoid of sense” (Bolzano 1929, p. 143). “The first and most distinctive” use of the term “true”, he wrote, is, instead, the one according to which we understand it as a certain characteristic of propositions “by means of which they state something as it is” (Bolzano 1929, p. 108). This was, indeed, as Bolzano added, also the opinion of Aristotle. He took the Aristotelian conception to mean moreover specifically that a proposition is true “whenever the object with which it deals really has the properties that it ascribes to it” (Bolzano 1929, p. 112). More precisely he said that “in every proposition there must be an object with which it deals (the subject) and also a certain something that is said of this object (the predicate). In a true proposition, moreover, that which is said of the object must really belong to it” (Bolzano 1929, p. 122). Bolzano subscribed thus not only to Aristotle’s general formula of what truth is but to that

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particular interpretation of it which assumed that judgments are essentially composed of subjects and predicates. Adopting a formula first proposed by Malebranche, he asserted that “veritas nil aliud est, quam relatio realis sive aequalitatis sive inaequalitatis” (Bolzano 1929, p. 119). He departed from the tradition only in one respect and that was in not wanting to call this view a correspondence theory of truth for the term “correspondence”, so he complained, lacked the appropriate logical precision. He wrote: I cannot omit the demand that one should indicate precisely what is meant to be understood by the correspondence (Übereinstimmung) which is supposed to obtain between ideas or propositions and their correlated objects. One can certainly not imagine here an absolute identity or sameness. For propositions or ideas are not absolutely the same as the objects to which they refer; nor are the properties of the former also properties of the latter. (Bolzano 1929, p. 128)

The relation that makes a proposition true must, in fact, not be conceived as one of similarity between our ideas and reality. Bolzano was worried at this point that such a characterization might introduce an entirely unwanted subjective element into the concept of truth. The threat of subjectivism constitutes, indeed, one of the major concerns of Bolzano’s Wissenschaftslehre. In order to escape it, he considered it necessary to “separate the logical from all admixture of the psychological” — a formula he derived from Herbart and which recalls for us Frege’s similar words in the Foundations of Arithmetic (cf. Frege 1997, p. 90)). In further agreement with Herbart, Bolzano wrote in the same passage also of the necessity to reveal “the judgment as no appearance in the mind, but as something objective” (Bolzano 1929, p. 85). His preferred term for the judgment conceived objectively was “proposition in itself ”. Such a proposition in itself was for him the genuine, actual bearer of truth and falsity. But Bolzano warned us not to interpret the term “proposition” here in the ordinary way. He wrote: “Through its derivation from the verb “to propose” the term “proposition” used here suggests admittedly an action, a something proposed by someone (in other words, something that is produced or altered in some way). But in the case of truths in themselves one must ignore this” (Bolzano 1929, p. 114f.). We must, for the same reason, also not conceive of a proposition as “saying” or “stating” something; those expressions are once again strictly speaking only figurative (uneigentlich) in meaning. When we speak of an assertion (Aussage) as true or false, that too must be considered a figurative expression. For in reality truth and falsity are not asserted.

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A proposition in itself in Bolzano’s sense must certainly not be identified with either a sentence or a subjective thought or idea associated with such a sentence. Hobbes may have believed that only verbal assertions can be considered true and false and that truth belongs to words and not to things and that therefore only beings capable of language can possess truths, Bolzano argued. But this view is either “merely love for the absurd” or rests on confusing representations (Vorstellungen) with the words we use to indicate them (cf. Bolzano (1929, p. 144)). A proposition is, in Bolzano’s competing view, rather “the sense which a certain combination of words can express” (Bolzano 1929, p. 121). Propositions, understood his way, “have no real existence, i.e., they are not something that is in any particular place, at any particular time, or is in any other way something real” (Bolzano 1929, p. 112). To illustrate this point, Bolzano wrote that “the number of blossoms that were on a certain tree last spring is a statable, if unknown figure. Thus, the proposition which states this figure I call an objective truth, even if nobody knows it” (Bolzano 1929, p. 112). For Bolzano there were thus, in other words, both non-propositional facts and propositions in themselves and these are ontologically distinct from each other. Does this mean that there might have been a world that contained those facts but no propositions in themselves or, more strangely, a world of propositions in themselves without any non-propositional facts? Might the world have contained that blossoming tree last spring with its specific number of flowers but no proposition stating that this was so? Or, alternatively, a world of propositions in themselves in which one of them concerns the blossoming tree last spring but no non-propositional facts and no blossoming tree? Bolzano appears to rule out both possibilities but on what grounds is unclear. He finds himself thereby caught in a puzzling and not further argued for duplication of entities that postulates in addition to every fact in the world a corresponding true proposition in itself. And for this reason there arises for him immediately the question of the nature of their relation. On Bolzano’s view, that relation is one of adequation which links the fact and the proposition and does so independently of our knowing of it. The fact and the proposition in itself thus each have a relational property whose existence has nothing whatever to do with human thought and our practice of making assertions. Logic is on his view not at all concerned with the assertoric force of our utterances but with certain timeless entities and their timeless relations. All this clearly separates Bolzano from Frege. For the latter logic proves to be ultimately concerned with the assertoric force, truth is not a relation, the truth of

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a thought is not one of its properties, and the fact p and the thought p coincide. It may strike us, of course, at this point that Bolzano’s position comes closer to common sense but that means, in effect, only that he comes closer to what our philosophical tradition has always taught us. Bolzano’s conception of truth certainly remains committed entirely to that tradition; he holds on for that reason also to the idea that judgments are essentially composed of subjects and predicates and for this reason he is, in the end, incapable of seeing the limitations of classical logic and of the need of getting beyond them. In all this he stands on the other bank of the great divide that splits nineteenth century philosophical thinking. He can, for that reason, not seriously be considered part of the analytic tradition in philosophy that emerged in the last decades of that century and of which Frege is surely its first representative. But all I have said so far characterizes Frege’s view on truth only negatively. His assertion that truth is simple and indefinable still demands positive explication. We still need to ask then how Frege actually conceived of truth and why he was not, in the end, a mere redundancy theorist. In order to answer those questions we must turn our attention to something else that separates Bolzano and Frege and that is the latter’s concern with the defects of ordinary language and with the need for the construction of a formalized logical language. An adequate language When Frege first set out to determine whether arithmetical propositions should be considered empirical, synthetic a priori, or perhaps even analytic truths, he quickly discovered a need to supplement the common arithmetical notation with symbols expressing logico-deductive relationships. In trying to supplement the resources of mathematics and ordinary language he proceded at first in a somewhat makeshift fashion. In the 1879 Begriffsschrift he confessed: “In my first draft of a formula language I was misled by the example of ordinary language into constructing judgments out of subject and predicate” (Frege 1997, p. 54). But he soon discovered this to be unsatisfactory and now set out to construct a notation in which “everything that is necessary for valid inference is fully expressed; but what is not necessary is mostly not even indicated” (Frege 1997, p. 54). He called this notation a conceptual script since in contrast to ordinary language it was not intended to stand in for spoken language “but directly

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expresses the facts without the intervention of speech” (Frege 1972, p. 88). We have by now, of course, become so familiar with logic presented in a symbolic notation that we no longer appreciate the remarkableness of Frege’s invention. It was Kurt Gödel who had to remind us that “the first comprehensive and thorough going presentation of a mathematical logic and the derivation of Mathematics from it” — i. e., Russell and Whitehead’s Principia Mathematica — was “so greatly lacking in formal precision in the foundations, that it presents in this respect a considerable step backwards as compared with Frege” (Gödel 1944, p. 126). But how did Frege come to his conception of a symbolic calculus in the first place, and what did this invention mean to him at the time? I have examined that story at some length in other places (Sluga 1980 and 1984) but since it has much to do with his thinking about the concept of truth, it will pay to highlight certain aspects of it once more. The intellectual world into which Frege grew up was characterized first and foremost by a revival of interest in Kantian philosophy which was at the time bringing a thirty-year period of philosophical decline and stagnation to an end (cf. Windelband (1909)). The only productive philosopher of the preceding age had been Hermann Lotze at Göttingen, a respected, knowledgeable, and even weighty professional, but also, indubitably, a transitional figure whose name has largely vanished from our consciousness and whose writings make today for difficult reading. Historically speaking, Lotze’s main function has been to bridge the abyss between the established concepts and presumptions of the philosophical past and the new ideas motivating the middle and late nineteenth century. In seeking to reconcile Kant and Hegel, idealism and naturalism, science and revelation, Lotze laid the ground on which a new generation of philosophers could build. He became, thus, a forerunner not only of the Neo-Kantians but also of the British idealists, of Brentano and Husserl with his work in psychology; and he also provided direction and inspiration for Frege’s logico-philosophical undertakings. Frege had begun his University education at Jena where he heard Cuno Fischer’s lectures on Kant. Fischer himself was one of the seminal figures in the emergent Neo-Kantian movement and Jena, where Frege was to spend his career teaching mathematics, was to become one of the havens of Neo-Kantian thought. Because Fischer published the second edition of his Kant book the year in which Frege attended his class, we can say with some assurance what he must have talked about in his lectures. Fischer’s central concern was to show that the status of mathematical truth was decisive for

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the entire Kantian system — a claim that would certainly have stirred the interest of Frege, the budding young mathematician. The most pressing issue for Kant was, according to Fischer, whether mathematical truths are synthetic a priori or empirical in character — since they could obviously not be analytical or logical truths. Transcendental idealism stood and fell, in any case, with this fundamental decision. When Frege transferred to Göttingen for his doctoral studies, he came into contact with Lotze and his very different conception of the nature of mathematical truth. In his Logic, Lotze restated the Leibnizian thesis that the propositions of mathematics are analytic and, indeed, logical truths. He considered it evident that “the principles of mathematics have their systematic place in logic” and that mathematics must be considered “an independently progressive branch of universal logic”. But he cautioned at the same time that the complexity of modern mathematics “forbids any attempt to re-insert it in universal logic” (Lotze 1888, p. 35). These remarks appear to have proved a challenge to Frege’s philosophical and mathematical ingenuity for shortly after the publication of Lotze’s Logic he embarked on the actual attempt to derive arithmetic from logic. In trying to carry this project through, Frege wrote, he was forced to see, however, “how far one could get in arithmetic by inferences alone, supported only by the laws of thought that transcend all particulars” (Frege 1997, p. 48). But he found it difficult to assure himself that his chains of inference were free of gaps and that no intuitive assumptions were being smuggled into his lengthy deductions. The difficulty, he concluded, was due to “the inadequacy”, “the cumbersomeness, and “lack of precision” of language (cf. Frege (1997, p. 48)). The idea of the imperfection of ordinary language was gaining once again ground at the time Frege was writing these words after it had been first voiced by Leibniz two hundred years earlier but then been set aside by later philosophers. Kant, in particular, had shown no interest in the Leibnizian project of a lingua characteristica and the philosophers after him from Hegel to Lotze had either ignored or downplayed its significance. Lotze’s dismissive remarks on Boolean algebra were wholly indicative of where philosophers stood on this issue. Meanwhile, however, mathematics was undergoing a process of rapid formalization and the culture at large was coming to concern itself with the values of the standardized, the rationally constructed, the industrially produced, the uniformly made, the mechanical, the purified and refined as against the raw, the grown, and merely natural. Frege’s thoughts on this question were affected by a discussion of Leibniz’s lingua characteristica that he found in Adolf Trendelenburg’s

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Historische Beiträge zur Philosophie, a three volume collection of essays on various themes in the history of philosophy. We may wonder why the young mathematician would have found reason to look at Trendelenburg’s book in the first instance. The answer is probably that the relevant third volume which Frege consulted also contained an attack on Cuno Fischer’s claims about the supposed link between the synthetic a priori truth of mathematical propositions and Kant’s transcendental idealism. Trendelenburg disagreed strongly with those claims and considered the Kantian view of mathematical truth to be fully compatible with a realist conception of space and time. Frege would have known about some of the earlier stages of the Fischer-Trendelenburg controversy from Fischer’s Kant book and possibly also from his lectures. That matter would have been of immediate interest to him since (as we realize now) his own earliest foundational interests were focused on geometry, a part of mathematics where he agreed with the Kantian position that truths are synthetic a priori. But it appears that Trendelenburg’s essay on Leibniz made a greater impression on him than his attack on Fischer. We can see this not only from Frege adoption of the term that Trendelenburg had coined as a name for Leibniz’s lingua characterica and that from now on he called his own symbolic logic a Begriffsschrift. Trendelenburg’s book is also the only work referred to in the Begriffsschrift of 1879 and when Frege sought to justify his new symbolism in philosophical terms three years later in the essay “On the Scientific Justification of a Begriffsschrift” he once more relied on Trendelenburg’s observations. According to the latter, Leibniz had endeavored to construct an artificial language specifically designed to make up for the shortcomings of ordinary language. Leibniz’s language, so Trendelenburg, was in contrast to natural languages meant to represent concepts directly and to do so in a written notation. Trendelenburg argued persuasively that signs are certainly necessary for human thinking, “both for the solitary process of thought by itself and the busy exchange of thought in human life” but he warned at once that in ordinary language the linguistic signs “have only to a small part an inner relation to the designated idea”. The connection between the sign and what it designated was, thus, ordinarily, as Trendelenburg put it, “one-sided”, “indeterminate and arbitrary”, “obscured”, and the result of “blind habit”, rather than the product of a discriminating consciousness; it was, in other words, “psychological, rather than logical” in character (cf. Trendelenburg (1867, p. 3)). The realization of these shortcomings had led Leibniz to conceive of the possibility of a characteristic language, but

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that project, though undoubtedly worthwhile, had remained incomplete in Leibniz’s hands. Having begun work on it around 1676, and thus shortly after his invention of the calculus, Leibniz had more or less abandoned it ten years later when he undertook his journey to Italy for the purpose of historical research. The construction of a true lingua characterica, of a fully realized Begriffsschrift, was thus still to be undertaken. Frege was to use exactly the same considerations in his essay “On the Scientific Justification of a Begriffsschrift” in order to explain and defend his new logic and its accompanying symbolism. Ordinary language, he wrote in that essay, “proves to be deficient… when it comes to protecting thought from error” (Frege 1972, p. 84). He noted also that “language is not governed by logical laws” and that its shortcomings “are rooted in a certain softness and instability” (Frege 1972, pp. 84 and 86). Its words are often ambiguous and this proves to be “most dangerous” in that it generates the possibility that the same word may designate a concept and a single object which falls under that concept. Generally, no strong distinction is made between concept and individual. ‘The horse’ can denote a single creature; it can also denote a species … Finally, ‘horse’ can denote a concept. (Frege 1972, p. 84)

The distinction between objects and concepts and consequently that between first level concepts that apply to objects and second level concepts that apply to first level concepts had already proved of major importance to Frege in his Begriffsschrift and it was to occupy him again and again in subsequent writings when he raised the question, for instance, whether the numbers were to be understood as concepts or as objects, or in his construction of the extensions of concepts or, more generally, of valueranges of functions as logical objects, and also finally in his philosophical reflection “On Concept and Object” — an essay in which he spoke of the distinction as categorical in nature and as one that we can grasp but not fully characterize in words. One of the decisive failures of ordinary language was then that it failed to mark properly this logically basic distinction. Three years earlier, Frege had already concluded in his Begriffsschrift that the reduction of arithmetic to logic demanded first of all a reform of logic itself and a break with the limitations of the Aristotelian theory and that it required secondly the replacement of ordinary language with an appropriate formal symbolism modeled on that of mathematics. When he had first sought to prove that arithmetic could be derived from pure logic,

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Frege wrote, “I found an obstacle in the inadequacy of language … Out of this came the idea of the present Begriffsschrift” (Frege 1997, p. 48). Doubts about the reliability of ordinary language were not confined to those who like Frege (and Trendelenburg) concerned themselves with mathematics, logic, and more broadly with science. We can see how far those doubts reached in this period when we put Frege and Nietzsche together. At first sight, those two may be thought to have almost nothing in common. Born only four years apart, they belonged admittedly to the same generation and they grew up in the same part of the world and were thus exposed to the same general ideas. But apart from these biographical circumstances the two seem to have shared very little. Nietzsche was, after all, a philologist by training, Frege a mathematician. One wrote on history, metaphysics, and morals, the other on mathematics, logic, and meaning. They certainly lived in different social spheres and just as certainly never knew of each other. But their distance is not as absolute as might be thought. The two shared, for instant, an intense preoccupation with the question of truth. They were both also critical of the correspondence conception of truth. And they were both convinced of the imperfection of our language. In Beyond Good and Evil Nietzsche spoke dramatically of truth as a mystery, a sphinx which our philosophers have hardly begun to decipher. He asked himself: What is truth? Are there indubitable, philosophical truths? Why do we want truth at all? What stands in the way of obtaining it? Every statement we make about the world, he argued, contains always already an interpretation and we have no reasons to assume that reality ever corresponds to the categories of our thinking. What stands in the way is the fact that our thought is so deeply determined by grammatical categories and by distinctions that are merely historical and psychological in origin. The philosopher says “I think” but “what gives him the right to speak of an ‘I’, and even of an ‘I’ as cause, and finally of an ‘I’ as cause of thought?” (Nietzsche 1973, section 16). We would be better advised to say “it thinks”, but even that would be misleading. This ‘it’ already contains an interpretation of the event and does not belong to the event itself. The inference here is in accordance with the habit of grammar: thinking is an activity, to every activity pertains one who acts, consequently —. (Nietzsche 1973, section 17)

Philosophers speak likewise of the will as if it were the best-known thing in the world. But willing is “a unity only as a word — and it is precisely in

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this one word that the popular prejudice resides which has overborne the always inadequate caution of the philosophers” (Nietzsche 1973, section 19). What stands in the way of truth is first of all our language and secondly our human psychology. In Beyond Good and Evil Nietzsche discerns a family resemblance “between all Indian, Greek, and German philosophizing” and this he considers to be due to a “language affinity” and thus to “the unconscious domination and directing by similar functions” (Nietzsche 1973, section 20). The subject-predicate distinction is built into the grammar of all Indo-European languages and with it our philosophical belief in the distinction between subjects and objects. Nietzsche writes: “Philosophers of the Ural-Altaic languages (in which the concept of the subject is least developed) will in all probability look ‘into the world’ differently and be found on different paths from the Indo-Germans and Moslems” (Nietzsche 1973, section 20). That our language says “Lightning strikes” may suggest to the Indian, Greek, and modern philosopher a distinction between a substance and its action but what he ends up with is only a linguistically induced metaphysical picture. Nietzsche expounds on this theme also in the essay-fragment “On Truth and Lie in a Nonmoral Sense” which probably served as the source for his reflections in part one of Beyond Good and Evil. He speaks there of concepts being formed through the recognition of the similarity of different things. “Every word instantly becomes a concept precisely insofar as it is not supposed to serve as a reminder of the unique and entirely individual original experience” (Nietzsche 1979, p. 83). But this demands at the same time a transformation of our perceptual metaphors into schemata, the construction of a great edifice of concepts which displays rigid regularity and “exhales in logic that strength and coolness which is characteristic of mathematics” (Nietzsche 1979, p. 85). How we organize these concepts will differ from language to language and from language-group to language-group. Hence, the variations between the Indo-European and the Ural-Altaic languages. These schematic forms are, of course, not entirely arbitrary. “The spell of grammatical functions is in the last resort the spell of physiological value-judgments and racial conditions”, or to speak more cautiously of psychological factors (Nietzsche 1973, section 20). Nietzsche’s remarks about the variability of grammatical forms refers us, of course, in the first instance to the nineteenth century rise of philology and its recognition of the variability of human languages, while his explanation of these variations as grounded in physiological and psychological facts refers us to the emergence of physiology, psychology, and anthropol-

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ogy in the same period. When Nietzsche speaks here of a psychological explanation of the linguistic forms, one must think, of course, not only in terms of modern empirical psychology but must recall also the philosophical psychology of the Kantian system. Nietzsche is, indeed, explicit on this. He allows provisionally that Kant may be right in suggesting that the forms of our judgments are determined by the pure concepts of the understanding and that synthetic a priori judgments depend on our specific intuition of space and time. But he uses these observations to draw the anti-Kantian conclusion that we have no reason to regard any of those judgments as true. “Synthetic judgments a priori should not ‘be possible’ at all; we have no right to them, in our mouths they are nothing but false judgments” (Nietzsche 1973, section 11). In the essay fragment “On Truth and Lies” he asked himself dramatically therefore: “What then is truth?” and answered equally dramatically: “A movable host of metaphors, metonymies, and anthropomorphisms: in short … truths are illusions which we have forgotten are illusions” (Nietzsche 1979, p. 84). Nietzsche’s whole philosophizing was ultimately based on this particular conception of truth. The conclusions he drew from his observations on language, its grammatical forms and their psychological foundation, were surely different from Frege’s but the remarkable thing is that the two thinkers shared a belief in the unreliability and untrustworthiness of language and that their entire philosophical projects were built on that presupposition. There are, of course, other ways to think about ordinary language and its relation to a symbolic notation. Thus Wittgenstein insisted in his Tractatus that ordinary language is perfectly all right as it stands, that it has, indeed, a precise logical structure, but that it hides that structure under the irregular surface of its grammar. The logical symbolism cannot, therefore, improve on ordinary language; it can only make the logic of our language more visible. Wittgenstein’s decisive idea is here the distinction between surface and deep structure — a distinction that has gained new life more recently in structural linguistics. The later Wittgenstein held on to the idea that ordinary language is all right but abandoned the surface-deep structure distinction. According to his later view — taken up and expanded by the so-called ordinary-language philosophers of the 1950’s and 60’s — ordinary language is, in fact, the only language we initially understand and any new idiom or notation will have to be explained in its terms. There is, thus, no escape from the assumptions that are built into our ordinary modes of speaking. The symbolic notations of modern logic can be thought of only as extensions of our old ways of speaking not as a replacement for them;

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they certainly must not be conceived as the substitution of something perfect for something mortally flawed. As Wittgenstein put it metaphorically in his Philosophical Investigations: “Our language can be seen as an ancient city: a maze of little streets and squares, of old and new houses with additions from various periods; and this surrounded by a multitude of new boroughs with straight regular streets and uniform houses” (Wittgenstein 1953, section 18). Frege often expressed himself in a similarly cautious manner about the meaning of his conceptual notation. Such a notation he wrote in the preface to the Begriffsschrift has only a limited and instrumental function. It is like the microscope in relation to the naked eye — sharper in its definition but also more limited in its uses. And in the essay “On the Scientific Justification of a Conceptual Notation” he spoke of it for similar reasons as an artificial hand compared to our real, more flexible ones. But under the influence of Leibniz and Trendelenburg, he also at other times conceived of it as a “symbolism suited to things themselves” (Frege 1997, p. 50). He admittedly chided Leibniz for his overly optimistic view on how much it would take to construct such a language but he thought at the same time that there was no reason to despair “if this great aim cannot be achieved at the first attempt” (Frege 1997, p. 50). What was needed, in fact, was “a slow, step by step approach”. Eventually, such a notation might even help “philosophy to break the power of words over the human mind, by uncovering illusions that through the use of language often almost unavoidably arise concerning the relations of concepts”. It might free thought “from the taint of ordinary linguistic means of expression” (Frege 1997, pp. 50–51). Frege was looking, in other words, for what Leibniz had called “an adequate language”, one that was suited to things themselves, first in arithmetic, then in physics and other sciences, and finally in philosophy. In such a language we would speak with absolute clarity; all its sentences would be determinately true or false; and through appropriate reasoning we could avoid saying things that are, in fact, false. For this we would not need to mobilize words like “true” and “false”. It would be sufficient that our language is adequate to the world. No explicitly formulated theory of truth would, in any case, guarantee this; our guarantee would be contained instead in the successful practice of making assertions. Frege understood, however, that we are far from having such an adequate language and he certainly did not consider his conceptual notation as providing us with such a language. His notation had, after all, as yet only

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a limited number of uses. Elsewhere we still had to mobilize the imperfect resources of our ordinary language. And in this language we find ourselves forced to say that the same sentence (e.g., “It’s raining”) may be true now but not later and that another one may be true when uttered by one person but not by another or in one place and not in another. We might have to say also that some of the statements in a collection of statements are true while others are not without being able to specify which are and which are not. In all these circumstances the use of such words as “true” or “false” cannot be avoided. The words “true” ands “false” are for those reasons far from redundant in our imperfect language. “If our language were more perfect”, we would, of course, have no need of the word “true”. In such a language its adequacy would show itself. We would have no use at that point of a logical theory, but would “read it off from the language”. But we are, naturally, far from that. In writing this Frege was, no doubt, painfully conscious of what had happened to him in trying to lay logical foundations for arithmetic. He had assumed at that time that one could turn a concept expression into the name for an extension of the concept and thus move from concepts to objects. From the concept fixed star one could in this way, for instance, move to the extension of the concept fixed star. Because of the definite article, this expression appears to designate an object; but there is no object for which this phrase could be a linguistically appropriate designation. From this have arisen the paradoxes of set theory which have dealt the death blow to set theory itself. I myself was under this illusion when, in attempting to provide a logical foundation for numbers, I tried to construe numbers as sets. (Frege 1997, p. 369)

We must realize then that “work in logic is, to a large extent, a struggle with the logical defects of language” — a struggle that is as yet by no means over. Only after this struggle has been completed, if it ever can be so, will we possess “a more perfect instrument”. Until that moment we have to recognize that language, i. e., ordinary language, “remains for us an indispensable tool” and in that language the word “true” is far from redundant (cf. Frege (1997, pp. 323–324)). The historical context When I put earlier Frege and Nietzsche together as sharing certain fundamental concerns with respect to language, meaning, and truth I was trying

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to indicate at the same time the scope of the newly revitalized philosophizing that appeared in the late nineteenth century. Looking back at what has happened since, we can see that philosophy has passed in the last century through a period of exceptional productivity from which only now it may be emerging into an era of less assured fortunes. It is difficult to survey the range of philosophical activity that unfolded in the age that began with Frege and Nietzsche and many others. The philosophical thought that began to emerge in the last quarter of the nineteenth century is notable for its diversity (and therefore for its internal squabbles) and it is not at all easy to describe the overall character of the renewal of philosophy that began at that time. Some commonalities reveal themselves, of course, even to the glancing eye. We easily notice the pervasive effort to accommodate philosophy in one way or other to the rapidly expanding sciences. We also notice a prevailing preoccupation with the mind, the subject, with consciousness and its states. What is more remarkable still is that philosophers from the various schools are all attracted to questions of language and meaning. Finally (and most significantly for our discussion) these philosophers share an active concern with the question and the concept of truth. To fully realize the scope of this concern we need to consider here in addition to Nietzsche’s reflections on the value of truth and Frege’s understanding of the logical laws as laws of truth: Bradley’s monistic theory of truth and Moore and Russell’s characterization of truth as a simple property of judgments or propositions, Wittgenstein’s picture conception of truth and Tarski’s definition of truth for formalized languages, the positivist critique of truth in the name of verifiability and falsifiability and Donald Davidson’s programmatic linkage of meaning and truth in ordinary language, as well as finally Heidegger’s redefinition of truth as aletheia and Foucault’s concatenation of truth and power. This intense preoccupation with the concept of truth contrasts sharply with the disregard of the question of truth in classical modern philosophy. Thus, Kant writes of it in a bemused tone as “the question famed of old, by which logicians were supposed to be driven into a corner”. For common purposes he is willing to take “the nominal definition of truth” as “agreement of knowledge with its object” for granted but also holds that with respect to the actual content of knowledge no general criterion of truth can be given. And he concludes shortly that “further than this logic cannot go” (Kant 1963, B82–84). This is not to suggest that Kant’s attitude is universally shared in modern philosophy. But there is no doubt that in this period the Aristotelian conception of truth remains

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largely untested. When Aristotle had declared that “to say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, or of what is not that it is not, is true,” he appeared to many of philosophers to have said as much as is possible on the matter. That Aristotle formulation implied, moreover, that every judgment has a subject and a predicate and that a judgment is true specifically when the object named by the subject term has the property (is in the state, is engaged in the activity, occupies the location, etc.) identified by the predicate struck these philosophers, moreover, as entirely plausible. It was only in the late nineteenth century that this attitude began to give way to substantial unrest over the question how truth should be understood. The new kind of philosophizing that emerges in the late nineteenth century can, indeed, be characterized comprehensively by its uncertainty over the classical correspondence conception of truth and the active search for alternatives. That does not mean that the correspondence conception of truth was completely abandoned. But it became now a topic for philosophical investigation and controversy and we can see that even those philosophers who are said to have maintained a correspondence theory of truth transformed it in ways their classical antecedents would hardly have understood. It has been said, for instance, that Wittgenstein’s picture conception of truth and of Tarski’s definition of truth in formalized languages constitute forms of the correspondence theory. But we must understand that what they substituted for the old theory was something that goes substantially beyond the traditional formulas. Wittgenstein’s picture conception of truth differs, indeed, radically from the traditional view in that it nowhere allows for the possibility of a comparison of our propositions with the facts. All we have available according to Wittgenstein are the propositions themselves and the a priori certainty that they must mirror something, if they are to have a definite meaning. Tarski has, admittedly, characterized his theory at times as a version of the classical correspondence view of truth. But if we are to speak of correspondence at all in Tarski’s account, it is only one between the sentences of the object- and those of the meta-language, not one between language and world. Donald Davidson is surely right when he noted that Tarski has given us a formal definition of truth (or, rather, a definition of true in some language L) but no comprehensive theory of truth since the latter would also have to speak of the relations between truth, on the one hand, and our beliefs and desires, on the other and that Tarski’s account provides us with neither.

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I continue thus to believe that the critique of the correspondence view of truth lies at the heart of the philosophical concern with truth that has developed over the last 130 years or so and that Frege’s critical confrontation with that view marks him as a characteristic thinker of that epoch. Frege is to many interpreters still a figure detached from his historical background. Thus, Michael Dummett once wrote that Frege’s new logic seems to have been born from his brain “unfertilized by external influences” (Dummett 1973, p. xvii). We can let this stand as a tribute to Frege’s originality, but the statement can hardly satisfy us as an objective assessment. By contrast, I find myself agreeing with Michel Foucault who insisted that truth is, after all, not “the reward of free spirits, the child of protracted solitude, nor the privilege of those who have succeeded in liberating themselves. Truth is a thing of this world” (Foucault 1980, p. 130). To recognize that Frege was after all a nineteenth century thinker and was tied to the currents of German thought in his period is by no means to relativize or diminish his substantial achievements. We are certainly not diminishing Plato by discovering that he was, after all, a man of fourth century Athens. Far from bringing Frege’s stature down, such a perspective can teach us that he was part of a heroic age in philosophy to which we have difficulty now of measuring up and in seeing him in this way we can learn to discriminate what was genuinely original in his thought from where he drew on the thought of others.

REFERENCES Bolzano, B., 1929. Wissenschaftslehre. Vol. 3, 2nd ed., W. Schultz, ed. Leipzig: Felix Meiner. Dummett, M., 1973. Frege. Philosophy of Language. London: Duckworth. Foucault, M., 1980. “Truth and Power”. In: C. Gordon, ed. Power/Knowledge. New York: Pantheon Books, 109–133. Frege, G., 1979. The Frege Reader. Ed. and transl. by M. Beaney. Oxford: Blackwell. — 1972. “On The Scientific Justification of a Conceptual Notation”. In: T. W. Bynum, ed. Conceptual Notation and related articles. Oxford: Clarendon Press, 83–89. — 1996. Lectures on Begriffsschrift. History and Philosophy of Logic 17. Gödel, K., 1944. “Russell’s Mathematical Logic”. In: P. A. Schilpp, ed. The Philosophy of Bertrand Russell. New York: Tudor Publishing Co, 123–153.

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Hare, R. M., 1952. The Language of Morals. Oxford: Clarendon Press. Heijenoort, J. van, 1993. “Logic as Calculus and Logic as Language”. In: H. Sluga, ed. The Philosophy of Frege. Garland: New York., vol. 1, 72–78. Kant, I., 1963. The Critique of Pure Reason. Transl. by N. Kemp Smith. Macmillan: London. Lotze, H., 1888. Logic. Vol. 1. Transl. by B. Bosanquet. Oxford: Clarendon Press. Moore, G. E., 1960. Principia Ethica. Cambridge: Cambridge University Press. Nietzsche, F., 1973. Beyond Good and Evil. Prelude to a Philosophy of the Future. Transl. by R. J. Hollingdale. Harmondsworth: Penguin Books. — 1979. “On Truth and Lie in a Nonmoral Sense”. In: F. Nietzsche, Philosophy and Truth. Selections from Nietzsche’s Notebooks of the Early1870’s. Transl. by D. Breazeale. Atlantic Highlands N.J.: Humanities Press International, 79–97. Sluga, H., 1980. Gottlob Frege. London: Routledge. — 1984. “Frege: the early years”. In: R. Rorty, J. B. Schneewind and Q. Sinner, eds. Philosophy in History. Cambridge: Cambridge University Press, 29–356. — 2001. “Frege and the Indefinability of Truth”. In: E. Reck, ed. From Frege to Wittgenstein. Oxford: Oxford University Press. — 2003. “Freges These von der Undefinierbarkeit der Wahrheit”. In: D. Greimann, ed. Das Wahre und das Falsche. Studien zu Freges Auffassung der Wahrheit. Hildesheim: Olms, 83–113. Trendelenburg, A., 1867. Historische Beiträge zur Philosophie. Vol 3. Berlin: G. Bethge. Windelband, W., 1909. Die Philosophie im Geistesleben des XIX. Jahrhunderts. Tübingen: J. C. B. Mohr. Wittgenstein, L., 1953. Philosophical Investigations. Transl. by G. E. M. Anscombe. Oxford: Blackwell.

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Grazer Philosophische Studien 75 (2007), 27–63.

FREGE AND SEMANTICS Richard G. HECK, Jr. Brown University Summary In recent work on Frege, one of the most salient issues has been whether he was prepared to make serious use of semantical notions such as reference and truth. I argue here Frege did make very serious use of semantical concepts. I argue, first, that Frege had reason to be interested in the question how the axioms and rules of his formal theory might be justified and, second, that he explicitly commits himself to offering a justification that appeals to the notion of reference. I then discuss the justifications Frege offered, focusing on his discussion of inferences involving free variables, in section 17 of Grundgesetze, and his argument, in sections 29–32, that every well-formed expression of his formal language has a unique reference.

1. Frege and the justification of logical laws In recent work on Frege, one of the most salient issues has been whether he was prepared to make serious use of semantical notions such as reference and truth. Those not familiar with this debate are often surprised to hear of it. Surely, they say, Frege’s post-1891 writings are replete with uses of ‘true’ and ‘refers’. But no-one wants to deny that Frege makes use of such terms: Rather, what is at issue is how Frege understood them; more precisely, what is at issue is whether Frege employed them for anything like the purposes for which philosophers now employ them. What these purposes are (or should be) is of course itself a matter of philosophical dispute, and, although I shall discuss some aspects of this issue, I will not be addressing it directly. My purpose here, rather, is to argue that Frege did make very serious use of semantical concepts: In particular, he offered informal mathematical arguments, making use of semantical notions, for semantical claims. For example, he argues that all of the axioms of the Begriffsschrift — the formal system1 in which he proves the basic laws of 1. Frege, like Tarski after him, does not clearly distinguish a formal language from a formal

arithmetic — are true, that its rules of inference are truth-preserving, and that every well-formed expression in Begriffsschrift has been assigned a reference by the stipulations he makes about the references of its primitive expressions. Let me say at the outset that Frege was not Tarski and did not produce, as Tarski (1958) did, a formal semantic theory, a mathematical definition of truth. But that is not of any significance here. One does not have to provide a formal semantic theory to make serious use of semantical notions. At most, the question is whether Frege would have been prepared to offer such a theory, or whether he would have accepted the sort of theory Tarski provided (or some alternative), had he known of it. On the other hand, the issue is not whether Frege would have accepted Tarski’s theory of truth, or Gödel’s proof that first-order logic is complete, as a piece of mathematics;2 it is whether he would have taken these results to have the kind of significance we (or at least some of us) would ascribe to them. Tarski argues in “The Concept of Truth in Formalized Languages” that all axioms of the calculus of classes are true; the completeness theorem shows that every valid first-order schema is provable in certain formal systems. The question is whether Frege could have accepted Tarski’s characterization of truth, or Gödel’s characterization of validity, or some alternative, as a characterization of truth or validity. The issue is sometimes framed as concerning whether Frege was interested in justifying the laws of logic. But it is unclear what it would be to ‘justify’ the laws of logic. On the one hand, the question might be whether Frege would have accepted a proof of the soundness of first-order logic as showing that every formula provable in a certain formal system is valid. Understood in this way, the question is no different from that mentioned in the previous paragraph. Another, more tendentious way to understand the issue is as concerning whether Frege believed the laws of logic could be justified ex nihilo: whether an argument in their favor could be produced that would (or should) convince someone antecedently skeptical of their truth or, worse, someone skeptical of the truth of any of the laws of logic. If this is what is supposed to be at issue,3 then let me say, as clearly as I can, that neither I nor anyone else, so far as I know, has ever held that theory formulated in that language, but we can make the distinction on his behalf. I shall therefore use “the Begriffsschrift” to refer to the theory, and “Begriffsschrift”, without the article, to refer to the language. 2. Burton Dreben was fond of making this point. 3. This notion of justification does seem to be the one some commentators have had in mind: See Ricketts (1986a, p. 190) and Weiner (1990, p. 277).

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Frege thought logical laws could be justified in this sense. Moreover, so far as I know, no one now does think that the laws of logic can be justified to a logical skeptic — and, to be honest, I doubt that anyone ever has.4 So in so far as Frege, or anyone else, thinks the laws of logic can be ‘justified’,5 the justification envisaged cannot be an argument designed to convince a logical skeptic. But what then might it be? This is a nice problem, and a very old one, namely, the problem of the Cartesian Circle. I am not going to solve this problem here (and not for lack of space), but there are a few things that should be said about it. The problem is that any justification of a logical law will have to involve some reasoning, which will depend for its correctness on the correctness of the inferences employed in it. Hence, any justification of the laws of logic must, from the point of view of a logical skeptical, be circular. Moreover, even if one were only attempting to justify, say, the law of excluded middle, no argument that appealed to that very law could have any probative force. But, although these considerations do show that no such justification could be used to convince someone of the truth of the law of excluded middle, the circularity is not of the usual sort. One is not assuming, as a premise, that the law of excluded middle is valid: If that were what one were doing, then the ‘justification’ could establish nothing, since one could not help but reach the conclusion one had assumed as a premise. What one is doing, rather, is appealing to certain instances of the law of excluded middle in an argument whose conclusion is that the law is valid. That one is prepared to appeal to (instances of ) excluded middle does not imply that one cannot but reach the conclusion that excluded middle is valid: A semantic theory for intuitionistic logic can be developed in a classical meta-language, and that semantic theory does not validate excluded middle. So the mere fact that one uses instances of excluded middle in the course of proving the soundness of classical logic need not imply that the justification of the 4. I have heard it suggested that Michael Dummett believes something like this. But he writes: “… [T]here is no skeptic who denies the validity of all principles of deductive reasoning, and, if there were, there would obviously be no reasoning with him” (Dummett, 1991, p. 204). 5. Note that I am not here intending to use this term in whatever sense Frege himself may have used it. I am not concerned, that is, with whether Frege would have said (in translation, of course), “It is (or is not) possible to justify the laws of logic”. I am concerned with the question whether Frege thought that the laws of logic can be justified and, if so, in what sense, not with whether he would have used (a translation of ) these words to make this claim. The point may seem obvious, but some commentators have displayed an extraordinary level of confusion about this simple distinction. But let me not name names.

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classical laws so provided is worthless. If one were trying to explain the universal validity of the law of excluded middle, for example, a justification that employed instances of that very law might suffice.6 That would be one way of understanding what justifications of logical laws are meant to accomplish: They answer the question why a given logical law is valid. It suggests another. The objection that justifications of logical laws are circular depends upon the assumption that their purpose is to show that the laws are true (or the rules, truth-preserving). It will be circular to appeal to instances of the law of excluded middle in a justification of that very law only if the truth of instances of the law is what is at issue. But justifications of logical laws need not be intended to demonstrate their truth. We might all be agreed that every instance of (say) the law of excluded middle is, as it happens, true but still disagree about whether those instances are logical truths.7 The purpose of a justification of a law of logic might be, not to show that it is true, but to uncover the source of its truth, to demonstrate that it is indeed a law of logic. It is far from obvious that an argument that assumed that all instances of excluded middle were true could not informatively prove that they were logically true.8 There is reason to suppose that Frege should have been interested in giving a justification at least of the validity of the axioms and rules of inference of the Begriffsschrift. Consider, for example, the following remark:9 I became aware of the need for a Begriffsschrift when I was looking for the fundamental principles or axioms upon which the whole of mathematics rests. Only after this question is answered can it be hoped to trace successfully the 6. The discussion in this paragraph is heavily indebted to Dummett’s (1991, pp. 200–4). It is also worth emphasizing, with Jamie Tappenden (1997), that an explanation of a fact need not amount to a reduction to simpler, or more basic, facts. 7. For example, intuitionists accept all instances of excluded middle for quantifier-free (and, indeed, bounded) formulae of the language of arithmetic, on the ground that any such formulae can, in principle, be proved or refuted. Now imagine a constructivist who was convinced, for whatever reason, that every statement could, in principle, either be verified or be refuted. She would accept all instances of excluded middle as true, but not as logical truths. 8. More generally, if one is to accept a proof that a particular sentence is logically true, one will have to agree that the principles from which the proof begins are true and that the means of inference used in it are truth-preserving. But one need not agree that the principles and means of inference are logical: The proof does not purport to establish that it is logically true that the particular sentence is logically true, only that the sentence is logically true. And in model-theoretic proofs of validity, one routinely employs premises that are obviously not logically true, such as axioms of set theory. 9. References to papers reprinted in Frege’s Collected Papers (1984) are given with the page number in the reprint (p. n) and the page number in the original publication (op. n).

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springs of knowledge upon which this science thrives. (Frege 1984c, p. 235, op. 362)

Frege’s life’s work was devoted to showing that the basic laws of arithmetic are truths of logic, and his strategy for doing this was to prove them in the Begriffsschrift. But no derivation of the basic laws of arithmetic will decide the epistemological status of arithmetic on its own: It will simply leave us with the question of the epistemological status of the axioms and rules used in that derivation. It thus must be at least an intelligible question whether the axioms and rules of the Begriffsschrift are logical in character. What other question could remain? The discussion that follows the passage just quoted reinforces these points. Frege first argues that epistemological questions about the source of mathematical knowledge are, at least in part, themselves mathematical in character, because the question what the fundamental principles of mathematics are is mathematical in character. In order to test whether a list of axioms is complete,10 we have to try and derive from them all the proofs of the branch of learning to which they relate. And in doing this it is imperative that we draw conclusions only in accordance with purely logical laws. … The reason why verbal languages are ill suited to this purpose lies not just in the occasional ambiguity of expressions, but above all in the absence of fixed forms for inferring. … If we try to list all the laws governing the inferences that occur when arguments are conducted in the usual way, we find an almost unsurveyable multitude which apparently has no precise limits. The reason for this, obviously, is that these inferences are composed of simpler ones. And hence it is easy for something to intrude which is not of a logical nature and which consequently ought to be specified as an axiom. This is where the difficulty of discerning the axioms lies: for this the inferences have to be resolved into their simpler components. By so doing we shall arrive at just a few modes of inference, with which we must then attempt to make do at all times. And if at some point this attempt fails, then we shall have to ask whether we have hit upon a truth issuing from a non-logical source of cognition, whether a new mode of inference has to be acknowledged, or whether perhaps the intended step ought not to have been taken at all. (Frege 1984c, p. 235, opp. 362–3)

Much of this passage will seem familiar, so strong is the echo of remarks Frege had made some years earlier, in the Preface to Begriffsschrift, regarding 10 Note that Frege uses this term in a way that is close to, but not identical to, how it is standardly used in contemporary logic.

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the need for a formalization of logic (Frege 1967, pp. 5–6). But the most interesting remark is the last one, which addresses the question what we should do if at some point we were to find ourselves unable to formalize the proof of a theorem previously proven informally. The most natural next step would be to try to isolate some principle on which the proof apparently depended, which principle would then be a candidate to be added to our list of fundamental principles of mathematics. Once we had isolated this principle, call it NewAx, there would be three possibilities among which we should have to decide: NewAx may be a “non-logical” truth, one derived from intuition or even from experience; NewAx may be a truth of logic, which is what Frege means when he says that we may have to recognize “a new mode of inference”; or NewAx may not be true at all, which is what Frege means when he says that the “intended step ought not to have been taken”. Frege is not just describing a hypothetical scenario here: Frege had encountered this sort of problem on at least two occasions. I have discussed these two occasions in more detail elsewhere (Boolos and Heck 1998, and Heck 1998b). Let me summarize those discussions. In Grundgesetze, Frege begins his explanation of the proof of the crucial theorem that every number has a successor by considering a way of attempting to prove it that ultimately does not work, namely, the way outlined in §§ 82–3 of Die Grundlagen. As part of that proof, one has to prove a proposition11 that, Frege remarks in a footnote, “is, as it seems, unprovable …” (Frege 1964, I § 114). It is notable that Frege does not say that this proposition is false, and there is good reason to think he regarded it as true and so true but unprovable in the Begriffsschrift: It follows immediately from the proposition Frege proves in its place, together with Dedekind’s result that every infinite set is Dedekind infinite (Dedekind 1963, § 159). Frege knew of Dedekind’s proof of this theorem and seems to have accepted it, although he complains in his review of Cantor’s Contributions to the Theory of the Transfinite that Dedekind’s proof “is hardly executed with sufficient rigour” (Frege 1984f, p. 180, op. 271). Frege apparently expended some effort trying to formalize Dedekind’s proof. In the course of doing so, he could hardly have avoided discovering the point at which Dedekind relies upon an assumption not obviously available in the Begriffsschrift, namely, the axiom of (countable) choice. One can thus think of the theorem whose proof we have been unable to formalize either 11. The proposition in question is that labeled (1) in § 82 of Die Grundlagen.

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as Dedekind’s result or as the unprovable proposition mentioned in section 114 of Grundgesetze and of NewAx as the axiom of choice. Remarks of Dummett’s suggest he would regard the foregoing as anachronistic: No doubt Frege would have claimed his axioms, taken together with the additional informal stipulations not embodied in them,12 as yielding a complete theory: to impute to him an awareness of the incompleteness of higher-order theories would be an anachronism. (Dummett 1981b, p. 423)

But I am suggesting only that Frege was prepared to consider the possibility that his formalization of logic (or arithmetic) was not complete: It is obvious that particular formalizations can be incomplete. What Gödel showed was that arithmetic (and therefore higher-order logic) is essentially incomplete, that is, that every consistent formal theory extending arithmetic is incomplete. Of that Frege surely had no suspicion, but that is not relevant here. In any event, the question whether a given (primitive) principle is a truth of logic is clearly one Frege regards as intelligible. And important. The question of the epistemological status of the basic laws of arithmetic is of central significance for Frege’s project: His uncovering the fundamental principles of arithmetic will not decide arithmetic’s epistemological status on its own. Though he did derive the axioms of arithmetic in the Begriffsschrift, that does not show that the basic laws of arithmetic are logical truths: That will follow only if the axioms of the Begriffsschrift are themselves logical laws and if its rules of inference are logically valid. The question of the epistemological status of arithmetic then reduces to that of the epistemological status of the axioms and rules of the Begriffsschrift — among other things, to the epistemological status of Frege’s infamous Basic Law V, which states that functions F and G have the same ‘value-range’ if, and only if, they are co-extensional. It is well-known that, even before receiving Russell’s letter informing him of the paradox, Frege was uncomfortable about Basic Law V. The passage usually quoted in this connection is this one:13 12. These are the stipulations made in section 10 of Grundgesetze, which we shall discuss below. 13. Frege also writes, in the appendix to Grundgesetze on Russell’s paradox: “I have never disguised from myself [Basic Law V’s] lack of the self-evidence that belongs to the other axioms and that must properly be demanded of a logical law” (Frege 1964, II, p. 253). The axiom’s lacking self-evidence is reason to doubt it is a logical law: Self-evidence can be demanded only of primitive logical laws, not, say, of the axioms of geometry, which are evident on the basis of intuition.

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A dispute can arise, so far as I can see, only with regard to my basic law (V) concerning value-ranges, which logicians perhaps have not yet expressly enunciated, and yet is what people have in mind, for example, where they speak of the extensions of concepts. I hold that it is a law of pure logic. In any event, the place is pointed out where the decision must be made. (Frege 1964, I, p. vii)

Although few commentators have said explicitly that Frege is here expressing doubt that Basic Law V is true, the view would nonetheless appear to be very widely held: It is probably expressed so rarely because it is thought that the point is too obvious to be worth stating.14 But we must be careful about reading our post-Russellian doubts about Basic Law V back into Frege: He thinks of Basic Law V as codifying something implicit, not only in the way logicians speak of the extensions of concepts, but in the way mathematicians speak of functions (Frege 1964, II § 147).15 And there is, so far as I can see, no reason to conclude, on the basis of the extant texts, that Frege had any doubts about the Law’s truth. The nature of the dispute Frege expects, and “the decision which must be made”, is clarified by what precedes the passage just quoted: Because there are no gaps in the chains of inference, every ‘axiom’ … upon which a proof is based is brought to light; and in this way we gain a basis upon which to judge the epistemological nature of the law that is proved. Of course the pronouncement is often made that arithmetic is merely a more highly developed logic; yet that remains disputable [bestreitbar] so long as transitions occur in proofs that are not made according to acknowledged laws of logic, but seem rather to be based upon something known by intuition. Only if these transitions are split up into logically simple steps can we be persuaded that the root of the matter is logic alone. I have drawn together everything that can facilitate a judgment as to whether the chains of inference are cohesive and the buttresses solid. If anyone should find anything defective, he must be able to state precisely where the error lies: in the Basic Laws, in 14. An exception is Tyler Burge. Though Burge speaks, at one point, of “Frege’s struggle to justify Law (V) as a logical law” (1984, pp. 30ff), what he actually discusses are grounds Frege might have had for doubting its truth. Burge (1984, pp. 12ff) claims that Frege’s considering alternatives to Basic Law V suggests that he thought it might be false. But given Frege’s commitment to logicism, doubts about its epistemological status would also motivate such investigations. 15. Treating concepts as functions then makes Basic Law V sufficient to yield extensions of concepts, too. And there is really nothing puzzling about this treatment of concepts: Technically, it amounts to identifying them with their characteristic functions. For more on this point, see Heck (1997, pp. 282ff).

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the Definitions, in the Rules, or in the application of the Rules at a definite point. If we find everything in order, then we have accurate knowledge of the grounds upon which an individual theorem is based. A dispute [Streit] can arise, so far as I can see, only with regard to my basic law (V) concerning value-ranges … I hold that it is a law of pure logic. In any event, the place is pointed out where the decision must be made. (Frege 1964, I, p. vii)

The dispute Frege envisions would concern the truth of Basic Law V were the correctness of the proofs all that was at issue here. But as I read this passage, Frege is attempting to explain how the long proofs he gives in Grundgesetze support his logicism,16 how he intends to persuade us “that the root of the matter is logic alone”. The three sentences beginning with “I have drawn” constitute a self-contained explanation of how the formal presentation of the proofs gives us “accurate knowledge of the grounds upon which an individual theorem is based”, that is, how the proofs provide “a basis upon which to judge the epistemological nature of ” arithmetic, by reducing that problem to one about the epistemological status of the axioms and rules. Of course, someone might well object to Frege’s proofs on the ground that Basic Law V is not true. But, although Frege must have been aware that this objection might be made, he thought the Law was widely, if implicitly, accepted. Moreover, as we shall see below, Frege took himself to have proven that Basic Law V is true in the intended interpretation of the Begriffsschrift.17 But, in spite of all of this, Basic Law V was not an acknowledged law of logic. The “dispute” Frege envisages thus concerns what other treatments have left “disputable” — and these words are cognates in Frege’s German, too — namely, whether “arithmetic is merely a more highly developed logic”. The objection Frege expects, and to which he has no adequate reply, is not that Basic Law V is not true, but that it is not “a law of pure logic”. All he can do is to record his own conviction that it is and to remark that, at least, the question of arithmetic’s epistemological status has been reduced to the question of Law V’s epistemological status. 16. This question is, in fact, taken up again in section 66. It is unfortunate that this wonderful passage is so little known. 17. I thus am not saying that Frege nowhere speaks to the question whether Basic Law V is true, even in Grundgesetze itself (compare Burge (1998, p. 337, fn 21)). What I am discussing here is where Frege thought matters stood after the arguments of Grundgesetze had been given. I am thus claiming that Frege thought he could answer the objection that Basic Law V is not true but would have had to acknowledge that he had no convincing response to the objection that it is not a law of logic. (The foregoing remarks, I believe, answer a criticism made by Burge.)

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The general question with which we are concerned here is thus what it is for an axiom of a given formal theory to be a logical truth, a logical axiom.18 Frege does not say much about this question. One might think that that is because he had no view about the matter, that he had, as Warren Goldfarb has put it, no “overarching view of the logical”.19 Goldfarb is not, of course, merely pointing out that Frege did not have any general account of what distinguishes logical from non-logical truths. Nor do I. His claim is that Frege’s philosophical views precluded him from so much as envisaging, attempting, or aspiring to such an account. But I find it hard to see how one can make that claim without committing oneself to the view that, for Frege, it is not even a substantive question whether Basic Law V is a truth of logic. Frege does insist that Basic Law V is a truth of logic, to be sure. But suppose that I were to deny that it is. Does Frege believe that this question is one that can be discussed and, hopefully, resolved rationally? If not, then Frege’s logicism is a merely verbal doctrine: It amounts to nothing more than a proposal that we should call Basic Law V a ‘truth of logic’. I for one cannot believe that Frege’s considered views could commit him to this position. But if Frege thinks the epistemological status of Basic Law V is subject to rational discussion, then any principles or claims to which he might be inclined to appeal in attempting to resolve the question of its status will constitute an inchoate (even if incomplete) conception of the logical. One thing that is clear is that the notion of generality plays a central role in Frege’s thought about the nature of logic.20 According to Frege, logic is the most general science, in the sense that it is universally applicable. There might be special rules one must follow when reasoning about geometry, or physics, or history, which do not apply outside that limited area: But the truths of logic govern reasoning of all sorts. And if this is to be the case, it would seem that there must be another respect in which logic is general: 18. Similarly, Frege writes in Die Grundlagen that the question whether a proposition is analytic is to be decided by “finding the proof of the proposition, and following it all the way back to the primitive truths”, those truths “which … neither need nor admit of proof ”. The proposition is analytic if, and only if, it can be derived, by means of logical inferences, from primitive truths that are “general logical laws and definitions”. An analytic truth is thus a truth that follows from primitive logical axioms by means of logical inferences (Frege 1980, § 3). The problem is to say what primitive logical truths and logical means of inference are. 19. Goldfarb expressed the point this way in a lecture based upon his paper “Frege’s Conception of Logic” (Goldfarb 2001). 20. Naturally enough, since his discovery of quantification is so central to his conception of logic. See Dummett (1981a, pp. 43ff) for a discussion close in spirit to that to follow.

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As Thomas Ricketts puts the point, “… the basic laws of logic [must] generalize over every thing and every property [and] not mention this or that thing …” (Ricketts 1986b, p. 76); there can be nothing topic-specific about their content. Thus, the laws of logic are “[m]aximally general 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” (Ricketts 1986b, p. 80).21 So there is reason to think that Frege thought it necessary, if something is to be a logical law, that it should be maximally general in this sense. Some commentators, however, have flirted with the idea that Frege also held the condition to be sufficient.22 Let us call this interpretation the ‘syntactic’ interpretation of Frege’s conception of logic. One difficulty with it is that such a characterization of the logical, even if extensionally correct, would not serve Frege’s purposes. For consider any truth at all and existentially generalize on all non-logical terms occurring in it. The result will be a truth that is, in the relevant sense, maximally general and so, on the syntactic interpretation, should be a logical truth. Thus, ‘xy(x zy)’ should be a logical truth, since it is the result of existentially generalizing on all the non-logical terms in ‘Caesar is not Brutus’. But the notion of a truth of logic plays a crucial epistemological role for Frege. In particular, logical truths are supposed to be analytic, in roughly Kant’s sense: Our knowledge of them is not supposed to depend upon intuition or experience. Why should the mere fact that a truth is maximally general imply that it is analytic? Were there no way of knowing the truth of ‘xy(x zy)’ except by deriving it from a sentence like ‘Caesar is not Brutus’, it certainly would not be analytic. More worryingly, consider ‘xF(x z |FH)’, which asserts that some object is not a value-range. This sentence is maximally general — if it is not, that is reason enough to deny that Basic Law V is a truth of logic — and, presumably, Frege regarded it as either true or false. But surely the question whether there are non-logical objects is not one in the province of logic itself. Still, we need not be attempting to explain what it is for any truth at all 21. For similar views, see van Heijenoort (1967), Goldfarb (1979), and Dreben and van Heijenoort (1986). 22. Ricketts speaks of Frege’s “identification of the laws of logic with maximally general truths” (Ricketts 1986b, p. 80), quoting Frege’s remark that “logic is the science of the most general laws of truth” (Frege 1979a, p. 128). He glosses the remark as follows: “To say that the laws of logic are the most general laws of truth is to say that they are the most general truths”. But whence the identification of the most general laws of truth with the most general truths? Ricketts later (1996, p. 124) disowns this suggestion, however.

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to be a truth of logic, only what it is for a primitive truth (see Frege (1980, § 3)), an axiom, to be a truth of logic. So perhaps the condition should apply only to primitive truths: The view should be that a primitive truth is logical just in case it is maximally general. And it is eminently plausible that maximally general primitive truths must be analytic, for it is very hard to see how our knowledge of such a truth could depend upon intuition or experience. Intuition and experience deliver, in the first instance, truths that are not maximally general but that concern specific matters of fact. Hence, in so far as they support our knowledge of truths that are maximally general, they apparently must do so by means of inference. But then maximally general truths established on the basis of intuition or experience are not primitive.23 It might seem, therefore, that semantical concepts will play no role in Frege’s conception of a truth of logic, that his conception is essentially syntactic. This, however, would be a hasty conclusion, for there are two respects in which the syntactic interpretation is incomplete, and these matter. First, our earlier statement of what maximally general truths are needs to be refined. Ricketts writes that “[m]aximally general truths … do not mention any particular thing or any particular property”. But reference to some specific concepts will be necessary for the expression of any truth at all, logical or otherwise. Frege himself remarks that “logic … has its own concepts and relations; and it is only in virtue of this that it can have a content” (Frege 1984e, p. 338, op. 428): The universal quantifier refers to a specific second-level concept; the negation-sign, a particular first-level concept; the conditional, a first-level relation. And when Frege offers his “emanation of the formal nature of logical laws” — an account not unlike a primitive version of the model-theoretic account of consequence, according to which logical laws are those whose truth does not depend upon what non-logical terms occur in them — the main problem he discusses is precisely that of deciding which notions are logical ones, whose interpretations must remain fixed : “It is true that in an inference we can replace Charlemagne by Sahara, and the concept king by the concept desert … But one may not thus replace the relation of identity by the lying of a point in a plane” (Frege 1984e, pp. 338-9, op. 428).24 23. Something like this line of thought is suggested by Ricketts (1986b, p. 81). 24. The question which concepts are logical is not likely to admit of an answer in nonsemantical terms. For some contemporary work, see Sher (1991). Sher’s theory relies crucially on model-theoretic notions, such as preservation of truth-value under permutations of the domain. Dummett (1981a, p. 22, fn) considers a similar proposal when discussing Frege’s conception of

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The problem of the logical constant — the question which concepts belong to logic — is, for this reason, central to Frege’s account of logic. His inability to resolve this problem may well have been one of the sources of his doubts about Basic Law V: Unlike the quantifiers and the propositional connectives, the smooth breathing — from which names of value-ranges are formed — is not obviously a logical constant. It is clear enough that what we now regard as logical constants have the generality of application Frege requires them to have: They appear in arguments within all fields of scientific enquiry, arguments that are, at least plausibly, universally governed by the laws of the logical fragment of the Begriffsschrift. It is far less clear that the smooth breathing — and the set-theoretic reasoning in which it would be employed — is similarly ubiquitous. It would therefore hardly have been absurd for one of Frege’s contemporaries to insist that the smooth breathing and Basic Law V are peculiar to the ‘special science’ of mathematics. Frege would have disagreed, to be sure. But the syntactic interpretation offers him no ground on which to do so and, worse, seems to preclude him from having any such ground. The second problem with the syntactic interpretation is that it places a great deal of weight on the notion of primitiveness, and we have not been told how that is to be explained. Our modification of the syntactic interpretation — which consisted in claiming only that maximally general primitive truths are logical — will be vacuous unless there are restrictions upon what can be taken as a primitive truth. Otherwise, we could take ‘xF(x z |FH))’ as an axiom and its being a logical truth (assuming it is a truth) would follow immediately. One might suppose that Frege’s remarks on the nature of analyticity, mentioned above, committed him to the view that certain truths, of their very nature, admit of no proof. But that would be a mistake. Frege is perfectly aware that, although some rules of inference, and some truths, must be taken as primitive, it may be a matter of choice which are taken as primitive. And since it is not obvious that there are any rules or truths that must be taken as primitive in every reasonable formalization, there need be none that are essentially primitive.25 So, if the notion of primitiveness is to help at all here, we need an account of what logic and, in particular, his conception of logic’s generality. 25. Thus, Frege writes: “… [I]t is really only relative to a particular system that one can speak of something as an axiom” (Frege 1979b, p. 206). See also Frege (1967, § 13), where Frege says, in effect, that he could have chosen other axioms for the theory and, indeed, that it might be essential to consider other axiomatizations if all relations between laws of thought are to be made clear.

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makes a truth a candidate for being a primitive truth in some formalization or other. A natural thought would be that the notion of self-evidence should play some role (see Frege (1964, II, p. 253)), but Frege says almost nothing directly about this question, either.26 One way to approach this issue would be via Frege’s claim that logical laws are fundamental to thought and reasoning, in the sense that, should we deny them, we would “reduce our thought to confusion” (Frege 1964, I, p. vii; see also Frege (1980, § 14)). I have no interpretation to offer of this claim. But I want to emphasize that it is not enough for Frege simply to assert that his axioms cannot coherently be denied. What Frege would have needed is an account of why the particular statements he thought were laws of logic were, in that sense, inalienable.27 The semantical concepts Frege uses in stating the intended interpretation of Begriffsschrift, which I shall discuss momentarily, also pervade his mature work on the philosophy of logic, and it is a nice question why Frege should have turned to the study of semantical notions at just this time. My hunch, and it is just a hunch, is that he did so because he was struggling with the very questions about the nature of logic we have been discussing: He was developing a conception of logic in which they would play a fundamental role. Frege argues, in the famous papers written around the time he was writing Grundgesetze, that semantical concepts are central to any adequate account of our understanding of language, of our capacity to express thoughts by means of sentences, to make judgements and assertions, and so forth.28 So, if Frege could have shown that negation, the conditional, and the quantifier were explicable in terms of these semantical concepts — and he might well have thought that the semantic theory for Begriffsschrift shows just this — he could then have argued that they are, in principle, available to anyone able to think and reason, that is, that these notions (and the fundamental truths about them) are, in that sense, implicit in our capacity for thought. Unfortunately, such an argument would not apply to Basic Law V: The 26. There has been some recent work on this matter: See Burge (1998) and Jeshion (2001). 27. Vann McGee (1985) at least claims to believe that there are counter-examples to modus ponens, and one would suppose that if any law of logic were inalienable, that would be the one. To be sure, it’s not clear what the right conception of inalienability is, but that only makes Frege’s burden more obvious. 28. Frege claims in “On Sense and Reference” that the truth-values “are recognized, if only implicitly, by everybody who judges something to be true …” (Frege 1984d, p. 163, op. 34). See also Frege’s flirtation with a transcendental argument for the laws of logic (Frege 1964, I, p. xvii).

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notion of a value-range does not seem to be fundamental to thought in this way, and, as we shall see, Frege’s semantic theory does not treat it the same way it treats the other primitives. So that might have provided a second reason for Frege to worry about its epistemological status. But I shall leave the matter here, for we are already well beyond anything Frege ever discussed explicitly. 2. Formalism and the significance of interpretation The discussion in the preceding section began with the question what it might mean to justify the laws of logic. I argued that justifications of logical laws intended to establish their truth must be circular. But the argument for that claim depended upon an assumption that I did not make explicit, namely, that the logical laws whose truth is in question are the thoughts expressed by certain sentences. It is quite possible to argue, without circularity, that certain sentences that in fact express (or are instances of ) laws of logic are true, say, to argue that every instance of ‘A ›™A’ is true. I do just that in my introductory logic classes. Of course, the arguments carry conviction only because my students are willing to accept certain claims that I state in English using sentences that are themselves instances of excluded middle. But that discloses no circularity: My purpose is just to convince them of the truth of all sentences of a certain form, and those are not English sentences. Semantic theories frequently have just this kind of purpose. A formal system is specified: A language is defined, certain sentences are stipulated as axioms, and rules governing the construction of proofs are laid down. The language is then given an interpretation: The references of primitive expressions of the language are specified, and rules are stated that determine the reference of a compound expression from the references of its parts. It is then argued — completely without circularity — that all of the sentences taken as axioms are true and that the rules of inference are truthpreserving. Of course, the argument carries conviction only because we are willing to accept certain claims stated in the meta-language — that is, the language in which the interpretation is given — claims that may well express precisely what the sentences in the formal language express. But that discloses no circularity: The purpose of the argument is to demonstrate the truth of the sentences taken as axioms and the truth-preserving character of the rules. Its purpose is to show not that the thoughts expressed

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by certain formal sentences are true but only that those sentences are true. The semantic theory Frege develops in Part I of Grundgesetze has the same purpose. In the case of each of the primitive expressions of Begriffsschrift, he states what its interpretation — that is, its reference — is to be. Thus, for example:29 “*= '” shall denote the True if * is the same as '; in all other cases it shall denote the False. (Frege 1964, I § 7) “a )(a)” is to denote the True if, for every argument, the value of the function )([) is the True, and otherwise it is to denote the False. (Frege 1964, I § 8)

Some of Frege’s stipulations — which I shall call his semantical stipulations regarding the primitive expressions — do not take such an explicitly semantical form. Thus, for example, in connection with the horizontal, Frege writes: I regard it as a function-name, as follows: —' is the True if ' is the True; on the other hand, it is the False if ' is not the True. (Frege 1964, I § 5)

Frege wanders back and forth between the explicitly semantical stipulations and ones like this: But the point, in each case, is to say what the reference of the expression is supposed to be, and Frege argues in section 31 of Grundgesetze that these stipulations do secure a reference for the primitives. And he argues, in section 30, that the stipulations suffice to assign references to all expressions if they assign references to all the primitive expressions.30 Frege goes on to argue that each axiom of the Begriffsschrift is true. Thus, about Axiom I he writes: By [the explanation of the conditional given in] § 12,   *o('o*) could be the False only if both * and ' were the True while * was not the True. This is impossible; therefore 29. I am silently converting some of Frege’s notation to ours and will continue to do so. 30. For discussion of these arguments, see Heck (1998a and 1999) and Linnebo (2004).

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  A*o('o*). (Frege 1964, I §18)

And, similarly, in the case of each of the rules of inference, he argues that it is truth-preserving. Thus, regarding transitivity for the conditional, he writes: From the two propositions   AΔ o*   A4oΔ we may infer the proposition   A4o* For 4o* is the False only if 4 is the True and * is not the True. But if 4 is the True, then ' too must be the True, for otherwise 4o' would be the False. But if ' is the True then if * were not the True then 'o* would be the False. Hence the case in which 4o* is not the True cannot arise; and 4o* is the True. (Frege 1964, I § 15)

These arguments — which, for the moment, I shall call elucidatory demonstrations — tend by and large not to be explicitly semantical: That is, Frege usually speaks not of what the premises and conclusion denote but rather of particular objects’ being the True or the False. One might suppose that this shows that Frege’s arguments should not be taken to be semantical in any sense at all. But, to my mind, the observation is of little significance: What it means is just that Frege is not being as careful about use and mention as he ought to be. It is sometimes said that Begriffsschrift is not an ‘interpreted language’: a syntactic object — a language, in the technical sense — that has been given an interpretation. Rather, it is a ‘meaningful formalism’, something like a language in the ordinary sense, but one that just happens to be written in funny symbols — something in connection with which it would be more appropriate to speak, as Ricketts does, of “foreign language instruction” than of interpretation (Ricketts, 1986a, p. 176). If so, then one might suppose that Frege could not have been interested in ‘interpretations’ of Begriffsschrift because, in his exchanges with Hilbert, he seems to be opposed to any consideration of varying interpretations of meaningful languages. But, as Jamie Tappenden has pointed out, Frege’s own mathematical work involved the provision of just such reinterpretations of, for

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example, complex number theory. What Frege objected to was Hilbert’s claim that content can be bestowed upon a sign simply by indicating a range of alternative interpretations (Tappenden 1995).31 In some sense, it seems to me, Frege thought that the concept of an interpreted language was more basic than that of an uninterpreted one — and it is hard not to be sympathetic. But it simply does not follow that one cannot intelligibly consider other interpretations of the dis-interpreted symbols of a given language. In any event, Frege was certainly aware that it would be possible to treat Begriffsschrift as an uninterpreted language, with nothing but rules specifying how one sentence may be constructed from others. For the central tenet of Formalism, as Frege understood the position, is precisely that arithmetic ought to be developed as a Formal theory,32 in the sense that the symbols that occur in it have no meaning (or that any meaning they may have is somehow irrelevant). Such a theory need not be lacking in mathematical interest: It can, in particular, be an object of mathematical investigation. There could, for example, be a mathematical theory that would prove such things as that this ‘figure’ (formula) can be ‘constructed’ (derived) from others using certain rules — or that a given figure cannot be so constructed (Frege 1964, II § 93). One can, if one likes, stipulate that certain figures are ‘axioms’, which specification one might compare to the stipulation of the initial position in chess, and take special interest in the question what figures can be derived from the ‘axioms’ (Frege 1964, II §§ 90–1). Frege’s fundamental objection to Formalism is that it cannot explain the applicability of arithmetic, and this needs to be explained, for “it is applicability alone which elevates arithmetic from a game to the rank of a science” (Frege 1964, II § 91). An examination of Frege’s development of this objection will thus reveal what he thought would have been lacking had Begriffsschrift been left uninterpreted — and so what purpose he intended his semantical stipulations to serve.33 31. For further consideration of this kind of question, see Tappenden (2000). And even if we were to accept this objection, it still would not follow that Frege was uninterested in semantics (Stanley 1996, p. 64). 32. For a discussion of this notion of a formal theory, see Frege (1984b). I shall capitalize the word “Formal” when I am using it in the sense explained here. 33. Frege’s discussion explicitly concerns the rules of arithmetic, not those of logic: But, of course, for Frege, arithmetic is logic, and his formal system of arithmetic, the Begriffsschrift, contains no axioms or rules that are (intended to be) non-logical. His discussion of what requirements the rules of arithmetic must meet therefore applies directly to the axioms and rules of inference of the Begriffsschrift itself. Thus, he writes: “Now it is quite true that we could have

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Frege distinguishes “Formal” from “Significant”34 arithmetic. He characterizes Significant arithmetic as the sort of arithmetic that concerns itself with the references of arithmetical signs, as well as with the signs themselves and with rules for their manipulation. Formal arithmetic is interested only in the signs and the rules: It treats Begriffsschrift as an uninterpreted language. On the Formalist view, the references of, say, numerals are of no importance to arithmetic itself, though they may be of significance for the application of arithmetic (Frege 1964, II § 88). And, according to Frege, this refusal to recognize the references of numerical terms is what is behind another of the central tenets of Formalism, that the rules35 of a system of arithmetic are, from the point of view of arithmetic proper, entirely arbitrary: “In Formal arithmetic we need no basis for the rules of the game — we simply stipulate them” (Frege 1964, II § 89). Though Formalists recognize that the rules of arithmetic cannot really be arbitrary, they take this fact to be of no significance for arithmetic but only for its applications: Thomae … contrasts the arbitrary rules of chess with the rules of arithmetic…. But this contrast first arises when the applications of arithmetic are in question. If we stay within its boundaries, its rules appear as arbitrary as those of chess. This applicability cannot be an accident — but in Formal arithmetic we absolve ourselves from accounting for one choice of the rules rather than another. (Frege 1964, II § 89)

It is important to remember that, throughout this discussion, Frege is contrasting Formal and Significant arithmetic. When he speaks of “absolv[ing] introduced our rules of inference and the other laws of the Begriffsschrift as arbitrary stipulations, without speaking of the reference and the sense of the signs. We would then have been treating the signs as figures” (Frege 1964, II § 90). That is to say, we should then have been adopting a Formalist perspective on the Begriffsschrift. 34. The German term is “inhaltlich”, which Geach and Black translate in the first edition of Translations as “meaningful”. While this was a reasonable translation then, it is now dangerous, since the cognate term “meaning” has become a common translation of Frege’s term “Bedeutung”. In the third edition, they translate “inhaltliche Arithmetik” as “arithmetic with content”; a literal translation would be “contentful arithmetic”. Both of these sound cumbersome to my ear. 35. Frege speaks, throughout these passages, of the “rules” of the Formal game, thereby meaning to include, I think, not just its ‘rules of inference’, but also its ‘axioms’ — though he does tend to focus more on the “rules permitting transformations” than on the stipulation of the initial position or “starting points” (Frege 1964, II § 90). The reason is that he tends to think even of the axioms of a Formal theory as rules saying, in effect, that certain things can always be written down. (See here Frege (1964, II § 109).) And, of course, one can think of axioms as a kind of degenerate inference rule.

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ourselves from accounting for one choice of the rules rather than another”, he is not just saying that the rules of arithmetic are non-arbitrary; he is implying that, if we are to formulate a system of Significant arithmetic, we must ourselves answer the question why we have formulated the rules as we have. Frege does not think of this account as a mere appendage to Significant arithmetic, but as a crucial part of the work of the arithmetician: It is likely that the problem of the usefulness of arithmetic is to be solved — in part, at least — independently of those sciences to which it is to be applied. Therefore it is reasonable to ask the arithmetician to undertake the task. … This much, it appears to me, can be demanded of arithmetic. Otherwise it might happen that, while [arithmetic] handled its formulas simply as groups of figures without sense, a physicist wishing to apply them might assume quite without justification that they expressed thoughts whose truth had been demonstrated. This would be — at best — to create the illusion of knowledge. The gulf between arithmetical formulas and their applications would not be bridged. In order to bridge it, it is necessary that the formulas express a sense and that the rules be grounded in the reference of the signs. (Frege 1964, II § 92)

The rules must be so grounded because arithmetic is expected to deliver truths — not just truths, in fact, but knowledge. As Frege concludes the passage: “The end must be knowledge, and it must determine everything that happens” (Frege 1964, II § 92). On the Formalist view, the numerals and other signs of a system of arithmetic can have no reference, as far as arithmetic itself is concerned: “If their reference were considered, the ground for the rules would be found in these same references …” (Frege 1964, II § 90). What is most important, for present purposes, is Frege’s conception of how the references of the expressions ground the rules: The question, ‘What is to be demanded of numbers in arithmetic?’ is, says Thomae, to be answered as follows: In arithmetic we require of numbers only their signs, which, however, are not treated as being signs of numbers, but solely as figures; and rules are needed to manipulate these figures. We do not take these rules from the reference of the signs, but lay them down on our own authority, retaining full freedom and acknowledging no necessity to justify the rules. (Frege 1964, II § 94)

Thus, not only do the references of the signs ground the rules that govern them, but, unless we are Formalists, we must recognize an obligation to

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justify these rules, presumably by showing that they are grounded in the references of the signs. Frege elsewhere specifies what condition rules of inference, in particular, must be shown to satisfy: Whereas in Significant arithmetic equations and inequations are sentences expressing thoughts, in Formal arithmetic they are comparable with the positions of chess pieces, transformed in accordance with certain rules without consideration for any sense. For if they were viewed as having a sense, the rules could not be arbitrarily stipulated; they would have to be so chosen that, from formulas expressing true thoughts, only formulas likewise expressing true thoughts could be derived. (Frege 1964, II § 94)

Thus, the rules of inference in a system of Significant arithmetic must be truth-preserving. And this condition — that the rules should be truthpreserving — is not arbitrarily stipulated, either. It follows from arithmetic’s ambition to contribute to the growth of knowledge: If in a sentence of Significant arithmetic the group ‘3 + 5’ occurs, we may substitute the sign ‘8’ without changing the truth-value, since both signs designate the same object, the same actual number, and therefore everything which is true of the object designated by ‘3 + 5’ must be true of the object designated by ‘8’. … It is therefore the goal of knowledge that determines the rule that the group ‘3 + 5’ may be replaced by the sign ‘8’. This goal requires the character of the rules to be such that, if in accordance with them a sentence is derived from true sentences, the new sentence will also be true. (Frege 1964, II § 104)

Derivation must preserve truth, for only if it does, and only if the axioms are themselves true, will the theorems of the system be guaranteed to express true thoughts; it is only because the thoughts expressed by these formulas are true — and, indeed, are known to be true — that their application contributes to the growth of knowledge, rather than producing a mere “illusion of knowledge” (see Frege (1964, II §§ 92, 140)).36 Since Frege is interested in developing a system of Significant arithmetic, he in particular owes some account of why the rules of the Begriffsschrift are non-arbitrary, that is, a demonstration that they are truth-preserv36. Note that Frege is arguing here not only that the rules are required to be truth-preserving if arithmetic is to deliver knowledge but, conversely, that the substitution of terms having the same reference is permissible because the goal of arithmetic is knowledge. Substitution of co-referential terms — indeed, even of terms with the same sense — is not permitted everywhere: It is not permitted in poetry or in comedy, for example.

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ing (and a similar demonstration that its axioms are true). Unless Frege flagrantly failed to do just what he is criticizing the Formalists for failing to do, he must somewhere have provided such an account. There is no option but to suppose that he does so in Part I of Grundgesetze and that the elucidatory demonstrations in particular are intended to show that the rules of the system are truth-preserving and that the axioms are true. Indeed, since Frege himself speaks of a need to justify the rules and of their being grounded in the references of the signs, we may dispense with our euphemism and speak, not of elucidatory demonstrations, but of Frege’s semantical justifications of the axioms and rules. 3. Frege’s semantical justifications I have argued that Frege’s semantical justifications of the axioms and rules of his system are intended to establish that, under the intended interpretation of the Begriffsschrift — this being given by the semantical stipulations governing the primitive expressions — its axioms are true and its rules are truth-preserving. But, according to Ricketts, they cannot have been intended to serve this purpose, because Frege’s “conception of judgment precludes any serious metalogical perspective” from which he could attempt to justify his axioms and rules (Ricketts 1986b, p. 76).37 His philosophical views “preclude ineliminable uses of a truth-predicate, including uses in bona fide generalizations”, such as would be necessary were one even to be able to say that a rule of inference is valid. Ricketts is 37. Van Heijenoort goes so far as to claim that Frege’s rules of inference “are void of any intuitive logic” (van Heijenoort 1967, p. 326). But Frege simply spends too much time explaining the intuitive basis for his rules for this claim to be plausible; and, if that weren’t enough, if the point were correct, it would make Frege a formalist. The following passage is often cited as expressing Frege’s opposition to meta-perspectives: We have already introduced a number of fundamental principles of thought in the first chapter in order to transform them into rules for the use of our signs. These rules and the laws whose transforms they are cannot be expressed in the Begriffsschrift, because they form its basis. (Frege 1967, § 13) But it would be absurd for Frege to suggest that the axioms cannot be expressed in Begriffsschrift. He is speaking here simply of rules, in particular, of rules of inference, and noting that they cannot be so expressed: In fact, in the first chapter, Frege does not introduce any of his system’s axioms but only its rules. He goes on to explain that he is out, in the second chapter, to find axioms from which all “judgements of pure thought” will follow by means of those rules. In the passage quoted, then, Frege is simply making the distinction between rules and axioms, not expressing his alleged opposition to meta-perspectives.

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not, of course, unaware of what goes on in Part I of Grundgesetze, but he claims that Frege’s sole purpose in Part I is to38 teach his audience Begriffsschrift. Frege’s stipulations, examples, and commentary function like foreign language instruction to put his readers in a position to know what would be affirmed by the assertion of any Begriffsschrift formula. The understanding produced by Frege’s elucidatory remarks should have two immediate upshots. First, it should lead to the affirmation of the formulas Frege propounds as axioms; second, it should prompt the appreciation of the validity of the inference rules Frege sets forth. (Ricketts 1986a, pp. 176–7)

Frege’s elucidations thus enable his reader to know what is expressed by any Begriffsschrift formula; so knowing, the reader can determine whether the formulae expressing the axioms are true by asking herself whether she is prepared to assert what they express. She may be aided by Frege’s examples, commentary, and so forth, but this heuristic purpose is the only purpose the elucidations serve: The semantical justifications are not demonstrations of the truth of the axioms, nor of the validity of the rules, but are meant to persuade. But it is unclear why, if Frege’s only purpose were to teach his audience Begriffsschrift, he should make use of such notions as that of an object, or of a truth-value, or of reference, and why his ‘explanations’ should be, in the usual sense, compositional. It would do as well (and be far simpler) just to explain how to translate a proposition of Begriffsschrift into English (or German).39 But Frege does not say simply that ‘* = '’ expresses the thought that *is the same as ': He says that it “shall denote the True if * is the same as ' [and] in all other cases … shall denote the False” (Frege 1964, I § 7). One might reply that natural languages do not perspicuously express what Frege wishes to express in the Begriffsschrift. But while this is fine so far as it goes, it suggests merely that some technical vocabulary might be needed to ‘teach Begriffsschrift’. It does not explain why that vocabulary should be semantical. Moreover, Frege’s semantical justifications become a great deal more 38. I have capitalized “Begriffsschrift” in both occurrences. I do not have the space to consider Ricketts’s reasons for ascribing this view to Frege, but see Stanley (1996), Tappenden (1997), and Burge (2005b) for extended discussion. 39. The contrast between a semantic theory and a translation manual is, of course, emphasized in Davidson (1984, pp. 129–30). And surely the contrast is obvious to anyone who has taught introductory logic. It is one thing to teach students how to ‘read’ logical notation and another to show them a semantics for it.

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complicated than those cited so far, particularly in cases in which free variables — which he calls Roman letters — occur in the premises and conclusion of an inference.40 But this has been obscured by an almost universal misunderstanding of Frege’s use of Roman letters. I just said that they are free variables, but it is widely held that there really aren’t any free variables in Begriffsschrift: that Roman letters are tacitly bound by invisible initial universal quantifiers. Frege does say that the scope of a Roman letter “shall comprise everything that occurs in the proposition” (Frege 1964, I § 17), which amounts to his stipulating that a formula containing free variables is true just in case its universal closure is true. But he rejects the interpretation of Roman letters as tacitly bound almost immediately thereafter: Our stipulation regarding the scope of a Roman letter is to set only a lower bound upon the scope, not an upper bound. Thus it remains permissible to extend such a scope over several propositions, and this renders the Roman letters suitable to do duty in inferences, which the Gothic letters, with the strict closure of their scopes, cannot. If we have the premises ‘A x 2 = 1 o x 4 = 1’ and ‘A x 4 = 1 o x 8 = 1’ and infer the proposition ‘A x 2 = 1 o x 8 = 1’, in making the transition we extend the scope of the ‘x’ over both of the premises and the conclusion, in order to perform the inference, although each of these propositions still holds good apart from this extension. (Frege 1964, I § 17)

There is, for Frege, an important difference between a proposition of the form ‘)(x)’ and its universal closure ‘x)(x)’.41 The nature of this difference, however, is puzzling: What could Frege mean by saying that, in making certain inferences, we must “extend the scope of the ‘x’ over 40. The interpretive claims made in the remainder of this section and the next are developed in more detail, and defended, in Heck (1998a). That paper limits itself to discussion of the technical details of Frege’s arguments in §§ 29–32 and does not, as the present paper does, discuss the bearing of my interpretation on questions about Frege’s conception of logic. This paper and that one are therefore companion pieces, to some extent, although the discussion here is independent of the messy details encountered there. A more unified discussion will appear in a book on Grundgesetze now in preparation. 41. Compare this remark: “Now when the scope of the generality is to extend over the whole of a sentence closed off by the judgement stroke, then as a rule I employ Latin letters. … But if generality is to extend over only part of the sentence, then I adopt German letters. … Instead of the German letters, I could have chosen Latin ones here, just as Mr. Peano does. But from the point of view of inference, generality which extends over the content of the entire sentence is of vitally different significance from that whose scope constitutes only a part of the sentence. Hence it contributes substantially to perspicuity that the eye discerns these different roles in the different sorts of letters, Latin and German” (Frege 1984c, p. 378; I have altered the translation slightly).

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both of the premises and the conclusion”? Surely he cannot mean that something like x{(A x 2 = 1 o x 4 = 1) š (A x 4 = 1 o x 8 = 1) o(A x 2 = 1 o x 8 = 1)} is supposed to be well-formed! Frege is concerned here with what licenses us to make the inference under discussion. There is a rule in his system, rule (7), that permits it.42 That rule — transitivity for the conditional — allows the inference from ‘'o*’ and ‘4o'’ to ‘4o*’. But if Roman letters were treated as tacitly bound, rule (7) would not apply: Rule (7) does not allow an inference from ‘x(x 2 = 1 o x 4 = 1)’ and ‘x(x 4 = 1 o x 8 = 1)’ to ‘x(x 2 = 1 o x 8 = 1)’. The point is not that this formal rule could not be made to apply: It can, if we introduce a notation in which initial universal quantifiers can be suppressed; some formal systems treat free variables in just that way. Nor is there any substantive worry about whether the inference is in fact valid. Rather, the problem is that we are at present without any argument that inferences of this form are valid when the premises and conclusion contain free variables:43 The semantical justification of rule (7) given in section 15 of Grundgesetze (and quoted earlier) did not allow for the possibility that ‘*’, ‘'’, and ‘4’ might contain free variables. That justification, which is essentially a justification in terms of truth-tables, presupposes that ‘*’, ‘'’, and ‘4’ have truth-values and, moreover, that the truth-values they have, when they occur in one premise, are the same as those they have when they occur in the other or in the conclusion. Only if we may speak of the truth-value of the occurrence of ‘x 2 = 1’ in the first premise, and only if it has the same truth-value in all of its occurrences, will the justification apply. And we cannot so speak. Nowadays, what we would say is that the inference is valid because, whenever we make a simultaneous assignment of objects to free variables in the premises and the conclusion, the usual argument on behalf of transitivity — the argument in terms of truth-tables — still goes through, if we replace occurrences of ‘true’ with occurrences of ‘true under that assignment’: That is to say, that argument can be adapted to show that, if 42. The rules of the system are listed in Frege (1964, I § 48). 43. It is worth emphasizing that free variable reasoning is distinctive of Frege’s new logic (polyadic quantification theory). There is no need for such reasoning in syllogistic logic (which is not to deny that monadic quantification theory can be formulated as a sub-theory of polyadic).

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the premises are both true under a given assignment, the conclusion must also be true under that same assignment. When Frege says that the scope of ‘x’ is to be “extend[ed] … over several propositions”, he is attempting to express the relevant notion of simultaneous assignment: The idea is that, as we perform the inference, we treat the variable as ‘indicating’ (as Frege puts it) the same object in every one of its occurrences, whether in one of the premises or in the conclusion. What Frege has said to this point speaks only to the notion of simultaneity and not to the notion of an assignment itself. But what follows the passage just discussed are further remarks on the nature of free variables and inferences involving them, including rule (5) of the Begriffsschrift, the rule of universal generalization: A Roman letter may be replaced at all of its occurrences in a proposition by one and the same Gothic letter. … The Gothic letter must then at the same time be inserted over a concavity in front of a main component outside which the Gothic letter does not occur. (Frege 1964, I § 48)

Decoding Frege’s terminology, what the rule says is that one can infer ‘A o xB(x)’ from ‘A o B(x)’ if ‘x’ is not free in A.44 Frege’s semantical justification of this rule is contained in section 17 of Grundgesetze and is in three stages. First, he notes that ‘*o )(x)’ is equivalent to ‘x[*o )(x)]’, since a formula containing a Roman letter is true just in case its universal closure is true. Secondly, he argues that, if ‘x’ is not free in * and no other variables are free in either * or )(x), then ‘x[*o )(x)]’ is equivalent to ‘*o x)(x)’: That is, he shows, by means of what is now a familiar argument, that ‘x(p o Fx)’ is equivalent to ‘p ox(Fx)’. The final stage of the argument is contained in the following passage: If for ‘*’ and ‘)(x)’, combinations of signs are substituted that do not refer to an object and a function respectively, but only indicate, because they contain Roman letters, then the foregoing still holds generally if for each Roman letter a name is substituted, whatever this may be. (Frege 1964, I § 17)

It is important to see how odd this final stage of the argument is. What Frege wants to show is that, if ‘x’ is not free in A, then ‘x(A o B(x))’ is equivalent to ‘A o x(B(x))’. But what he says is that, if we substitute 44. A Roman letter is a free variable; a Gothic letter, a bound one; and the concavity, the universal quantifier. To say that the quantifier must appear “in front of a main component outside which the Gothic letter does not occur” is to say that it need not contain the antecedent of the conditional in its scope if the Roman letter in question does not occur in the antecedent.

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names for all free variables, other than ‘x’, in A and B(x), the argument that establishes that ‘x(p oFx)’ is equivalent to ‘p ox(Fx)’ will go through. It is not immediately obvious why that should suffice. What we would say nowadays is that, if we make a simultaneous assignment to the free variables other than ‘x’ in A and B(x), that same argument will go through, ‘true’ again being replaced by ‘true under the assignment’. The only difference between this argument and Frege’s is that, where we speak of assignments, he speaks of substitutions. Frege does not, however, mean to speak here of substitutions of actual terms of Begriffsschrift for the variable,45 but of auxiliary names assumed only to denote some object in the domain. What Frege is assuming, in the argument at which we have just looked, is that the inference from ‘M(x)’ to ‘\(x)’ will be valid just in case \(') is true whenever I(') is true, ' being a name new to the language and subject only to the condition that it must denote a member of the domain. This idea can be made precise: Applied to quantification, it constitutes a coherent alternative to Tarski’s treatment in terms of satisfaction.46 It is a mark of the depth of Frege’s understanding of logic that he realized that the presence of free variables in the language implies that the validity of rules of inference belonging to its propositional fragment — rules like modus ponens and transitivity for the conditional — cannot be justified simply in terms of the truth-tables. It is all the more remarkable that, in thinking about this problem, Frege was led to produce this alternative to Tarski’s treatment of the quantifiers. And I, for one, find it hard to believe that the arguments at which we have just looked are but part of an attempt to ‘teach Begriffsschrift’. The argument Frege gives in favor of the validity of universal generalization is surely not intended merely to encourage the reader not to object to the applications he makes of it. If that were all he wanted, he could have had it far more easily.

45. If he were so to speak, the argument would show only (to put the point in Tarskian language) that the conclusion is true under any assignment that makes the premise true and that assigns objects denoted by terms in the language to the free variables. Compare Dummett (1981a, p. 17). For discussion of how Frege’s argument leads to the conclusion that the inference is valid, see Heck (1998a, p. 446). 46. See Heck (1998a, appendix) for a sketch of such a theory and for references. A similar treatment of quantification is given in Benson Mates’s textbook Elementary Logic (1972).

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4. Grundgesetze der Arithmetik I §§ 30-31 Matters become yet more complicated with Basic Law V.47 The semantical stipulation governing the smooth breathing is not like the stipulations Frege gives for the other primitives: He does not directly stipulate what its reference is to be. Of course, it would not have been difficult for him to do so: He need only have said that a term of the form )() denotes the value-range of )([);48 he could then have argued that, since the valuerange of )([) is the same as the value-range of <([) just in case the same objects fall under )([) and <([), Basic Law V holds. Now, in fact, Frege does consider such a stipulation at one point (Frege 1964, I § 9), but all we are told about value-ranges is that the value-range of the function )([) is the same as that of <([) just in case they have the same values for the same arguments (Frege 1964, I § 3). In effect, then, the only stipulation Frege makes about the smooth breathing — and, more importantly, the only one he uses in his arguments — is that )() = <() has the same truth-value as x()(x) = <(x)). Frege (1964, I § 20) notes that the truth of Basic Law V follows immediately from this stipulation (or from the combined effect of those made in sections 3 and 9). But it will do so only if the stipulation is in good order, only if it suffices to assign a reference to the smooth breathing. The problem, however, is that this stipulation does not directly assign a reference to the smooth breathing. And unless it somehow succeeds in doing so indirectly, Basic Law V cannot be justified in terms of it: Officially, the axiom ought then to be declared neither true nor false, on the ground that it contains an expression that has no reference. Frege therefore needs to argue that his stipulation, augmented by others to be mentioned shortly, does indeed secure a reference for the smooth breathing: That argument comprises most of section 31 of Grundgesetze. Had it been successful, Frege would have proven that Basic Law V is true in the intended interpretation of the system. That is why I said earlier that Frege could have had no real doubts about the truth of Basic Law V.49 47. The discussion in this section summarizes some of the results of earlier papers (Heck 1998a and 1999), which should be consulted for defenses of claims that are not defended here. 48. I’ll write quotation marks and corner quotes with invisible ink in this section, to avoid cluttering the exposition. 49. Frege also speaks of the “legitimacy” of the semantical stipulation as having been “established once for all” and makes reference to his intention to “develop the whole wealth of objects and functions treated of in mathematics out of the germ of the eight functions whose

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The question whether the smooth breathing has been assigned a reference is made pressing by the peculiar nature of the semantical stipulation governing it. But Frege still argues that a reference has been assigned to the other primitives of Begriffsschrift.50 The complete argument of sections 30–31 has a more general conclusion: That the stipulations provide every well-formed expression with a reference — and not just a reference (that is, at least one reference) but a unique reference. Since Frege argues in section 31 that a reference has been assigned to each of the primitive expressions, he need only show that, if every primitive expression of the language has a reference, then every expression that can be formed from these primitives also has a reference. That argument, which is probably the first proof by induction on the complexity of expressions ever given, is contained in section 30. In fact, that section contains two things, which are not separated in Frege’s exposition: A reasonably precise account of the syntax of Begriffsschrift and a demonstration that every expression correctly formed from referring expressions refers. Frege explains that complex names are formed by applying certain combinatorial operations to the primitive expressions of the language and that every name is formed by successive applications of these operations. This ‘closure clause’ serves to define the class of well-formed expressions by means of the ancestral and so implies the validity of proof by induction on the complexity of expressions:51 It is this that allows Frege to argue that, if all primitive expressions of Begriffsschrift refer, then every well-formed expression refers, by arguing that the two ways of forming complex expressions from simpler ones preserve referentiality. The proof is not trivial: The argument that complex predicates such as ‘[= [’ denote is both subtle and elegant.52 names are enumerated in vol. I, § 31” (Frege 1964, II § 147). The primitive expressions of the Begriffsschrift are indeed listed in section 31, but it is hard to believe that Frege refers to it at this point simply for that reason: Rather, the argument given in section 31 is what shows that all of these expressions refer, and that is what makes them legitimate. 50. That Frege should argue for this claim contradicts Weiner’s view that, for Frege, “no work is required to show that primitive terms have Bedeutung” (Weiner 1990, p. 129). To be sure, not much work is required to show that most of them refer, but a lot of work is required to show that the smooth breathing does. 51. Thus, Weiner’s objection that the induction principle employed in this proof is never stated (Weiner 1990, p. 240) is met, since no special induction principle needs to be stated here. 52. A complex predicate is one formed by omitting occurrences of one term from another, leaving argument-places in its wake. Thus, one can form the complex predicate ‘[= [’ by omitting both occurrences of ‘t’ from ‘t = t’. See, again, Heck (1998a, pp. 439–40) for discussion of the argument.

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Frege’s argument that the smooth breathing denotes is complex and difficult to interpret. For present purposes, we do not need to discuss its details, but there is a feature of the argument that is worth mentioning. Frege takes it to be enough to prove:53 (I) If )([) and <([) denote, then )() = <() denotes; (II) If )([) denotes, and if p denotes a truth-value, then p = )() denotes. Claim (I) is supposed to follow from the semantical stipulation governing the smooth breathing, that )() = <() has the same reference as x[)(x) = <(x)], the latter formula itself having a reference because the expressions from which it is constructed do. To establish (II), Frege needs to specify whether the truth-values are value-ranges and, if so, which ones they are: If they are not value-ranges, p = )() will always be false (and so will denote); if they are, then p will have the same reference as some expression of the form <(), whence p = )() will have the same reference as some sentence of the form <() = )(), and case (II) will reduce to case (I). In section 10, Frege argues that it is consistent with the other semantical stipulations that the truth-values are their own unit classes and then stipulates that they are. It is often said that Frege needs to make this stipulation because he requires every predicate to denote a total function, one that has a value for every argument. This is right, but we are now in a position to appreciate the reason for this requirement: It is imposed by the purpose of the proof being given in section 31 and, more generally, by the fact that Begriffsschrift is supposed to have a classical semantics. The truth-values of complex sentences are specified in terms of the references of their simpler components, by means of the truth-tables and the usual sorts of (objectual) stipulations for the quantifiers. If ' = )() did not have a reference when ' denotes a truth-value, x(x = )()) would not have a reference, and the argument would collapse.54 The stipulation that the truth-values are their own unit 53. What Frege needs to show is that '= )() denotes, so long as ' and )([) do. His assumption that these two cases are the only ones that need to be considered involves a tacit restriction of the domain to truth-values and value-ranges. (Thus, the oft-heard claim that, for Frege, the quantifiers always have an unrestricted range is false: see Heck (1999, pp. 271–4)). If the domain contains only such objects, then each of them is either the value of p, for some assignment of a truth-value to p, or the reference of )(), for some assignment of a function to )([), since every value-range is the value-range of some function. 54. And its reason for collapsing would be quite independent of whether any term of the

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classes thus plays an essential role in Frege’s proof that every well-formed expression denotes, and it is not mentioned outside section 10, where it is made, and section 31, where it is applied, except in a handful of sections that themselves refer to one of these. In particular, the stipulation is not embodied in the axioms and rules of the Begriffsschrift. The sentence stating that the truth-values are their unit classes is neither provable nor refutable in the Begriffsschrift, as Frege essentially shows in section 10. Of course, Frege could have adopted additional axioms embodying the stipulation. But he doesn’t bother, because the reason he needs to make the stipulation has nothing to do with the syntax of the formal theory but rather concerns its semantics.55 The purpose of sections 30–31 is thus to prove56 that every well-formed expression in Begriffsschrift refers (and, in particular, that the smooth breathing does). It follows (or would follow, were the argument not fatally flawed) that Basic Law V is true and, moreover, that the system is consistent, since all axioms of the theory are true, the rules are truth-preserving — whence every theorem has the value True — and there is a sentence, e.g., ‘x(x zx)’, that is assigned the value False by the stipulations and so is not a theorem. As we have seen, the argument makes heavy use of semantical notions, in particular, of the notion of reference. Moreover, although the argument that the smooth breathing refers is flawed, there is nothing wrong with the remainder of the proof: The remainder of sections 30–31 constitutes a correct proof that the semantical stipulations governing the primitive logical expressions suffice to assign each of them a unique reference and so to assign a unique reference to every expression properly formed from them. Since the semantical justifications really do show that the axioms and rules of the Begriffsschrift, other than Basic Laws V and VI, are true and truth-preserving, respectively, Part I of Grundgesetze conlanguage — let alone any primitive term — denotes a truth-value (Heck 1999, p. 272). 55. Parallel remarks could be made about Frege’s stipulation, in section 11, concerning the references of improper descriptions: It too is embodied in no axiom or rule of the system. 56. Weiner has argued that “there are serious obstacles to reading sections 28–31 as the presentation of a proof ” (Weiner 1990, p. 240). She notes that the conclusion of the proof is not used in Frege’s proofs of the axioms of arithmetic (Weiner 1990, p. 242). But the proof is meta-theoretic: Its conclusion is a claim about Begriffsschrift; there is no reason that appeal need be made to it in later proofs. She also says that the argument does not meet the standards for “a metatheoretic proof in an introductory logic course” (Weiner 1990, p. 240). But it should not be surprising if Frege is unclear about the conceptual underpinnings of the argument, since it is likely the first meta-theoretic argument ever given. And, with the exception of the failed proof that the smooth breathing denotes, I’d have to disagree: It’s a very sophisticated proof, especially the part concerning complex predicates.

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tains a correct proof that the logical fragment of the Begriffsschrift — that is, Frege’s formulation of second-order logic — is sound, that is, that all of its theorems are true.57 5. Closing We have thus seen that, in Grundgesetze, Frege gives a number of arguments whose purpose is to show that the axioms and rules of the Begriffsschrift are, respectively, true and truth-preserving. There are the semantical justifications of the axioms and rules, found scattered throughout Part I; and there is the argument of sections 30–31, which is supposed to show that every well-formed expression has a reference. These arguments have explicitly semantical conclusions, and they make heavy use of semantical notions. Their character makes it extremely unlikely that they are intended merely as a peculiar sort of foreign language instruction. Such oft-heard claims as that “Frege never raises any metasystematic question” (van Heijenoort 1967, p. 326) or, more strongly, that “metasystematic questions as such … could not meaningfully be raised” by him (Dreben and van Heijenoort 1986, p. 44) seem simply to be wrong. One might yet question how seriously these apparently semantical arguments are to be taken, on the ground that, if they are to be understood as ‘properly scientific’, rather than as merely ‘elucidatory’, they would have to be formalizable in the Begriffsschrift itself. And perhaps, for some reason or other, Frege would have denied that semantical arguments could be formalized in the Begriffsschrift. But why?58 Of course, it follows from Tarski’s theorem on the indefinability of truth that, since the Begriffsschrift formalizes a classical theory sufficient for arithmetic, it is inconsistent if its own truth-predicate is definable in it. But Frege had no reason to think this and so no reason to think that the semantical arguments he gives in 57. And note again that this proof does not depend upon any assumptions about what is in the domain. Granted, then, that all of Frege’s ‘logical’ primitives really are logical primitives, whose interpretation may remain fixed, the proof shows that all of the system’s theorems are logically valid, not just true, where the notion of ‘logical validity’ here is the one we have inherited (more or less) from Tarski. But see below for some warnings about how far this point can be extended. 58. One might have thought that the concept horse problem would pose technical difficulties: But that problem does not arise when the argument is carried out in a higher-order formal theory, but only when one is attempting to talk about the semantics of Begriffsschrift in natural language.

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Grundgesetze could not be formalized in the Begriffsschrift. Indeed, the natural view would surely have been that such reasoning can be reproduced within the Begriffsschrift — as, indeed, it can.59 So the Begriffsschrift is inconsistent. Again. Such terms as ‘metalogical perspective’, ‘semantical metaperspective’, and ‘metasystematic standpoint’ — these being the buzzwords of a now familiar tradition in Frege scholarship — are deeply misleading:60 There is an almost subliminal suggestion that semantical reasoning requires a perspective beyond the Begriffsschrift, that such reasoning cannot be carried out within it. But the mere fact that the conclusion of an argument concerns the semantical properties of a particular theory does not show that it cannot be formalized within it: Though not all arguments for semantical claims concerning Peano Arithmetic can be formalized within Peano Arithmetic, many can be.61 Nor are semantical claims about PA the only ones that cannot be proven in PA: That PA is consistent is a syntactic claim, purely syntactic proofs of which (for example, Gentzen’s) cannot be carried out in PA. But we do need to be careful here. Ricketts claims, at one point, 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” (Ricketts 1986b, p. 67). This claim I think I have shown to be untenable. But I have not argued that formal semantics, as it has come to be understood in the light of Tarski’s work on logical consequence, is not foreign to Frege. The mathematical work of Frege’s at which we have looked is concerned with such questions as whether the axioms are true, or whether the rules are truth-preserving, or whether the primitive expressions of Begriffsschrift refer. None of the work at which we have looked was intended to address such questions as whether the axioms are logically true or the rules are logically valid. And although I have argued that Frege ought to have been, 59. Tarski shows us how to formulate a definition of truth for a second-order language in a third-order language. But Basic Law V can be used to reduce quantification over third-level concepts to quantification over second-level concepts — or, indeed, over objects. 60. Jamie Tappenden (1997) has well documented the extent to which certain forms of argument have become something akin to secret handshakes among the members of this tradition. The terms I’ve just mentioned are among those that signal the occurrences of such arguments. 61. For example, a materially adequate definition of truth for 6n sentences, for any n you like, can be formulated within PA, and using these definitions one can then give a semantical proof in PA of the consistency of 6n arithmetic, for every n. (Inconsistency is averted because PA gives us no way to paste all these definitions together into a definition of truth for the whole of the language of arithmetic.) Or again: PA proves that Q (and therefore PA) proves every true 61 sentence.

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and was, interested in these questions, it is unclear whether he thought mathematical work might bear upon them, let alone whether he would have accepted Tarski’s characterization of the notion of logical consequence (or some alternative).62 Though there are indications that, a few years after the publication of Grundgesetze, Frege was beginning to think about logical consequence in mathematical terms,63 we do not, in my opinion, yet know enough to decide this interpretive issue.64

REFERENCES Boolos, George and Heck, Richard G., 1998. Die Grundlagen der Arithmetik §§ 82–83. In: Logic, Logic, and Logic. Cambridge MA: Harvard University Press, 315–338. Burge, Tyler, 1984. Frege on extensions of concepts from 1884 to 1903. Philosophical Review 93, 3–34. Reprinted in: Burge (2005a), 273–98. — 1998. Frege on knowing the foundation. Mind 107, 305–347. Reprinted in: Burge (2005a), 317–55. 62. It is worth remembering that some contemporary philosophers have also rejected Tarski’s characterization of consequence, notably, John Etchemendy (1990). 63. The relevant discussion is in Frege (1984e, Part III). For discussion of these passages, see Ricketts (1997) and Tappenden (1997). 64. I am fortunate to have had very helpful comments on various drafts of this paper: Thanks are due to George Boolos, Tyler Burge, Warren Goldfarb, Michael Kremer, Ian Proops, Thomas Ricketts, Alison Simmons, and Joan Weiner. And thanks, too, to Dirk Greimann for suggesting the paper be included in the present volume. I have two very large debts, which I decided not to acknowledge at every point at which they were felt, as that would have cluttered the paper. The first is owed to Jamie Tappenden. While he was visiting at Harvard, during the 1994–95 academic year, we had an extraordinarily fruitful, year-long discussion about Frege and, in particular, about the issues with which this paper is concerned. Those conversations were crucial to the development of my views on these matters. The second such debt is to Jason Stanley. Much of the first half of this paper was born in conversation with him. It is difficult to remember which ideas originated with whom, and so he deserves some of the credit for what may be of value here and all of the blame for anything that may not be. The first draft of this paper was written in the summer of 1995; it reached essentially its current form in the summer of 1997. That the paper has remained unpublished for nearly a decade is due to circumstances over which I had no control. So, if the paper seems a little out of date, that is why. I did once consider trying to take more serious account of papers published or written since, but it quickly became clear that I would essentially have to re-write the entire paper. I have therefore added references to some more recent work but otherwise have chosen to be silent. That silence should not itself be interpreted.

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— 2005a. Truth, Thought, Reason: Essays on Frege. Oxford: Clarendon Press. — 2005b. Postscript to ‘Frege on Truth’. In: Burge (2005a), 133–52. Davidson, Donald, 1984. Radical interpretation. In: Inquiries Into Truth and Interpretation. Oxford: Clarendon Press, 125–139. Dedekind, Richard, 1963. The nature and meaning of numbers. In: Essays on the Theory of Numbers. Dover: New York. Tr. by W.W. Beman. Dreben, Burton, and van Heijenoort, Jean, 1986. Introductory note to 1929, 1930, and 1930a. In: S. Feferman, J. Dawson, and St. Kleene, eds. Collected Works, volume 1. Oxford: Oxford University Press, 44–59. Dummett, Michael, 1981a. Frege: Philosophy of Language. 2d edition. Cambridge MA: Harvard University Press. — 1981b. The Interpretation of Frege’s Philosophy. Cambridge MA: Harvard University Press. — 1991. The Logical Basis of Metaphysics. Cambridge MA: Harvard University Press. Etchemendy, John, 1990. The Concept of Logical Consequence. Cambridge MA: Harvard University Press. Frege, Gottlob, 1962. Grundgesetze der Arithmetik. Hildesheim: Georg Olms Verlagsbuchhandlung. Translations of Part I are based upon Frege (1964); of the prose parts of Part III, upon Frege (1970). — 1964. The Basic Laws of Arithmetic: Exposition of the System. M. Furth, ed. and tr. Berkeley CA: University of California Press. — 1967. Begriffsschrift: A formula language modeled upon that of arithmetic, for pure thought. J. van Heijenoort, tr. In: J. van Heijenoort, ed. From Frege to Gödel: A Sourcebook in Mathematical Logic. Cambridge MA: Harvard University Press, 5–82. — 1970. Translations from the Philosophical Writings of Gottlob Frege. P. Geach and M. Black, eds. Oxford: Blackwell. — 1979a. Logic (1897). In: Frege (1979c), 126–151. — 1979b. Logic in mathematics. In: Frege (1979c), 203–250. — 1979c. Posthumous Writings. H. Hermes, F. Kambartel, and F. Kaulbach, eds. P. Long and R. White, trs. Chicago: University of Chicago Press. — 1980. The Foundations of Arithmetic. 2d edition. J. Austin, tr. Evanston IL: Northwestern University Press. — 1984a. Collected Papers on Mathematics, Logic, and Philosophy. B. McGuiness, ed. Oxford: Basil Blackwell. — 1984b. Formal theories of arithmetic. In: Frege (1984a), 112–21. — 1984c. On Mr. Peano’s conceptual notation and my own. In: Frege (1984a), 234–248. — 1984d. On sense and meaning. In: Frege (1984a), 157–177.

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— 1984e. On the foundations of geometry: Second series. In: Frege (1984a), 293–340. — 1984f. Review of Georg Cantor, Zur Lehre vom Transfiniten. In: Frege (1984a), 178–81. Goldfarb, Warren, 1979. Logic in the twenties. Journal of Symbolic Logic 44, 351–68. — 2001. Frege’s conception of logic. In: J. Floyd and S. Shieh, eds. Future Pasts: The Analytic Tradition in Twentieth-Century Philosophy. New York: Oxford University Press, 25–41. Heck, Richard G., 1997. The Julius Caesar objection. In: R.G. Heck, ed. Language, Thought, and Logic: Essays in Honour of Michael Dummett. Oxford: Clarendon Press, Oxford, 273–308. — 1998a. Grundgesetze der Arithmetik I §§ 29–32. Notre Dame Journal of Formal Logic 38, 437–74. — 1998b. The finite and the infinite in Frege’s Grundgesetze der Arithmetik. In: M. Schirn, ed. Philosophy of Mathematics Today. Oxford: Oxford University Press, 429–466. — 1999. Grundgesetze der Arithmetik I § 10. Philosophia Mathematica 7, 258– 92. Jeshion, Robin, 2001. Frege’s notion of self-evidence. Mind 110, 937–76. Linnebo, Øystein, 2004. Frege’s proof of referentiality. Notre Dame Journal of Formal Logic 45, 73–98. Mates, Benson, 1972. Elementary Logic. 2nd edition. Oxford: Oxford University Press. McGee, Vann, 1985. A counter-example to modus ponens. Journal of Philosophy 82, 462-71. Ricketts, Thomas, 1986a. Generality, sense, and meaning in Frege. Pacific Philosophical Quarterly 67, 172–95. — 1986b. Objectivity and objecthood: Frege’s metaphysics of judgement. In: L. Haaparanta and J. Hintikka, eds. Frege Synthesized: Essays on the Philosophical and Foundational Work of Gottlob Frege. Dordrecht: Reidel, 65–95. — 1996. Logic and truth in Frege. Proceedings of the Aristotelian Society 70, 12140. — 1997. Frege’s 1906 foray into meta-logic. Philosophical Topics 25, 169–88. Sher, Gila, 1991. The Bounds of Logic: A Generalized Viewpoint. Cambridge MA: MIT Press. Stanley, Jason, 1996. Truth and metatheory in Frege. Pacific Philosophical Quarterly 77, 45–70. Tappenden, Jamie, 1995. Geometry and generality in Frege. Synthese 102, 319– 61.

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— 1997. Metatheory and mathematical practice in Frege. Philosophical Topics 25, 213–64. — 2000. Frege on axioms, indirect proof, and independence arguments in geometry: Did Frege reject independence arguments? Notre Dame Journal of Formal Logic 41, 271–315. Tarski, Alfred, 1958. The concept of truth in formalized languages. In: J. Corcoran, ed. Logic, Semantics, and Metamathematics. Indianapolis: Hackett, 152–278. Van Heijenoort, Jean, 1967. Logic as calculus and logic as language. Synthese 17, 324–30. Weiner, Joan, 1990. Frege in Perspective. Ithaca NY: Cornell University Press.

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Grazer Philosophische Studien 75 (2007), 65–92.

STRIVING FOR TRUTH IN THE PRACTICE OF MATHEMATICS: KANT AND FREGE 1 Danielle MACBETH Haverford College Summary My aim is to understand the practice of mathematics in a way that sheds light on the fact that it is at once a priori and capable of extending our knowledge. The account that is sketched draws first on the idea, derived from Kant, that a calculation or demonstration can yield new knowledge in virtue of the fact that the system of signs it employs involves primitive parts (e.g., the ten digits of arithmetic or the points, lines, angles, and areas of Euclidean geometry) that combine into wholes (numerals or drawn Euclidean figures) that are themselves parts of larger wholes (the array of written numerals in a calculation or the diagram of a Euclidean demonstration). Because wholes such as numerals and Euclidean figures both have parts and are parts of larger wholes, their parts can be recombined into new wholes in ways that enable extensions of our knowledge. I show that sentences of Frege’s Begriffsschrift can also be read as involving three such levels of articulation; because they have these three levels, we can understand in essentially the same way how a proof from concepts alone can extend our knowledge.

Mathematics, the most venerable of the exact sciences, is also the most puzzling. How are we to understand a discipline that is, as mathematics seems to be, at once a priori and capable of yielding substantive knowledge, that is, judgments that are objectively true?2 The most detailed, and in its way 1. This essay introduces a number of themes that are central to, and more adequately defended in, a much larger work-in-progress entitled The Metaphysics of Judgment: Truth and Knowledge in the Exact Sciences. What I aim to do here is only to sketch a way of thinking about the practice of mathematics that promises to resolve some of our most fundamental philosophical perplexities about that practice. 2. It was, of course, Kant who first formulated the problem, in the form of the question how synthetic a priori judgments are possible. A more recent formulation is due to Benacerraf (1973).

compelling, answer is Kant’s: because intuition is involved in mathematical practice, according to Kant, mathematical practice can yield substantive knowledge; because the relevant intuition is pure, mathematical knowledge is nonetheless a priori. This answer is unacceptable, philosophically insofar as it entails transcendental idealism, and mathematically because, although it would have been less obvious to Kant in his time than it is to us in ours, at least some mathematics does not in any way involve the construction of concepts in pure intuition but only reasoning from concepts. How then are we to understand the striving for truth in mathematics? I will suggest that, with a little help from Kant, Frege’s formula language of pure thought, his Begriffsschrift, provides the key. Before Kant, logic was not conceived as something merely formal, that is, empty of all content.3 For Kant’s predecessors, then, the idea that mathematical knowledge might be by means of reason alone did not entail that it is merely explicative. It is only in the context of Kant’s own conception of cognition in terms of intuitions through which objects are given and concepts through which those objects are thought that there is a contradiction in the idea that one might, by reason alone, extend one’s knowledge. For Kant, the fact that true mathematical judgments are ampliative, that they constitute real extensions of our knowledge, entails that they are not known by logic alone. Instead, he came to think, mathematical reasoning involves also appeal to constructions in pure intuition. The clue to understanding how this is to work is to be found in Kant’s “Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality” written in 1764. Kant (1764, p. 250) observes in the “Inquiry” that the mathematician, unlike the philosopher, “examines the universal under signs in concreto”; that is, instead of focusing on the thing itself, say time, as the philosopher does, the mathematician is most immediately concerned with signs. Beginning with signs for simple objects, the mathematician creates signs for distinct concepts of the objects of interest by combining those signs.4 In arithmetic and algebra, for example, “there are posited first of all not things themselves but their signs, together with the special designations of their increase or decrease, their relations etc.” (Kant 1764, p. 250). 3. See MacFarlane (2000). 4. A concept is distinct just if its characteristic marks and their relations one to another are clearly apprehended. Because in mathematics one creates definitions of one’s objects through combining well-understood simples, according to Kant, even complex mathematical concepts are distinct.

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Similarly, in geometry, one operates with diagrams, proving things about, say, triangles by considering one drawn triangle, that is, the concept in concreto. As a recent author has put what I take to be Kant’s essential point here, “one doesn’t speak mathematics but writes it. Equally important, one doesn’t write it as one writes or notates speech; rather one ‘writes’ in some other, more originating and constitutive sense” (Rotman 2000, p. ix). Mathematics is essentially written. But the interest of symbolism in arithmetic and algebra, and of ostensive constructions in Euclidean geometry, does not lie merely in the fact that it involves the manipulation of marks according to Kant. What is equally significant is just how, and why, this works. First, as Kant (1764, p. 251) remarks, signs in these contexts “show in their composition the constituent concepts of which the whole idea … consists”. The Arabic numeral ‘278’, for example, shows that the number designated consists of two hundreds, seven tens, and eight units. A drawn triangle similarly is manifestly a threesided closed plane figure; like the numeral ‘278’, it is a whole made up of simple parts. These complexes are then further combined to show “in their combinations the relations of the . . . thoughts to each other” (Kant 1764, p. 251).5 In mathematics, one combines the wholes that are created out of simples into larger wholes that exhibit relations among them. The systems of signs used in mathematics, then, have three distinct levels of articulation: first there are the primitive signs out of which everything else is composed; then there are the wholes formed out of those primitives, wholes that constitute the subject-matter of the relevant part of mathematics, the numbers of arithmetic, say, or the figures of Euclidean geometry; and finally there are the largest wholes (e.g., a Euclidean diagram, or a calculation in Arabic numeration) that are wholes of the (intermediate) wholes of primitive parts. It is just this feature of the systems of signs found in mathematics that is critical, I think, to mathematical practice as Kant understands it in the first Critique. Consider Kant’s famous example in the B Introduction, that of ‘7
5  12’. Leibniz had argued that such a truth is provable by means of logic and definitions alone, and before the critical period Kant had concurred.6 In the Critique, the view is rejected. Kant argues: 5. Kant is in fact describing what the words of natural language that are used in philosophy cannot do. It is clear that he means indirectly to say what the marks used in mathematics can do. 6. The evidence for Kant’s early Leibnizean view is to be found in Kant (1980, 49–66), which is Herder’s notes on Kant’s lectures on mathematics. (See especially § 36.) Herder attended Kant’s lectures from 1762 to 1764.

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… the concept of twelve is by no means already thought merely by my thinking of that unification of seven and five, and no matter how long I analyze my concept of such a possible sum I will still not find twelve in it. One must go beyond these concepts, seeking assistance in the intuition that corresponds to one of the two, one’s five fingers, say, or (as in Segner’s arithmetic) five points, and one after another add the units of the five given in the intuition to the concept of seven … I … add the units that I have previously taken together in order to constitute the number 5 one after another to the number 7, and thus see the number 12 arise. (B15–16)

The number twelve, Kant suggests, must be constructed by the stepwise addition of units: a unit that is initially given as a part of the number 5 (defined as 4
1) must come to be seen instead as a part of the given number 7 to yield thereby the number 8 (defined as 7
1), and so on. The three levels of articulation that Kant identifies in the “Inquiry” are manifest. First there are the primitive parts, the units; next are the wholes of these parts, the given numbers seven and five conceived as collections of units; and finally we have the larger whole (potentially, the sum that is wanted) of which those given numbers are parts, the whole in virtue of which the parts of those numbers, the individual units, can be reconceived as parts of different wholes. Kant’s point against Leibniz is that the fact that 7
1 equals eight by definition cannot help in the demonstration until and unless we reconceive a unit given as a part of the number five instead as a part of a collection that includes the seven units of seven.7 According to Kant, essentially the same point can be made for the case of calculations involving larger numbers in the Arabic numeration system. Suppose that one wished to determine, say, the product of twenty-seven and forty-four. The Arabic numeration system provides the means as follows. First one writes signs for the two numbers to be multiplied in a particular array: 27 u44 The first number is that given by the first line and the second is written directly below it. As in the case of the sum of seven and five as Kant conceives it, three levels of articulation are discernable, first, the primitive 7. Frege (1884, § 6) makes a related point by appeal to the use of brackets in the formula language of arithmetic.

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parts, that is, the primitive signs ‘2’, ‘4’, and ‘7’, then the wholes of these parts, namely the signs ‘27’ and ‘44’ that are signs for the numbers given as the terms of the problem, and finally, the whole display. The calculation is enabled by this three-leveled structure as follows. First, one reconfigures at the second level of articulation, taking a part of the whole ‘44’, namely the rightmost ‘4’, and reconceiving it as belonging with the ‘7’ in ‘27’. Multiplying the two numbers symbolized in this new whole yields the number twenty-eight so one puts a new primitive sign ‘8’ under the rightmost column and the sign ‘2’ above the leftmost column. Next one takes the same sign ‘4’ and considers it together with the ‘2’ in ‘27’, and so on in a familiar series of steps that result in the following: 27 u44 108 1080 1188 The last line is of course arrived at by the successive addition of the numbers given in the columns at the third and fourth rows. It is by reading down that one understands why just those signs appear in the bottom row; but it is by reading across that one knows the product that is wanted. In this way, through simple calculations on successive reconfigurations of various parts of the original display, one achieves the result that is wanted. Exactly the same point applies to calculations in algebra.8 In both the examples just considered, the three levels that Kant identifies in his “Inquiry” are discernable. But there is also an important difference between the two cases. In the first example, which we might think of as an instance of pebble arithmetic (as Frege would have called it), the task is merely to take a unit, that is, a stroke or pebble, first seen as a part of one whole to be a part of another. In the Arabic numeration system one does not merely reconfigure parts of wholes in this way; rather those reconfigura8. “Even the way algebraists proceed with their equations, from which by means of reduction they bring forth the truth together with the proof, is not a geometrical construction, but it is still a characteristic construction, in which one displays by signs in intuition the concepts, especially of relations of quantities, and, without even regarding the heuristic, secures all inferences against mistakes by placing each of them before one’s eyes”(A734/B762). One solves the problem by manipulating symbols “either through mere imagination, in pure intuition, or on paper, in empirical intuition, but in both cases completely a priori, without having to borrow the pattern for it from any experience” (A713/B741).

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tions themselves set a task for the mathematician. One must add or multiply the numbers that are now taken together. To calculate in the Arabic numeration system, the array of numbers with which one begins must be seen now this way now that in a particular stepwise sequence that breaks the problem down into a series of simple steps. This difference between the two cases is, furthermore, a function of an essential difference between the way the signs operate in the two systems. Whereas in pebble arithmetic a stroke or a pebble is merely a unit that can be combined with other units in collections, the primitive signs of the Arabic numeration system, that is, the ten digits, have different meanings in different contexts. The Arabic numeration system is a positional system; what a digit means depends on the context in which it occurs (that is, on its position within the whole). In ‘14’, for example, the digit ‘4’ designates four, but in ‘41’ it instead designates forty.9 Similarly, in our calculation above, when we multiplied the parts, we took the ‘2’ in ‘27’ to designate not two but twenty. The positional Arabic numeration system is fundamentally different from pebble arithmetic, and from a system such as that of Roman numeration: whereas in the latter systems the primitive signs designate what they designate independent of the context of use (in Roman numeration, ‘V’ always means five, ‘X’ ten, and so on), in the Arabic system, what the primitive signs designate is determined only relative to a context of use, by their positions in the whole. Expressions involving both digits and signs for the basic arithmetic operations reinforces the point. Kant (1788, p. 283) explains in a letter to Schultz written shortly after the appearance of the second edition of the Critique: I can form a concept of one and the same magnitude by means of several different kinds of composition and separation, (notice, however, that both addition and subtraction are syntheses). Objectively, the concept I form is indeed identical (as in every equation). But subjectively, depending on the type of composition that I think, in order to arrive at that concept, the concepts are very different. So that at any rate my judgment goes beyond the concept I have of the synthesis, in that the judgment substitutes another kind of synthesis (simpler and more appropriate to the construction) in place of the first synthesis, though it always determines the object in the same way. 9. Perhaps it will be objected that forty is just four tens, that is, that we should think of ‘41’ as ‘101 u 4 100 u 1’. There are a number of problems with this line of thought, not least of which is the fact that it leaves us with no account of ‘10’. It also entails that calculations in Arabic numeration involve what are in fact quite complex algebraic manipulations when in fact they do not — as is shown by our example above.

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Thus I can arrive at a single determination of a magnitude  8 by means of 3
5, or 12 
4, or 2 u4, or 23, namely 8. But my thought “3
5” did not include the thought “2 u4.” Just as little did it include the concept “8,” which is equal in value to both of these.

A concept, Kant suggests, can be considered either objectively or subjectively, and two concepts can be objectively the same, that is, concepts of one and the same object, but subjectively different. In ‘2 u4’, for instance, the number eight is thought as a product; in ‘4 4’ that same object is thought instead as a sum. It follows that in ‘4 4’, say, the digit ‘4’ does not designate the number four; if it did, the whole could not be taken to designate eight, which Kant clearly thinks it does. Objectively, the two expressions ‘2 u4’ and ‘4 4’ are the same because in both cases the number designated is eight; but subjectively they are not the same because although what I think of, namely the number eight, is the same in the two cases, what I think, in the one case the sum of four and four and in the other the product of two and four, is quite different in the two cases. As Frege would put the point, while the two expressions designate one and the same object, they do so under different modes of determination; they express different senses. In the “Inquiry”, Kant notes that arithmetic, algebra, and Euclidean geometry all employ systems of signs that involve three levels of articulation. By the time of the writing of the Critique, this insight is combined with his understanding of the distinction between intuitions through which objects are given and concepts through which (given) objects are thought to yield a much more detailed understanding of these mathematical practices. As Kant explains to Schultz, arithmetic expressions give objects under concepts in a way that requires that the primitives of the language to be variously interpretable depending on the context. What we need now to see is that just the same is true in Euclidean geometry. Like the other systems we have considered, Euclidean geometry involves three levels of articulation. At the most basic level are the primitives of the system, the points, lines, angles, and areas, out of which everything else is constructed. At the second level are the objects we are interested in, those that form the subject matter of geometry, all of which are wholes of the primitive parts. At this level we find points as endpoints of lines, as points of intersection of lines, and as centers of circles; we find angles of various sorts that are limited by lines that are also parts of those angles; and we find figures of various sorts. A drawn figure such as (say) a square

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has as parts: four straight line lengths, four points connecting them, four angles all of which are right, and the area that is bounded by those four lines. Of course, in the figure as actually drawn, the lines will not be truly straight or equal, and they will not meet at a point; the angles will not be right or all equal to one another. But this is immaterial because the drawn square is not merely a picture or instance of a square (any more than an Arabic numeral is merely a picture or instance of a number); it is a means of encoding information that is relevant to the demonstrations that will be made on the basis of it. Suppose, for instance, that one constructs a square on a line by appeal to the fact that all radii of a circle are equal, and the fact that given a line and a point not on the line one can draw a line through the point that is parallel to the given line. One then knows, on the basis of that construction, that the sides are all equal in length, that the angles are all right, and so on. It is the construction (together with the definitions) that determines what is true of the square, not what the drawn square looks like. And this is generally true of the figures that are discernable at the second level in Euclidean geometry. At the third level, finally, is the whole diagram within which can be discerned various second-level objects depending on how one configures various collections of drawn lines (depending on how, as Kant would say, one synthesizes their manifold under a concept). As we will see, the cogency of Euclidean demonstrations crucially depends not only on our being able to see a given point, line, angle, or area now as a part of one figure and now as a part of another, but on our being able to see it as meaning something different in these different contexts, a given line, for instance, now as a radius of a circle and now as a side of a triangle. The very first proposition in Euclid’s Elements illustrates the essential point. The problem is to construct an equilateral triangle on a given finite straight line, and the demonstration is essentially as follows. First one constructs one circle with one endpoint of the given line as center and the line itself as radius, and another circle with the other endpoint as center and the line as radius. Then, from one of the two points of intersection of the two circles, one draws two lines, one to each of the endpoints of the original line. Now one reasons on the basis of the drawn diagram: two of the three lines are radii of one circle and so are known to be equal in length, and one of those radii along with the third line are radii of the other circle and so are known to be equal in length. But if the two lines in each of the two pairs are equal in length, and there is one line that is in both pairs, then all three lines must be equal in length. Those very same

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lines, however, can also be conceived as the sides of a triangle. Because they can, we know that the triangle so constructed is equilateral. The practice of mathematics is distinctive, Kant thinks, because it involves not merely reasoning from concepts, as in philosophy, but instead what Kant thinks of as the construction of concepts, either ostensively, as in Euclidean geometry, or symbolically, as in arithmetic, algebra, and higher analysis. It is through the mediation of constructions that one is able to discover truths that go well beyond what lay in the concepts with which one began. This works, I have suggested, in virtue of the three levels of articulation in these systems of signs. The objects of interest are at the second level but because they not only have parts but are themselves parts of a larger whole, their parts can be variously recombined and reconceived in ways that reveal new truths. Unfortunately, even as Kant was working all this out, mathematicians were beginning more and more to eschew the sort of constructive approach that Kant focuses on and to adopt instead a conceptual approach, more and more, as one author puts it, “to conquer the problems with a minimum of blind calculation, a maximum of clear seeing thoughts” (Minkowski 1911 cited Stein 1988, p. 241).10 For example, whereas Euler (1748 cited in Boyer 1991, p. 443), one of the great masters of algebraic symbol manipulation, had defined a function of a variable quantity as “any analytic expression whatsoever made up from that variable quantity and from numbers or constant quantities”, that is as a kind of algebraic expression, Riemann (1826–1866), following Gauss (1777–1855), understands a function by way of its intrinsic properties. Instead of conceiving a function as an algebraic expression, Riemann focuses on the function’s “behavior”, whether or not the function is expressible algebraically.11 Whereas early modern 10. Minkowski called this “the other Dirichlet Principle” in recognition of Dirichlet’s role in the development and implementation of this essentially new mathematical practice. Essentially new, but not completely unprecedented. It is not by diagrammatic reasoning that one knows that the square root of two cannot be expressed as a ratio of two whole numbers but by reasoning from concepts, and this was known even to the Pythagoreans. 11. Wilson (1992, p. 151, n. 42) provides this analogy: “build a bathtub and specify the ‘sources’ and ‘sinks’ where that water enters and leaves the tub. These conditions will completely fix how the water will flow in the rest of the tub. For Riemann a ‘complex function’ automatically corresponds to the ‘flow’ induced by its singularities, et al., whether or not there is any formula that everywhere matches such a flow”. This nineteenth century trend away from (constructive) problem solving and towards a more conceptual approach is evident also in Dedekind’s definition of the real numbers as “cuts”. Like Riemann, Dedekind’s dissertation director, and Gauss, with whom Dedekind habilitated, Dedekind (1932 cited Stein 1988, p. 245) aims to get at the concepts underlying various mathematical formulations, “to draw demonstrations, no longer

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mathematics had focused on finding, that is, constructing, solutions to problems through the manipulation of algebraic expressions, mathematicians in Kant’s time, and on through the nineteenth century, were more and more concerned to prove theorems on the basis of given definitions. By the beginning of the twentieth century, two very different post-Kantian philosophies of mathematics had begun to emerge. The first and less radical approach, that which would come to dominate twentieth century thought, jettisons Kantian pure intuition, and with it the construction of concepts, but retains the Kantian conception of concepts as empty independent of relation to given objects and of logic as merely formal, that is, empty of all content. Because it was pure intuition that gave mathematics its content on Kant’s view, it follows that without pure intuition mathematics must be understood in terms of the notion of logical form alone, that independent of any empirical interpretation mathematics must be conceived as a merely formal science without content or truth. It is but a small step to seeing that this conception of the propositions of mathematics is essentially that of standard model theory. According to the standard model theoretic conception of language, a sentence, whether of natural language or of the symbolic language of arithmetic and algebra, is to be understood in terms of, on the one hand, its logical form, and on the other, its empirical content, if any, where that content is provided by an interpretation, or model or semantics that assigns a semantic value or designation to all the non-logical constants of the language. An axiomatization such as, for instance, Hilbert’s of geometry, is taken to provide an implicit definition of the non-logical constants it involves, but only as to their logical form. Independent of an interpretation, which assigns a semantic value or designation to each primitive sign of the language, these axioms are neither true nor false. They are, as Kant would put it, thinkable independent of any relation to an object; but they can be cognized or known only in light of an interpretation that relates them to objects given in sensory experience.12 There are, then, only two from calculations, but directly from the characteristic fundamental concepts, and to construct a theory in such a way that it will, on the contrary, be in a position to predict the results of the calculation”. In the case of the reals, Dedekind’s aim is not to construct the numbers out of other numbers but instead to develop an adequate conception of them. 12. Friedman (1992, p. 55) outlines just this view for the case of geometry: “pure geometry is the study of the formal or logical relations between propositions in a particular axiomatic system, an axiomatic system for Euclidean geometry, say. As such it is indeed a priori and certain (as a priori and certain as logic is, anyway) but it involves no appeal to spatial intuition or any other kind of experience. Applied geometry, on the other hand, concerns the truth or falsity

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possibilities: either mathematics belongs wholly to the realm of thought (that is, it is independent of any relation to objects given in intuition) in which case it is merely formal, empty, devoid of all content and truth; or it is contentful in virtue of its relation to ordinary material objects given in sense experience. Neither of these options is satisfactory. The first provides an account of the a priori character of mathematical practice but at the expense of an account of it as yielding substantive truths. The second, according to which mathematics concerns ordinary empirical objects, gives up on the a priori character of mathematical knowledge. The problem of understanding the striving for truth in mathematics that is set by this model theoretic conception of language in terms of logical form and empirical content is utterly intractable. We need another way. Frege, prefigured by Bolzano, provides one. Rather than merely jettisoning Kantian pure intuition, Frege makes the more radical move of jettisoning also the Kantian conception of concepts according to which they are empty, mere forms, independent of relation to any object. Concepts, Frege argues, are fully objective entities in their own right, entities about which truths can be discovered independent of all reference to objects. In effect, Frege splits Kant’s distinction between concept and intuition into two distinctions, that between Sinn and Bedeutung, on the one hand, and that between concept and object, on the other: objectivity is not inevitably grounded in relation to an object, and cognitive significance, Sinn, is not due to the involvement of concepts (see Macbeth (2005)). As a result, we will see, he is able to introduce a purely logical language that involves the same three levels of articulation that Kant had discerned in arithmetic, algebra, and Euclidean geometry. Because it has these three levels of articulation, Frege’s Begriffsschrift, by contrast with our logical languages conceived model theoretically, enables reasoning from concepts that is at once a priori and ampliative. We have seen that for Kant an arithmetic identity gives an object in two ways, or, as Kant thinks of it, as thought under two concepts. Frege, as already noted, thinks something similar. But, despite the impression one might get from reading “On Sense and Meaning”, this cannot be the whole story for Frege. Consider, for example, the simple arithmetic identity ‘1 1 1 
3’ that Frege discusses in Grundlagen. This sentence obviously of such systems of axioms under a particular interpretation in the real world … the truth (or approximate truth) of any particular axiom system [under an interpretation] is neither a priori nor certain but rather a matter for empirical investigation”. Friedman aims here to be criticizing Kant but it is easy to see this view as essentially Kantian in its overall orientation.

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expresses a truth of arithmetic. It presents, Kant thinks, a single object as thought under two concepts. If we instead read the sentence model theoretically, in terms of the idea that each primitive sign of the language designates prior to and independent of any context of use, then we will take each tokening of the numeral ‘1’ to designate the number one, and the sentence itself to express the fact (reminiscent of pebble arithmetic) that the number one and the number one and the number one together equal three. But as Frege argues in Grundlagen, that cannot be right: there is only one one and no amount of putting one together with itself will produce anything other than one. There is only one one and yet the sentence ‘1 1 1 
3’ expresses a truth of arithmetic. We need to understand how this can be. Frege suggests the following. Instead of taking the primitive signs of the Arabic numeration system to designate numbers independent of any use — the numeral ‘1’, for example, to designate the number one whatever the context — we should understand such signs as only expressing a sense prior to their use. The signs are then put together to form a sentence that expresses a thought, and that thought can be analyzed into function and argument in various ways, none of which are privileged. The sentence ‘1 1 1 
3’ can be taken to involve, for instance, the function [ 1 1 
3 with the number one as argument. Alternatively, we can, following Kant, take the object names ‘1 1 1’ and ‘3’, both of which designate the number three (though they do so under different modes of presentation), to designate the arguments for the two-place relation [ 
]; or alternatively, we can take the object name ‘1 1’ to designate the argument for the function [ 1 
3. Clearly other analyses are possible as well. So read the sentence does not present objects as thus and so. Instead it expresses a sense. Only relative to an analysis into function and argument are objects and concepts designated by the subsentential expressions of the language so conceived.13 13. Given this understanding of Frege’s mature conception of the workings of a symbolic language, it is easy to see that a well-known Dummettian criticism of Frege’s conception of the sense of predicate expressions is unfounded. The claim is that Frege puts two incompatible demands on the senses of predicate expressions: (1) that the compositionality of thought requires that the senses of predicate expressions be functions from the senses of singular terms to thoughts, which are the senses of sentences, and (2) that the senses of predicate expressions contain modes of presentation of functions from objects to truth-values. In fact, it is the senses of primitive expressions that are relevant to compositionality and the senses of those concept words that are the result of analysis that contain modes of presentation of functions from objects to truth-values. It is also, I think, easy now to see how there could be sentences that express thoughts though they fail to designate any truth-value. Consider the sentence ‘2/0 
5’. This

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According to Frege’s analysis in Grundlagen, as, in a different way, on Kant’s, a sentence of arithmetic should be understood to involve not merely two levels of articulation as on the model theoretic conception but instead three, that of the primitive signs which are understood to express only a sense independent of their occurrence in a proposition, that of the sentence as a whole which expresses a thought and designates a truthvalue, and between these two levels, that of the concept words and object names that are given relative to an analysis of the sentence into function and argument. Just the same is true of Frege’s logical language Begriffsschrift. In Frege’s logic we begin with primitive signs, the conditional and negation strokes, the concavity, and the content and judgment strokes, as well as two kinds of letters, Latin italic and German lending generality of content to sentences in which they occur. Frege’s elucidations are to give us an understanding of what these signs mean, the senses they express, so that when we encounter them in a sentence we can grasp the thought that is expressed. Such sentences can then be analyzed into function and argument in various ways. As should be clear, this conception of language is quite different from that of standard model theory. On the model theoretic conception, we have seen, primitive signs designate independent of any context of use. Indeed, the interpretation or model or semantics is to serve precisely this purpose on the model theoretic conception; the interpretation, semantics, or model is to determine the (context-free) meanings or designations of the primitive expressions in a way that fixes the truth conditions of all sentences in which they occur. There are, then, only two levels of articulation in language conceived model theoretically, that of the primitive signs and that of whole sentences within which those signs occur. In Frege’s logical language Begriffsschrift, I have suggested, primitive signs only express a sense independent of a context of use. Only relative to an analysis of a sentence into function and argument can we ask after the designations of various sub-sentential parts. It follows directly that one can form concept words of arbitrary complexity in Frege’s logical language. sentence is perfectly contentful; it expresses a thought that we grasp through our grasp of the senses of the primitive signs it involves. But this sentence does not designate any truth-value; it is neither true nor false because the sign ‘2/0’ does not designate any number. Because that sign does not designate any number, there is no analysis of the sentence that yields expressions that succeed in designating an argument and a function. One or other of the relevant expressions will inevitably fail to denote anything. It follows that the sentence itself fails to denote any truth-value—despite the fact that it succeeds in expressing a thought.

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Consider, to begin with, the judgment that is expressed in Begriffsschrift thus: a

C(a) R(a)

This sentence, like any sentence of Begriffsschrift, can be variously analyzed. It can, for instance, be read as expressing the thought that the concepts (say) red and colored are related by subordination; it can be read, that is, as involving the second-level relation of subordination, designated by the expression a

(a) (a),

for arguments C[ and R[. This concept word formed out of the concavity with German letter and the conditional stroke is a concept word for the second-level relation of subordination; it designates that relation, and also, of course, expresses a sense. But we can also read our original sentence differently. For example, we can take it to involve the second-level concept word a

(a)

designating the second-level property universally instantiated, for argument

Here we have a sign for a first-level conditional property: being colored on the condition of being red. This concept is, of course, universally instantiated; that is, it is a property of any object that might be given as argument for the function. We can also take our original sentence to involve the second-level concept a

(a) R(a)

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for argument C[. Clearly other analyses are possible as well. We have seen that on Frege’s mature conception of a sentence of Begriffsschrift three different levels of structure are discernable. At the lowest level a sentence is a collection of primitive signs of the language arranged in a certain way; at the highest level the sentence is itself a single unit, a whole that expresses a thought and designates a truth-value; and in between these levels are the object names and concept words that are revealed on an analysis. As has also been noted, a sentence on the model theoretic account has only two levels, that of its meaningful primitive parts (relative to a semantics or interpretation), parts that designate independent of any context of use in a sentence, and that of the whole that is made up of those parts. It follows that in Frege’s logical language, though not in language conceived model theoretically, one can form concept words of arbitrary complexity, words that in the context of a whole sentence can be variously analyzed. Consider the notion of continuity, which Weierstrass was the first clearly to articulate. As it is conceived quantificationally, Weierstrass’s analysis provides not the content of the concept of continuity but instead the truth conditions of the claim that a function f is continuous at a point a: that claim is true if and only if (H 0)(G 0)(x)(|x|  G „ |f(x a)  f(a)|  H). This sentence, as it is normally read, is composed of antecedently meaningful parts as specified in a standard semantics for the language, parts that are combined into a whole according to the syntactic rules of the language. Because it is so composed, the concept of continuity, on this account, has not being explained but reduced, that is, explained away. It is nothing over and above its parts in a given logical array. Compare, now, that same concept as Frege (1880/81, p. 24) conceives it: that a function ' is continuous at a point A is defined in Begriffsschrift as: n

g

d

n A d) a d g g 0 n 0

(A) n

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This sentence, read as Frege comes to read such sentences, only exhibits a sense, a Fregean thought, independent of any analysis—though in fact, in this case, an analysis is already indicated. We are to take the Greek letters ‘'’ and ‘A’ to mark the argument places for a higher-level concept word. What we have in this totality of signs, then, is an expression that designates (relative to the given analysis) the concept of continuity; we have a sign, a name, for this concept. The concept is not explained away, reduced to something else, on this account. It is instead explained; its cognitive content is exhibited in a way making it abundantly clear just what that content is. 14 But of course, this content can, in the context of a sentence, be variously analyzed. It is precisely this feature of expressions in Begriffsschrift that will enable us to understand how a proof from concepts alone can be nonetheless fruitful, or ampliative, an extension of our knowledge. Much as a Euclidean diagram provides a medium within which to demonstrate the theorems of geometry by providing a means of exhibiting geometrical features of objects, and Arabic numeration provides us with a language within which to calculate by providing a means of exhibiting what we can think of as the computational contents of numbers, so Frege, we will see, provides us a language within which to reason by providing a means of exhibiting the cognitive contents of concepts. Consider, first, a proof in Begriffsschrift of a simple theorem of logic, the rule that if a particular object has the property G but does not have the property F then it may be inferred generally that some G is not F. We begin with two axioms, though more will be added later, and one rule of inference, modus ponens. Given our one rule, it follows that the axioms and theorems of the system can serve in an inference in only two ways, either as the conditional or rule licensing the inference, or as marking the satisfaction of the condition, that is, as an instance to which the rule is to be applied. The two axioms, which following Frege we will understand as acknowledged truths, are: 1.

a b a

Axiom: If you know something then you know it on any condition. We can also read this axiom as saying that if a and b are true, then a is true.

14. This difference between the two sorts of language is developed in further detail in Macbeth (forthcoming).

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2.

Axiom: If something (here, c) is a condition on a conditional (here, a-on-condition-that-b), then it is a condition of both condition and conditioned. Again we can analyze the sentence differently, for instance, as saying that if c and b imply a, and c implies b, then c implies a.

a c b c a b c

Suppose now that (2) is treated as satisfying the condition in (1), treated, that is, as an instance to which the rule expressed in (1) applies. It follows that we can put any condition we like on (2), given that it is true. That is, we treat all of (2), save for the judgment stroke, as standing in for ‘a’ in (1). For ‘b’ we can put anything we like, and then infer by modus ponens, with (1) as rule and (2) as instance, the relevant theorem. In particular, we can infer: 3.

a c b c a b c a b

From (1) and (2): This is the result of putting (2) for the two occurrences of ‘a’ in (1) and the Begriffsschrift equivalent of ‘a-on-condition-that-b’ for ‘b’ in (1). Those substitutions give us a conditional of which (2) is the condition. (3) follows by modus ponens.

(2) tells us that if c is a condition on a conditional then it is a condition of both condition and conditioned. In (3) the first, or lowest, condition (a-on-condition-that-b) can be seen as a condition on a conditional, and so as satisfying the condition of (2). It therefore follows that:

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4.

a c b c a b a b c a b

From (2) and (3): Here we have made the condition a-on-condition-that-b a condition of both condition and conditioned (given a particular analysis). This is justified by the rule expressed in (2) as applied to (3).

Notice now that the first, or lowest, condition of this sentence, (4), can be read as a substitution instance of (1); that is, it has, on one analysis, the logical form of (1), as can be seen by putting ‘a-on-condition-that-b’ for ‘a’ and ‘c’ for ‘b’ in (1). So the first (lowest) condition of (4) is satisfied and we can detach it to yield: 5.

a c b c a b

From (4) and (1): We can think of this sentence as telling us that if we have a conditional then we can attach any condition we like to both condition (i.e., b) and conditioned (i.e., a).

Now we need another axiom that says, in effect, that the order of conditions is immaterial. 6.

a d b a b d

Axiom: If you have two conditions, b and d, on some sentence, then those conditions can be reversed.

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So, by (6), we can reverse the order of the conditions in (5): 7.

From (6) and (5): This gives as a formula the rule of hypothetical syllogism: if bon-condition-that-c and a-on-conditionthat-b, then a-on-condition-that-c.

a c a b b c

Now we treat (6), which before we read as a rule, as instead satisfying the first, that is, lowest, condition in (7); that is, we treat the first (lowest) condition in (6) as the condition c in (7) and the rest of (6) as standing in for ‘b’, while changing letters appropriately (because otherwise different tokenings of ‘a’ would be playing different, yet unmarked, roles). 8.

a b e d a b d e

From (7) and (6): This seems intuitively to be true: if a on the condition that bgiven-e-and-d, then a on the condition that b-given-d-and-e.

We introduce another axiom. 9.

b a a b

Axiom: This is just the rule of contraposition stated as a formula. It makes explicit the fundamental inferential significance of the negation stroke relative to the conditional stroke.

From (5) and (9), with (5) providing the rule and (9) the instance, that is, the condition that needs to be satisfied in (5), we can infer:

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10.

b a c a b c

From (5) and (9): Here we have just added c as a condition on both the condition and the conditioned in (9). This is licensed by the rule expressed in (5).

(10), as instance, together with (8), as rule, yields: 11.

b a c a c b

From (8) and (10): Licensed by (8), we have merely switched the order of ‘b’ and ‘c’ as they appear in the condition of (10). ‘a’ in (8), as that rule has been applied here, is everything to the right of the leftmost conditional stroke, that is, all of the Begriffsschrift equivalent of ‘not-b on condition that not-a and c’.

Again we introduce another axiom, that which makes explicit the basic inference potential of the concavity. 12. a

f (c) f (a)

Axiom: If a property has the second-level property universally instantiated then it is a property of any particular thing.

But we want this axiom in a particular form, namely, as: 13. a

f (b) g(b) f (a) g(a)

Instance of (12): Here we merely substituted a conditional property, beingf-on-condition-of-being-g, for the two occurrences of ‘f’ in (12).

From (13) as satisfying the condition on (11) conceived as a rule, finally, we get the theorem we set out to prove.

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14.

From (11) and (13): If some particular object b has the property g but not the property f then it can be inferred that some g is not f.

f (a) g(a) f (b) g(b)

a

Now we need to see how things might work in mathematics, and for that we need some definitions. What we want to prove is that if an object x has some property that is hereditary in the f-sequence and y follows x in the f-sequence, then y has that property. To prove this we need definitions both of the concept hereditary in a sequence and of the concept following in a sequence. 15.

16.

d

F(a) f (d,a) F(d)

a

F

F(y) F(a) f (x,a)

a

(



F( ) f(

(

F( ) f( , )

f (x ,y )

)

Definition of a property F being hereditary in the f-sequence, for some given f.

Definition of following in a sequence: y follows x in the f-sequence iff for all properties hereditary in the f-sequence if all objects bearing f to x have that property then y has that property.

Though we will not derive it formally, it should be relatively obvious that from (15) we can derive: 17.

F ( y) f (x,y)

(

F

If x has the property F, and F is hereditary in the f-sequence, and y bears f to x, then y has the property F.

f

F (x)

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Similarly, from (16) we can derive the following conditional: 18. a

If y follows x in the f-sequence, then if F is hereditary in the f-sequence, and anything bearing f to x is F, then y has the property F.

F (y) F (a) f (x,a)

( 

F( ) f( , )

f (x ,y )

Licensed by the rule in (6), we can switch around the three conditions in (18): 19.

From (6) and (18): Here we have collapsed a couple of steps into one. The validity of the inference is obvious, and in Grundgesetze Frege introduces a rule licensing interchange of what he there calls subcomponents, that is, conditions.

F (y)  f (x ,y ) a

F (a) f (x,a)

(

F( ) f( , )

Licensed by (2), we can see the lowest condition in (19) instead as a condition on both condition and conditioned (which is itself a conditional):

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20.

From (20 and (19): Not an especially pretty formula, but one should be able to see why it is true given that (19) is true. We should also by now have a pretty good idea of what is coming next.

F (y)  f (x ,y )

( a

F( ) f( , )

F (a) f (x,a)

(

F( ) f( , )

Licensed by (5), we add a condition to both condition and conditioned in (20): 21.

From (5) and (20): If it is true that any object bearing f to x has the property F if x has that property and that F is hereditary in the f-sequence, then it is true that y, which follows x in the f-sequence, has the property F if x has that property and it is hereditary in the f-sequence.

F (y)  f (x ,y )

( a

F( ) f( , )

F (x) F (a) f (x,a)

(

F( ) f( , )

F (x) Because (17) provides an instance for the application of the rule expressed in (21), we can now infer the theorem that is wanted:

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22.

From (21) and (17): If x has the property F which is hereditary in the f-sequence then if y follows x in the f-sequence, y has the property F.

F (y)  f

(

F f

F (x) By appeal only to definitions and the axioms of pure logic, we have proved a theorem in the theory of sequences. There is, of course, an analogous proof in standard (second-order) quantificational logic. The point is not that one can prove, in a merely technical sense, something in Frege’s logic, as if it might not be provable in that sense in our current logics; it is rather that given Frege’s conception of language, and the essentially two-dimensional notation that puts that conception before our eyes, we can see how a purely logical proof can extend our knowledge. Much as in a Euclidean demonstration, one begins with some simple axioms and some definitions. But where Euclid also has postulates governing the construction of diagrams, we were able to work in the language itself. The sentences of Begriffsschrift that express our axioms and theorems are essentially twodimensional, as of course Euclidean diagrams are, and because they are it is easy to see that they are variously analyzable, that they can be conceived in a variety of ways. In particular, a collection of primitive signs can be seen both as a single unit, as a concept word for some concept, and as providing a series of conditions on a judgment taken as a whole. Thus, we can consider the parts of a concept word as parts of different wholes and draw inferences on that basis. We can show, as we have seen, that given what it means to follow in a sequence or to be hereditary in a sequence, certain theorems are true. Of course here we have concepts that are purely logical, defined using only the primitive signs of logic. In mathematics, the concepts of interest will generally involve also primitive mathematical notions, for instance, that of a mathematical function in the definition of a group. The basic procedure, however, is the same: proof by appeal only to logic and definitions, where the proof essentially involves conceiving sentences now this way, now that. But why are definitions, with their defined signs, needed at all in this process? Why not omit the definition and simply exhibit the full content of, say, the concept following in a sequence everywhere in the proof? After

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all, if we did, everything would follow just as before. Everything would follow as before but it would be much harder to see the result as an interesting theorem in the theory of sequences, as something following from particular concepts. As Frege (1879, § 24) says, definitions “serve to call special attention to a particular combination of symbols from the abundance of possible ones”. Furthermore, such definitions, in the most interesting cases, are themselves the result of logical analysis. One begins with some familiar but as yet unanalyzed mathematical notion, say, that of being a prime number, or being a function continuous at a point; the task is to set out its precise content as it matters to judgment and inference, that is, “to articulate the sense clearly” (Frege 1914, p. 211). Having so articulated the sense, we “[obtain] a complex expression which in our opinion has the same sense” as the (unanalyzed) expression with which we began (Frege 1914, p. 210). Because, in Begriffsschrift, concept words are not one and all primitive signs, but can be formed out of combinations of primitive signs, we can express precisely that sense in Frege’s logical language, and then derive theorems from it. More specifically, we can consider parts of concept words now as parts of such (intermediate) wholes, now as parts of other (intermediate) wholes, and on that basis derive new truths. It is for just this reason that mathematics, on this account, is a substantive, albeit a priori, science. Indeed, so is logic. Frege’s conception of language and of the nature of reasoning, I have suggested, is something essentially new. We have already seen that it is very different from the standard model theoretic conception; but it is also quite different even from the inferentialism of Sellars, and of Brandom following him. For Sellars and Brandom, the formal or purely logical rules of language must be supplemented with material rules, rules that are not formally valid but nonetheless license inferences. Conceptual contents, on their view, are inferential roles where “the inferences that matter for such content in general must be conceived to include those that are in some sense materially correct, not just those that are formally valid” (Brandom 1994, p. 105; see also Sellars 1953). For example, it follows logically, or formally, from something’s being red (all over) that it is colored because red just is a color of a certain kind; and it follows materially from something’s being red (all over) that it is not blue because although the concept red contains nothing involving being blue, nonetheless, the two are materially, that is, necessarily though not logically, incompatible. The concept involves, as Kant would put it, both what is analytic with respect to it and its synthetic a priori connections to other concepts. Grasping such a concept, then,

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involves knowing both what follows formally from it and what follows materially from it. Concepts, on this view, have their own internal content and also necessary relations to other concepts, relations that might be made explicit in the axioms of a theory.15 Frege’s conception of a concept is not inferentialist in this sense. For Frege the whole content of a concept is given by an adequate definition of it; no material rules are needed. The role of axioms is rather to set out the basic truths of the system, thoughts that can be seen to be true (assuming we have gotten things right) on the basis of our grasp of the senses expressed by the primitives of the language. And yet the judgments that can be derived on the basis of the definitions are, or at least can be, ampliative because, as we have seen, “often we need several definitions for the proof of some proposition, which consequently is not contained in any one of them alone, yet does follow purely logically from all of them together” (Frege 1884, § 88). This is exactly how one proceeds in abstract algebra, and in modern mathematics more generally. Our problem was to understand the striving for truth in mathematics and in particular how the practice of mathematics can be at once a priori and capable of yielding substantive, objective truths. The solution came in two parts, one due to Kant and one due to Frege. From Kant we learned that a calculation or demonstration is ampliative in virtue of a fundamental feature of the systems of signs it employs, namely, its having three different levels or tiers of articulation: that of the primitive signs of the system, that of the largest whole of signs within which one works, and between the two both the signs for the objects that form the subject matter of the discipline and the various reconfigurations that enable one to see how, through a stepwise progression, one can establish the truth of the judgment one is interested in. What Kant did not have, and seems not to have thought it possible to have, was a system of signs that would serve in the expression, at the second level, of the contents of concepts. Frege provides exactly that in his formula language of pure thought. Because in Begriffsschrift primitive signs only express senses prior to their involvement in sentences, those sentences can be variously conceived in much the way the different parts of a Euclidean diagram can be variously conceived. The 15. Compare Hilbert’s (1903, p. 51) remark in a letter to Frege, that “a concept can be fixed logically only by its relations to other concepts. These relations, formulated in certain statements, I call axioms, thus arriving at the view that axioms (perhaps together with propositions assigning names to concepts) are the definitions of the concepts”. Axioms so conceived set out what Sellars and Brandom think of as the material inferential connections that are supposed to hold between concepts.

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conception of language that is embodied in Frege’s notation is not merely a conservative extension of Kant’s insight, however. As I have indicated, Frege’s logical language Begriffsschrift is something essentially new.16 It is also the language that is needed to understand the striving for truth in the practice of modern mathematics.

BIBLIOGRAPHY Benacerraf, Paul, 1973. Mathematical Truth. Journal of Philosophy 70, 661–679. Boyer, Carl B., 1991. A History of Mathematics, 2nd ed. New York: John Wiley and Sons. Brandom, Robert B., 1994. Making It Explicit: Reasoning, Representing, Discursive Commitment, Cambridge, Mass.: Harvard University Press. Frege, Gottlob, 1879. Begriffsschrift, a Formula Language of Pure Thought Modeled upon the Formula Language of Arithmetic. In: Conceptual Notation and Related Articles, trans. and ed. T. W. Bynum, Oxford: Clarendon Press, 1972. — 1880/81. Boole’s Logical Calculus and the Concept-script. In: Posthumous Writings, ed. H. Hermes, F. Kambartel, and F. Kaulbach, and trans. P. Long and R. White, Chicago: University of Chicago Press, 1979. — 1884. The Foundations of Arithmetic, trans. J. L. Austin, Evanston, Ill.: Northwestern University Press, 1980. — 1914. Logic in Mathematics. In: Posthumous Writings, ed. H. Hermes, F. Kambartel, and F. Kaulbach, and trans. P. Long and R. White, Chicago: University of Chicago Press, 1979. Friedman, Michael, 1992. Kant and the Exact Sciences, Cambridge, Mass.: Harvard University Press. Hilbert, David, 1903. Letter to Frege, November 7, 1903. In: Philosophical and Mathematical Correspondence, ed. B. McGuinness and trans. H. Kaal, Chicago: University of Chicago Press, 1980. Kant, Immanuel, 1764. Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality. In: Theoretical Philosophy, 1755–1770, trans. and ed. D. Walford with R. Meerbote. New York: Cambridge University Press, 1992. Kant, Immanuel, 1781/1787. Critique of Pure Reason, trans. and ed. P. Guyer and A. W. Wood, New York: Cambridge University Press, 1998. 16. Even Frege himself fully grasped the way his Begriffsschrift notation actually works only decades after its first introduction. See Macbeth (2005), especially Chapters Three and Four.

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Kant, Immanuel, 1788. Letter to Johann Schultz, November 25, 1788. In: Correspondence, trans. and ed. A. Zweig, New York: Cambridge University Press, 1999. — 1980. Kant’s gesammelte Schriften, vol. 29, ed. Deutsche Akademie der Wissenschaften, Berlin: Walter de Gruyter. Macbeth, Danielle, 2005. Frege’s Logic, Cambridge, Mass.: Harvard University Press. — forthcoming. Logical Analysis, Reduction, and Philosophical Understanding. The Croatian Journal of Philosophy. MacFarlane, John, 2000. What Does It Mean to Say that Logic is Formal? Thesis (PhD). University of Pittsburgh. Rotman, Brian, 2000. Mathematics as Sign: Writing, Imagining, Counting. Stanford: Stanford University Press. Sellars, Wilfrid, 1953. Inference and Meaning. Mind 62, 313–338. Stein, Howard, 1988. Logos, Logic, and Logistiké: Some Philosophical Remarks on Nineteenth Century Transformation of Mathematics. In: W. Aspray and Ph. Kitcher, ed. History and Philosophy of Modern Mathematics, Minnesota Studies in the Philosophy of Science, vol. XI, Minneapolis: University of Minnesota Press. Wilson, Mark, 1992. Frege: The Royal Road from Geometry. In: W. Demopoulos, ed. Frege’s Philosophy of Mathematics. Cambridge, Mass.: Harvard University Press, 1995.

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Grazer Philosophische Studien 75 (2007), 93–123.

FREGE’S USE OF FUNCTIONARGUMENT ANALYSIS AND HIS INTRODUCTION OF TRUTHVALUES AS OBJECTS Michael BEANEY University of York Summary One of Frege’s most characteristic ideas is his conception of truth-values as objects. On his account (from 1891 onwards), concepts are functions that map objects onto one of the two truth-values, the True and the False. These two truth-values are also seen as objects, an implication of Frege’s sharp distinction between objects and functions. Crucial to this account is his use of functionargument analysis, and in this paper I explore the relationship between this use and his introduction of truth-values as objects. In the first section I look at Frege’s use of function-argument analysis in his first work, the Begriffsschrift, and stress the importance of the idea that such a use permits alternative analyses. In the second section I examine his early notion of conceptual content, and argue that there is a problem in understanding that notion once alternative analyses are allowed. In the third section I turn to his key 1891 paper, ‘Function and Concept’, where the idea of truth-values as objects first appears, and consider its motivation. In the concluding section I comment on Frege’s general philosophical approach, which allowed objects to be readily ‘analyzed out’ in transforming one sentence into another.

1. Function-argument analysis in the Begriffsschrift Frege’s most important innovation was the use of function-argument analysis in developing his logical notation, which he called ‘Begriffsschrift’, as first set out in his book of that name in 1879.1 Indeed, it is no exaggeration to say that without this innovation Frege would have had no place at all in the history of logic, none of his characteristic doctrines would have been formulated, and twentieth-century analytic philosophy would have 1. In what follows, I shall use ‘Begriffsschrift’ to refer to Frege’s logical notation, and ‘Begriffsschrift’ (in italics) to refer to the book.

been very different. In his preface to the Begriffsschrift, Frege characterized his achievement as follows: The very invention of this Begriffsschrift, it seems to me, has advanced logic. I hope that logicians, if they are not put off by first impressions of unfamiliarity, will not repudiate the innovations to which I was driven by a necessity inherent in the subject matter itself. These deviations from what is traditional find their justification in the fact that logic hitherto has always followed ordinary language and grammar too closely. In particular, I believe that the replacement of the concepts subject and predicate by argument and function will prove itself in the long run. It is easy to see how taking a content as a function of an argument gives rise to concept formation. What also deserves notice is the demonstration of the connection between the meanings of the words: if, and, not, or, there is, some, all, etc. (1879, p. vii/1997, p. 51)

Several of the key ideas and themes in Frege’s philosophy are reflected here. The importance of function-argument analysis is emphasized, but this is done in the context of remarking on the logical inadequacy of ordinary language, a continual complaint throughout Frege’s work. Since function-argument analysis permits alternative ways of ‘splitting up’ content, we can see how Frege thought that concept formation is easily explained. His construals of the various logical connectives are also intimately bound up with his use of function-argument analysis. In writing of a “necessity inherent in the subject matter itself ”, we can also see Frege’s inclination to regard his logical categories as reflecting deep ontological truths. This was to find later expression in his talk of the distinction between first-level and second-level functions, for example, as being “founded deep in the nature of things” (1891, p. 31/1997, p. 148), and in his suggestion that he had ‘discovered’ the two objects, the True and the False, in the same sense in which one ‘discovers’ two new chemical elements (quoted on p. 120 below). However, at the time of the Begriffsschrift, Frege did not appreciate the full significance of his achievement, or draw out more than a few of the implications of his emphasis on function-argument analysis. Indeed, these implications only gradually dawned on him as he sought to develop and defend his ideas, and drawing out these implications constitutes a major part of his later philosophy. That philosophy can be seen as articulating a metaphysics and epistemology that justifies and reinforces his use of function-argument analysis, in the context of the project that governed his life’s work — the attempt to demonstrate the logicist thesis that arithmetic is

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reducible to logic. His construal of concepts as functions, his fundamental distinction between function and object, and his introduction of truthvalues as objects, were all consequences of his use of function-argument analysis in logic, the development of a more powerful logic being essential for his logicist project. According to Frege, a simple proposition such as ‘Socrates is mortal’ should be analyzed not in subject-predicate terms, taking ‘Socrates’ as the subject and ‘mortal’ as the predicate, the two terms joined by the copula, but in function-argument terms, taking ‘Socrates’ as the argument and ‘x is mortal’ as the function, the ‘x’ in ‘x is mortal’ indicating the place where the argument goes.2 At this simplest level, there might seem little to decide between the two analyses — the issue hinging, perhaps, on one’s view of the role of the copula. The superior power of function-argument analysis comes out when we turn to propositions involving quantifiers such as ‘all’ or ‘some’. In traditional (Aristotelian) logic, both ‘All humans are mortal’ and ‘Some humans are mortal’ were also analyzed in subjectpredicate terms, ‘All humans’ and ‘Some humans’ being the subjects, with ‘mortal’ as the predicate. As Frege sees it, however, such propositions have a greater complexity than subject-predicate analysis reveals. Take the syllogistic universal proposition ‘All humans are mortal’. Frege construes this proposition as ‘If anything is a human, then it is mortal’, in other words, as two more primitive ‘propositions’ (‘anything is a human’ and ‘it is mortal’) joined together by the propositional connective ‘if … then’. This can be readily expressed in function-argument form, replacing ‘anything’ and ‘it’ by the variable ‘x’ and binding the whole by the quantifier ‘for all x’: ‘For all x, if x is a human, then x is mortal’. Indeed, it is difficult to see how ‘If anything is a human, then it is mortal’ can be formalized without making use of the idea of a bound variable. All this is now thoroughly familiar and taken for granted, of course. But it is important to appreciate just how liberating and powerful function-argument analysis must have seemed to Frege. In particular, it allowed 2. I use the word ‘proposition’ in this paper in (roughly) the sense in which Frege uses the word ‘Satz’. It means more than just ‘sentence’, understood as a mere linguistic expression, but as something like ‘sentence with a content’. I also gloss over here the issue as to whether, at the time of Begriffsschrift, Frege understood functions and arguments as expressions or as what those expressions stand for. I agree with Baker (2001, 2005) and Baker and Hacker (2003) that Frege did not understand functions as merely linguistic items but saw them essentially as the ‘contents’ of functional expressions — a view he clarifies later by characterizing them as the Bedeutungen of functional expressions. I return to this, and take up the related issue as to what he regards as the value of a function for a given argument, shortly.

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him to analyze propositions involving multiple generality (i.e., with more than one quantifier) with relative ease, and propositions of multiple generality — such as ‘Every natural number has a successor’ — are prevalent in mathematics. Mathematical propositions were precisely what Frege needed to analyze to pursue his logicist project, and the success with which he managed to do so no doubt made that project seem feasible. Nor is it surprising that he should have felt that the possibility of function-argument analysis was grounded “deep in the nature of things”. For how could function-argument analysis be so successful if it did not carve reality at its joints? According to Frege, a simple proposition such as ‘Socrates is mortal’ is to be analyzed into the argument ‘Socrates’ and the function ‘x is mortal’. But what is the value of such a function for this argument? In § 9 of the Begriffsschrift, where Frege explains his conception of a function, he is not as explicit about this as one would have liked. But the implication is that the value of a function, in the case of propositions, is what Frege calls the ‘conceptual content’ (‘begrifflicher Inhalt’) of the proposition. This notion was introduced in § 3 of the Begriffsschrift, where it is characterized as that part of the content of a proposition that influences its possible consequences. On Frege’s view, the following two propositions have the same conceptual content: (GP) At Plataea the Greeks defeated the Persians. (PG) At Plataea the Persians were defeated by the Greeks. While we might discern “a slight difference of sense” between these two propositions, Frege writes, the content that they have in common is what predominates: “I call that part of the content that is the same in both the conceptual content” (1879, p. 3/1997, p. 53). Essentially, two propositions have the same conceptual content if and only if they are logically equivalent. I shall say more about this notion in § 2 of this paper. Frege appeals to the notion of conceptual content in discussing his main example in § 9, the proposition ‘Hydrogen is lighter than carbon dioxide’. According to Frege, this proposition can be analyzed in a number of different ways. If we take ‘hydrogen’ as the argument, then ‘x is lighter than carbon dioxide’ is the function. But we can also take ‘carbon dioxide’ as the argument, in which case ‘x is heavier than hydrogen’ is the function. Frege writes that in doing this we “grasp the same conceptual content” (1879, p. 15/1997, p. 66). It would be natural to interpret Frege, then, as

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saying that the value of the two functions for the arguments ‘hydrogen’ and ‘carbon dioxide’, respectively, is the same, this value being the conceptual content of the proposition. Of course, we might express this conceptual content in different ways, using two different propositions to reflect the different function-argument analyses: (HC) Hydrogen is lighter than carbon dioxide. (CH) Carbon dioxide is heavier than hydrogen. In (HC) ‘hydrogen’ is the subject, while in (CH) ‘carbon dioxide’ is the subject, and we could adopt the convention that the subject-predicate structure of a proposition signals our intended or preferred mode of function-argument analysis — the subject of the proposition indicating the argument, and the rest of the proposition the function.3 But (HC) and (CH) nevertheless have something in common, which is what Frege calls their ‘conceptual content’. It makes sense to regard this common conceptual content, then, as the value that the two different functions have for their respective arguments. Just as the same number 7 is the value both of the function ‘x + 3’ for the argument 4 and of the function ‘x – 3’ for the argument 10, so too one and the same conceptual content is the value both of the function ‘x is lighter than carbon dioxide’ for the argument ‘hydrogen’ and of the function ‘x is heavier than hydrogen’ for the argument ‘carbon dioxide’. In fact, these are not the only possible function-argument analyses, or even, arguably, the most basic. Frege writes: If, in a function, a symbol that has up to now been viewed as not replaceable is thought of as replaceable at some or all of the places at which it occurs, then, by being grasped in this way, a function is obtained that has another argument besides the previous one. In this way functions of two or more arguments arise. Thus, for example, ‘the circumstance that hydrogen is lighter than carbon dioxide’ can be taken as a function of the two arguments ‘hydrogen’ and ‘carbon dioxide’. (1879, § 9, pp. 17–18/1997, p. 68) 3. Cf. Frege (1879, § 9, p. 18/1997, p. 68): “The subject [of a proposition] is usually intended by the speaker to be the principal argument; the next most important often appears as the object. Through the choice of [grammatical] forms such as active and passive, and words such as ‘heavier’ and ‘lighter’, ‘give’ and ‘receive’, ordinary language has the freedom of allowing whatever part of the proposition it wishes to appear as the principal argument, a freedom, however, that is limited by the paucity of words.”

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A relational proposition such as ‘Hydrogen is lighter than carbon dioxide’ can be analyzed not only into an argument and a one-place function (in two different ways) but also into two arguments and a two-place function. Indeed, it might be suggested that this latter analysis is the most basic analysis, since it identifies all the fundamental objects.4 But even here there are two possible analyses, since what is important is not just the objects identified but also the order in which they are represented in a proposition, reflecting the precise relation (i.e., two-place function) involved. If we formalize the proposition as ‘aRb’ or ‘Rab’, where ‘R’ denotes the relation ‘x is lighter than y’, ‘a’ hydrogen, and ‘b’ carbon dioxide, then given the equivalence between (HC) and (CH) above, we can also formalize it as ‘bRca’ or ‘Rcba’, where ‘Rc’ denotes the converse relation ‘x is heavier than y’. Since R and Rc are different relations, there is no single function-argument analysis that can justifiably be regarded as the most fundamental, even if some analyses are more fundamental than others. In the passage just quoted, Frege also talks of the ‘circumstance’ (‘Umstand ’) that hydrogen is lighter than carbon dioxide, and this is a term he uses elsewhere. But it is clear from § 2 of the Begriffsschrift, where the term is introduced, that the phrase ‘the circumstance that’ is understood as equivalent to ‘the proposition that’, offered as an ordinary language translation of his content stroke ‘–––’. In Frege’s Begriffsschrift, ‘––– A’ represents the proposition or circumstance that A. (Cf. 1879, p. 2/1997, p. 53.) So does this mean that the value of a function for an argument or arguments, in the case of propositions, is the proposition or circumstance itself? The text of the Begriffsschrift does not give a clear answer. Frege alternates between talking of propositions or circumstances being functions (i.e., values of functions) of an argument or arguments (see e.g. 1879, pp. 16, 18) and talking of contents or conceptual contents being functions of an argument or arguments (see e.g. 1879, pp. vii, 15, 17, 19). Given that a ‘proposition’ involves more than just its ‘conceptual content’, which is merely its ‘logical core’, as it might be put (i.e., that part of its content that influences its possible consequences, as noted above), it would be most accurate to say that it is conceptual contents that are analyzed into functions and arguments, i.e., that are the values of functions of an argument or arguments. This interpretation would seem to be confirmed by ‘Boole’s rechnende Logik und die Begriffsschrift’, written shortly after the Begriffsschrift, where Frege does indeed talk of a 4. For this view, see Dummett (1981, ch. 15). I return to this below.

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‘content’, or more specifically, a ‘judgeable content’ (‘beurteilbarer Inhalt’), as what is analyzed (see e.g. 1969, pp. 17–19), although here too he is still not explicit that this is the value of a function of an argument or arguments. The position at the time of the Begriffsschrift, then, seems to be this. Propositions are to be analyzed in function-argument rather than subjectpredicate terms, and central to the idea of function-argument analysis is that two different functions can yield the same value for appropriate arguments, suggesting that propositions too can have alternative analyses. Propositions are also seen as having a conceptual content, two propositions having the same conceptual content when they are logically equivalent. Putting the two ideas together, then, leads naturally to the thought that conceptual contents are the values of functions. If it is the conceptual content of a proposition that is analyzed in function-argument terms, then the obvious implication is that the value that the relevant function has for the appropriate argument or arguments is the conceptual content. However, Frege does not explicitly draw what might seem to be the obvious implication, and in his later work he abandons the notion of conceptual content, precisely for the reason, as we will see, that it does not fit comfortably with his understanding of function-argument analysis. At the time of the Begriffsschrift, then, Frege had not yet thought through all the consequences of his use of function-argument analysis. But two key ideas from that time did remain with him for the rest of his life. The first is the idea, central to function-argument analysis, that alternative analyses are always possible. In a letter dated 29 August 1882, shortly after the publication of the Begriffsschrift, Frege writes: I think of a concept as having arisen by decomposition [Zerfallen] from a judgeable content. I do not believe that for any judgeable content there is only one way in which it can be decomposed, or that one of these possible ways can always claim objective pre-eminence. (1997, p. 81)

In his later work he talks not of ‘judgeable contents’ but of ‘thoughts’ being analyzed in different ways, but these too are seen as prior to concepts, which are arrived at by analysis. In his ‘Notes for Ludwig Darmstaedter’, composed in 1919, at the end of his life, he writes: “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 analysing the thought.” (1969, p. 273/1997, p. 362.) He does not state explicitly here that the same thought permits alternative analyses, but this is a view that he did hold in his later

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work. In ‘On Concept and Object’, published in 1892, for example, he writes: a thought can be split up in many ways, so that now one thing, now another, appears as subject or predicate. The thought itself does not yet determine what is to be regarded as the subject. If we say ‘the subject of this judgement’, we do not designate anything definite unless at the same time we indicate a definite kind of analysis; as a rule, we do this in connection with a definite wording. But we must never forget that different sentences may express the same thought. (1892b, p. 199/1997, p. 188)

The idea that there can be alternative analyses, then, survives the abandonment of the notion of conceptual content. But as the above passage shows, Frege also continued to emphasize the related idea that two different propositions can have something in common — a ‘logical core’, to put it in neutral terms. In his early work, the claim was that two different propositions can have the same conceptual content; in his later work, this became the claim that two different propositions can express the same thought. The question, then, is what motivated the abandonment of the notion of conceptual content, and what this had to do with Frege’s use of functionargument analysis and his introduction of truth-values as objects. 2. The notion of conceptual content According to Frege at the time of the Begriffsschrift, two propositions have the same conceptual content if and only if they have the same possible consequences (cf. 1879, p. 3/1997, p. 54). To say that two propositions P and Q have the same possible consequences is to say that they are logically equivalent, i.e., that P implies Q, and Q implies P. So Frege’s criterion can be formulated thus: (CC) Two propositions have the same conceptual content iff they are logically equivalent. Both (GP) and (PG), and (HC) and (CH), as given above, are logically equivalent, and hence count as having the same conceptual content, on Frege’s account. But it is not just propositions that are regarded as having conceptual content. In § 8 of the Begriffsschrift, in explaining his symbol for identity of

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content (‘{’), Frege makes clear that names have content, too, the content of a name being the object denoted.5 This suggests the following criterion in the case of names:6 (CCn) Two names have the same (conceptual) content iff they denote the same object. Notoriously, however, Frege regards the relation of identity of content itself as relating to names, not contents: While elsewhere symbols simply represent their contents, so that each combination into which they enter merely expresses a relation between their contents, they suddenly reveal their own selves as soon as they are combined by the symbol for identity of content; for this signifies [bezeichnet] the circumstance that two names have the same content. (1879, pp. 13–14/1997, p. 64; tr. modified slightly)

This leads Frege to offer a metalinguistic criterion for sameness of conceptual content, which can be formulated as follows: (CCs) Two symbols A and B have the same conceptual content iff A can always be replaced by B and vice versa. (Cf. 1879, p. 15/1997, p. 65) I shall return to these criteria in the next section. Here I want to consider what Frege might have had in mind in talking of the ‘conceptual content’ of a proposition. We may have criteria of identity, but what exactly are conceptual contents? To press the objection that Frege himself later raised in relation to his attempted contextual definition of number, how can we distinguish a conceptual content from an object of some other kind such as Julius Caesar?7 To explore this question, let us first take the simple proposition considered earlier: ‘Socrates is mortal’. What is the conceptual content of this proposition? If we play down talk of ‘conceptual’, the obvi5. The term that Frege uses here is ‘bedeutet’ (1879, p. 14). I consider the significance of this in the next section. 6. In the case of names, Frege tends to talk just of ‘content’, but § 8 does end with an explicit reference to conceptual content. 7. Cf. Frege (1884, §§ 56, 66). The objection is not misplaced, for the criteria for sameness of content just offered can themselves be taken as contextual definitions: they are analogous to the Cantor-Hume Principle to which Frege appeals in § 63. I discuss this principle in the next section.

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ous suggestion would be that the content is the fact that Socrates is mortal or the state of affairs of Socrates being mortal. The relevant fact or state of affairs comprises an object or particular, Socrates, and a property or universal, mortality, such that the instantiation of the property by the object is the fact or state of affairs concerned. While the content of ‘Socrates’ is Socrates himself, the content of ‘Socrates is mortal’ is the complex entity composed of Socrates and mortality related in some way. There are two obvious problems with this view. Given that Socrates no longer exists, is there really a fact or state of affairs in such a case, and if so, what is it? And in general, what are we to say about false propositions? Are there negative facts or non-actual states of affairs? In the case of names that fail to denote, Frege tended to brush the problems that arise here under the carpet, from the time of Begriffsschrift onwards. He simply held the view that, at least as far as science is concerned, all proper names (i.e., singular terms) must denote an object (and all concept words a well-defined concept). As he writes in his posthumously published dialogue with the theologian Pünjer on existence, which took place in the early 1880s, “The rules of logic always presuppose that the words we use are not empty, that our sentences express judgements, that one is not playing a mere game with words” (1979, p. 60). If ‘Socrates’ fails to denote, according to Frege, then ‘Socrates is mortal’ will express no judgement and so the issue of its having content will not arise.8 However, if we wanted to accommodate subject terms that fail to denote, then there is another option open to us, which might seem attractive anyway, if we take talk of ‘conceptual content’ more seriously. Perhaps the ‘conceptual content’ of ‘Socrates is mortal’ should be seen as involving two concepts — not only the concept ‘( ) is mortal’ but also the concept ‘( ) is Socrates’, as it might be put to abbreviate a fuller specification. We might take this latter concept as the concept ‘( ) is identical with Socrates’, for example, or regard it as a complex concept composed of all the concepts which specify his ‘essence’, or some subset of concepts which are jointly uniquely applicable to Socrates. The proposition ‘Socrates is mortal’ would then be treated as equivalent to ‘Whatever falls under the concept ‘( ) is Socrates’ also falls under the concept ‘( ) is mortal’’. The proposition would thus be treated as analogous to the proposition ‘All humans are mortal’, which Frege does indeed analyze as ‘Whatever falls under the concept ‘( ) 8. Of course, we might insist that such a sentence still has ‘meaning’, understood linguistically, but giving an account of linguistic meaning was (arguably) never Frege’s concern.

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is human’ also falls under the concept ‘( ) is mortal’’. However, it is clear that this would not have been Frege’s view, given the emphasis he placed on the distinction between object and concept, and the associated distinction between subsumption, which holds between object and concept, and subordination, which holds between concepts. Although these distinctions are not drawn explicitly in the Begriffsschrift, they are obvious implications of his use of function-argument analysis, and he does indeed stress them in several works written shortly after the Begriffsschrift.9 Turning to the second problem, what are we to say in the case of false propositions? Perhaps, as Wittgenstein did in the Tractatus, we should appeal to possible states of affairs, and regard these as ultimately resolvable into necessarily existent objects. There is no talk of necessarily existent objects in the Begriffsschrift — although, as we have just noted, Frege does assume that objects must exist in the context of science. But Frege does tacitly distinguish between fact (Tatsache) and circumstance (Umstand), a fact being understood, it seems, as a circumstance that obtains. In § 3 of the Begriffsschrift, for example, he imagines a language “in which the proposition [Satz] ‘Archimedes was killed at the capture of Syracuse’ is expressed in the following way: ‘The violent death of Archimedes at the capture of Syracuse is a fact [Tatsache]’”. Here the subject of the latter proposition, Frege writes, “contains the whole content, and the predicate [‘is a fact’] serves only to present it as a judgement”. “Our Begriffsschrift”, he goes on to say, “is such a language”. (1879, pp. 3–4/1997, p. 54.) Given the equivalence, noted above, between ‘the proposition that’ and ‘the circumstance that’, Frege’s remark might be glossed as saying that in making a judgement, we are asserting that a (possible) circumstance does indeed obtain, i.e., is an (actual) fact. So a false proposition, on Frege’s account, is just a circumstance (as expressed in a sentence) that does not in fact obtain. There is some looseness in Frege’s terminology at the time of Begriffsschrift, but reconstructed in the light of the above, his position seems 9. In his long piece on Boolean logic, written in 1880/81, for example, Frege criticizes Boole for obliterating the distinction between concept and individual, and writes that “We must likewise distinguish the case of one concept being subordinate to another from that of a thing falling under a concept although the same form of words is used for both” (1969, p. 19/1979, p. 18). He also emphasizes both distinctions in his letter to Marty of 29 August 1882 (1976, pp. 164–5/1997, pp. 80–2). The distinction between concept and object is stressed, most famously, in the introduction to the Grundlagen of 1884, where respect for it is formulated as the third of his three fundamental principles (1884, p. x/1997, p. 90). On the distinction between subsumption and subordination, cf. (1884, §§ 47, 53), and the further references cited in (1997, p. 81, fn. 10).

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to be this. For his logical system to be applicable within science, i.e., for judgements to be formalizable within his Begriffsschrift, every proper name must denote an object. The conceptual content of a proposition is the (possible) circumstance it represents. Such a circumstance may either obtain, in which case the proposition is true, or not obtain, in which case the proposition is false. In the first case, we can make a judgement that the circumstance is a fact; in the second case, we can make a judgement that the circumstance is not a fact, i.e., that it does not obtain.10 In the case of a simple proposition such as ‘Socrates is mortal’, then, the content of such a proposition (assuming that the subject term denotes) is the circumstance that a particular object falls under the relevant concept (or the circumstance constituted by the object’s instantiating the relevant concept).11 The idea of circumstances as conceptual contents, however, raises problems of its own, which can be appreciated when we consider more complex propositions. Take Frege’s example of a relational proposition, ‘Hydrogen is lighter than carbon dioxide’. We have already seen how this can be analysed in alternative ways, even at the supposedly most fundamental level. The problem for the notion of conceptual content can now be stated as follows. Does the circumstance that hydrogen is lighter than carbon dioxide (or the circumstance of hydrogen being lighter than carbon dioxide) involve the relation ‘( ) is lighter than ( )’ or the converse relation ‘( ) is heavier than ( )’, or both? It would seem arbitrary to regard one but not the other, together with the two ‘objects’, hydrogen and carbon dioxide, as constituting the circumstance. But are both relations then involved? Does a full analysis of the conceptual content have to specify both relations? But if so, then what relation do these relations themselves have in the supposed constitution of the circumstance? And are the two concepts ‘( ) is lighter than carbon dioxide’ and ‘( ) is heavier than hydrogen’ also involved, and how are they related to everything else? If they are indeed involved, then how many other concepts and relations are involved? ‘Hydrogen is lighter than carbon dioxide’ is also equivalent to the following proposition: (HC*) Hydrogen falls under the concept ‘( ) is lighter than carbon dioxide’. 10. Cf. Frege (1879, § 7), where Frege gives his account of negation. 11. Cf. also Frege’s letter to Marty of 29 August 1882, where Frege writes: “That an individual falls under [a concept] is a judgeable content” (1976, p. 164/1997, p. 81).

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So is the relation ‘( ) falls under ( )’ also involved? All sorts of further analyses might be offered, even if some are regarded as more fundamental than others. So where are we to draw the line? Clearly, many questions arise concerning the nature of relational circumstances. The problem is exacerbated when we bring in Frege’s appeal to the notion of conceptual content in the central sections of Die Grundlagen der Arithmetik of 1884, where Frege considers the possibility of defining abstract objects such as numbers and directions contextually. Here are the two relevant pairs of propositions: (Da) Line a is parallel to line b. (Db) The direction of line a is identical with the direction of line b. (Na) The concept F is equinumerous to the concept G. (There are as many objects falling under concept F as under concept G, i.e., there are just as many Fs as Gs.) (Nb) The number of Fs is identical with the number of Gs. According to Frege, in moving from (Da) to (Db), “We split up the content in a different way from the original way and thereby acquire a new concept”, namely, the concept of direction (1884, § 64/1997, p. 111). Analogously, starting from (Na), the crucial feature of which is that it can be defined purely logically (since one-one correlation is involved), we can acquire the concept of number. Presupposed here is that (Da) and (Db), and (Na) and (Nb), respectively, have the same conceptual content. That (Na) and (Nb) are equivalent is what is asserted by the Cantor-Hume Principle,12 which plays a key role in Frege’s logicist project. But if conceptual contents are circumstances, then we really do have a problem in specifying just what such circumstances involve. Does the circumstance in the case of (Da) and (Db), for example, involve the lines a and b, the relation of parallelism, and the relevant abstract object, the direction of a and b? What about the relation of identity? Is this, too, involved? Admittedly, Frege goes on to raise doubts about the legitimacy of contextual definition (because it fails to solve the Julius Caesar problem). But he does not deny that (Na) and (Nb) have the same content. Indeed, he 12. This has often just been called ‘Hume’s Principle’, but this does not do justice to Cantor’s role in the use of the principle. Cantor is mentioned as well as Hume by Frege himself in introducing the principle (1884, § 63/1997, p. 110). Cf. Beaney (2005, p. 307, fn. 4); Reck and Beaney (2005, p. 1).

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makes use of the same idea in suggesting that (Na) is also equivalent to the following proposition (cf. 1884, § 68/1997, pp. 114–17): (Nd) The extension of the concept ‘equinumerous to the concept F ’ is identical with the extension of the concept ‘equinumerous to the concept G’. It is this that leads him to offer his definition of number as an extension of a concept, which can be expressed as follows: (Ne) The number of Fs is the extension of the concept ‘equinumerous to the concept F ’. (Cf. 1884, § 68/1997, p. 115) In fact, (Nd) stands in the same relation to the following proposition as (Nb) does to (Na): (Nc) The concept ‘equinumerous to the concept F ’ is coextensive with the concept ‘equinumerous to the concept G’. But (Nc) itself can be derived from (Na) by ‘splitting up the content in a new way’. So all four propositions — (Na), (Nb), (Nc) and (Nd) — have the same conceptual content, on Frege’s view. Certainly, they are all logically equivalent, and hence by the criterion (CC), possess the same conceptual content. In fact, there is no limit to the number of further ways that the content can be split up. Here are two more: (Nf ) The concept ‘coextensive with the concept “equinumerous to the concept F ”’ is coextensive with the concept ‘coextensive with the concept “equinumerous to the concept G”’. (Ng) The extension of the concept ‘coextensive with the concept “equinumerous to the concept F”’ is identical with the extension of the concept ‘coextensive with the concept “equinumerous to the concept G”’. There is an obvious sense in which these further ‘analyses’ are not ‘real’: to use Dummett’s term, they are ‘degenerate decompositions’ rather than genuine analyses.13 But the question remains: how much of all the objects, 13. Cf. Dummett (1981, pp. 288–90). Cf. Beaney (1996, pp. 238ff.).

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concepts and relations supposedly denoted in these propositions are really contained in the conceptual content? Now the fact that Frege identifies numbers with extensions of concepts suggests that (Nb) and (Nd) are the very same identity, so that there is only one object involved here. But what about all the concepts and relations? How many of these are involved, and how are they all related? What all these considerations suggest is that there is a real problem in specifying what the conceptual content of a proposition involves, a problem that arises just because Frege allows alternative analyses. As the cases of (Da) and Db), and (Na) and (Nb), show, the propositions that reflect these analyses can look very different, even though they are regarded as having the same content. Such propositions can seem to have different ontological commitments. (Da) seems to refer to lines and parallelism, for example, while (Db) seems to refer to a direction and the relation of identity. Frege thinks that directions and numbers are genuine, albeit abstract, objects. So are both lines a and b, their direction, and the relations of parallelism and identity all involved in the content of (Da) and (Db)? There is no evidence in Frege’s writings that he ever explicitly addressed the metaphysical question of the composition of conceptual contents. Part of the reason may have been his assumption that judgements have priority over concepts, concepts being extracted by function-argument analysis rather than judgements being formed by composition of (independently existing) concepts (or of concepts and objects). But the notion of conceptual content does seem to be inherently unstable and to break down as soon as we press the question of precisely what is involved. Since the nature of concepts was one of his main philosophical concerns, it is reasonable to suppose that Frege became aware of the problem, and that this contributed to his abandoning the notion. Certainly, he continued to build on his use of function-argument analysis and to think through its implications, leading to revision of his philosophical ideas. 3. The introduction of truth-values as objects The key paper in the development of Frege’s philosophy from the time of the Begriffsschrift to his later work is ‘Function and Concept’, given as a lecture to the Jenaische Gesellschaft für Medicin und Naturwissenschaft on 9 January 1891. Frege introduces this paper by remarking that “My starting-point is what is called a function in mathematics” (1891, p. 1/1997,

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p. 131), and the paper makes clear just how important the notion of a function is in Frege’s thought. Virtually all of the characteristic doctrines of his mature philosophy are present in this paper, in particular, the conception of concepts as functions whose values are truth-values, the infamous ‘fundamental law of logic’ that becomes Axiom V of the Grundgesetze, and the distinction between Sinn and Bedeutung, all of which appear here for the first time. All these doctrines result from Frege’s thinking through the implications of his use of function-argument analysis. I shall focus here on the first. Frege starts by emphasizing what he calls the distinction “between form [Form] and content [Inhalt], sign [Zeichen] and thing signified [Bezeichnetes]” (1891, p. 2/1997, p. 131). ‘2 5’ and ‘3 4’, for example, are different signs (they have different forms), but signify the same thing, which Frege expresses by saying that they have the same content or ‘Bedeutung’ (1891, p. 3/1997, p. 131). This marks the first appearance of Frege’s mature notion of Bedeutung.14 At the level of proper names (i.e., singular terms), we can then say, Frege’s earlier notion of content has become the notion of Bedeutung: what Frege now calls the ‘Bedeutung’ of a name is what he earlier called — and is still happy to call — its ‘content’. Frege writes: “What is expressed in the equation ‘2.23 2 = 18’ is that the right-hand complex of signs has the same Bedeutung as the left-hand one.” (Ibid.) He goes on: “The different expressions correspond to different conceptions and aspects, but nevertheless always to the same thing.” (1891, p. 5/1997, p. 132.) We can already see here an anticipation of the distinction between Sinn and Bedeutung, although the notion of Sinn is not officially introduced until half-way through the paper. His initial concern is just to establish that numerals and other numerical singular terms signify (bedeuten) numbers. This distinguishes them from functional expressions, which is what Frege turns to next. If we consider the numerical singular terms ‘2.13 1’, ‘2.43 4’, and ‘2.53 5’, Frege writes, we can recognise a common form, which we might express as ‘2.x 3 x’. We can regard this as representing the function, which yields what those singular terms signify (bedeuten) for the arguments 1, 4 14. This is not, of course, the first time that the term ‘Bedeutung’ or any of its cognates is used, even in ‘Function and Concept’ itself, where Frege uses the term twice in talking of the Bedeutung of the word ‘function’. But many of these earlier uses cannot be regarded as expressing the technical sense that the term and its cognates later had. There are a few passages, such as in (1879, § 8) (cf. fn. 5 above), where ‘bedeutet’ is indeed used in a way consistent with his later technical sense, but it is not used to mark a deliberate contrast between Bedeutung and Sinn.

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and 5, respectively. But we have to understand that the ‘x’ here merely indicates the place where the argument term is to go, and that the functional expression might be better given as ‘2.( )3 ( )’ with gaps to indicate the argument places. A function, Frege emphasises, “must be called incomplete, in need of supplementation, or unsaturated [ungesättigt]” (1891, p. 6/1997, p. 133). I will return to the issue of unsaturatedness shortly. Frege goes on to note that “We give the name ‘the value of a function for an argument’ to the result of completing the function with the argument” (1891, p. 8/1997, p. 134), and he introduces his conception of the value-range (Wertverlauf) of a function, which is the set of pairings of the arguments of the function with their corresponding values. It is at this point that Frege offers the first formulation of what will become Axiom V of the Grundgesetze: two functions have the same value-range if and only if they have the same value for each argument. (Cf. 1891, pp. 9–10/1997, p. 135.) With this explanation of the notion of a function in place, Frege then asks: “how has the Bedeutung of the word ‘function’ been extended by the progress of science?” (1891, p. 12/1997, p. 137). In answer, he distinguishes two directions in which this has happened, first, extending the field of mathematical operations to include “the various means of transition to the limit”, and second, extending the field of possible arguments and values for functions by admitting complex numbers (ibid.). The three paragraphs that then follow constitute, in my view, the three most important paragraphs in Frege’s writings: they reveal the motivation for both his appeal to truth-values and his seminal distinction between Sinn and Bedeutung. Here are the first two paragraphs (on which I will concentrate here), in which Frege explains how he has extended the notion of a function even further, and introduces his idea of truth-values: In both directions I go still further. I begin by adding to the signs , , etc., which serve for constructing a functional expression, also signs such as , , , so that I can speak, e.g., of the function x²  1, where x takes the place of the argument as before. The first question that arises here is what the values of this function are for different arguments. Now if we replace x successively by 1, 0, 1, 2, we get: (–1)²  1, 0²  1, 1²  1, 2²  1.

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Of these equations the first and third are true, the others false. I now say: ‘the value of our function is a truth-value’, and distinguish between the truthvalues of what is true and what is false. I call the first, for short, the True; and the second, the False. Consequently, e.g., what ‘2²  4’ stands for [bedeutet] is the True just as, say, ‘2²’ stands for [bedeutet] 4. And ‘2²  1’ stands for [bedeutet] the False. Accordingly, ‘2²  4’, ‘2  1’, ‘24  4²’, all stand for the same thing [bedeuten dasselbe], viz. the True, so that in (2²  4)  (2  1) we have a correct equation. (1891, pp. 12–13/1997, p. 137)

This is where Frege introduces his idea that the value of certain kinds of function (propositional functions, as Russell will call them) is a truthvalue. But it is important to appreciate that it is identity statements that motivate the idea. As Frege says, he extends the notion of a function to allow functional expressions to be constructed, in particular, with the identity sign. This naturally raises the question of what the value of a function such as ‘x 2  1’ is. Frege considers the four identity statements that result from taking 1, 0, 1 and 2 as arguments for the function, and points out that two of these identity statements are true and two are false. He then seems to simply stipulate that the value of such a function is a truth-value. Identity statements, just like other statements, may indeed have a truth-value, but why should this be regarded as the value of the relevant function? Frege does not even consider possible alternatives. In particular, what is wrong with taking the ‘conceptual content’ of an identity statement as the value of the relevant function, as he appears to have done in his earlier work? Frege makes no mention of conceptual content in ‘Function and Concept’. So why has this notion apparently been abandoned? We have considered problems with this notion, but no such problems are identified by Frege. The function ‘x 2  1’ might also be expressed, in ordinary language, as ‘has the property that its square is 1’ or ‘is a square root of 1’, as Frege himself later notes (1891, p. 15/1997, pp. 138–9). But if so, and we take, say, the argument 1, then we have the following proposition: (1a) 1 is a square root of 1.

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This is indeed true, but why can we no longer say that the ‘content’ of this proposition is the circumstance that 1 is a square root of 1 (or the circumstance of 1 being a square root of 1)? However, it is here, I think, that the significance of Frege’s concern with identity statements comes out. For in the passage quoted above, Frege construes (i.e., analyzes) what has just been expressed as (1a) not as saying that the number 1 falls under the concept ‘is a square root of 1’ but as an identity statement, saying that the square of 1 is identical with the number 1, i.e., as Frege formulates it: (1b) (1)²  1. So our question now is: what would the ‘conceptual content’ of an identity statement such as (1b) be? In the Begriffsschrift, as noted above, the content of an identity statement was taken to signify (bezeichnen) “the circumstance that two names have the same content” (1879, § 8/1997, p. 64). But we can immediately see that this is problematic. For (1a) and (1b) are logically equivalent, and hence have the same conceptual content, according to Frege’s criterion (CC). But if (1b) is not about the content of the terms flanking the identity sign but about the terms themselves, then how can this have the same content as (1a), which is indeed about the content at least of the term ‘1’? The circumstance that two names have the same content seems very different, in other words, to the circumstance that an object has a certain property. Even if we give up Frege’s earlier metalinguistic construal of identity statements,15 however, we still have a problem. For if we now construe (1b) as somehow about the content of the two terms flanking the identity sign, stating that the number 1 is identical with itself, or something, then this still seems to have a very different content to that of (1a). For (1a) seems to be about 1 while (1b) now seems to be about 1. So does the conceptual content involve both the number 1 and the number 1? Indeed, does it also involve the number 2, implicated in the square sign? 15. This is just what Frege comes to do, of course, in distinguishing between Sinn and Bedeutung. In the paragraph immediately following the two with which I am concerned here (i.e., in the third of the three that I have suggested are the key paragraphs), Frege introduces that distinction, and his later paper ‘On Sinn and Bedeutung’ opens with criticism of his earlier metalinguistic construal of identity statements (1892a, pp. 25–6/1997, pp. 151–2). We can see why he had to reject that construal. The Begriffsschrift offered a hybrid account of the conceptual content of propositions. But if identity statements can have the same content as other types of proposition, then that account cannot work.

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And what about the concept ‘is a square root of 1’ and the relation of identity? Are these, too, involved? We are back with all the problems we considered in the previous section. In fact, the problem is now even worse than that considered in the previous section. For if (1b) is true, then so is the following proposition, and vice versa: (2b) (1)² 1  2. (1b) and (2b), in other words, are logically equivalent, so by Frege’s criterion (CC), they too have the same content. But now it seems that all (true) arithmetical identities have the same conceptual content, rendering the notion of conceptual content useless. Indeed, all (true) identities whatsoever turn out to have the same content, and if all propositions can be transformed into identity statements — as (Na) can be transformed into (Nb), for example, or (1a) into (1b) — then the notion of content is itself emptied of all content.16 There is a genuine problem, then, in specifying what is involved in the ‘conceptual content’ of a proposition, and this problem is highlighted by the case of identity statements. Indeed, it does look as if the only content that can be ascribed to a proposition, if (CC) is the criterion, is its truth-value. And this is just what Frege does, although he recognizes that ‘content’ (and especially ‘conceptual content’) is no longer an appropriate term and uses ‘Bedeutung’ instead. Given that he uses ‘content’ and ‘Bedeutung’ synonymously in the case of names, it makes sense to shift to ‘Bedeutung’ rather than any other term, though the verb ‘bezeichnen’ is also used as an alternative to the verb ‘bedeuten’. But it is here that function-argument analysis, too, plays a role. For it is natural to think that the relation between the argument of a function and a term for that argument (e.g. the relation between the argument 1 and the numeral ‘1’) is the same as the relation between the value of a function (for a given argument) and a term for that value. So if the value of a function is a truth-value, then any expression for that truth-value will signify or denote it in the same way as the name of an argument signifies or denotes that argument. As Frege says in the second paragraph of the passage quoted above, “what ‘2²  4’ stands for [bedeutet] is the True just as, say, ‘2²’ stands for [bedeutet] 16. What we have here is a version of what is now known as the slingshot argument. Frege’s work has often been identified as a source of this argument, but I will not pursue this further here.

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4”. In mathematics, functions map numbers onto numbers: the values of those functions for given arguments have the same ontological status as the arguments. The two paragraphs from ‘Function and Concept’ confirm, I think, that Frege did regard conceptual contents, in his early work, as the values of (propositional) functions. What has changed is that he now believes that these contents — qua values — should be taken as truth-values.17 The term ‘truth-value’ is entirely appropriate, of course, and only reinforces the new view. With this new view in place, in the rest of ‘Function and Concept’ Frege draws out the implications and elaborates on his extended notion of a function. (The only exception is the paragraph that immediately follows the passage quoted above, which introduces the distinction between Sinn and Bedeutung.18) In particular, Frege defines a concept as a function whose value is always a truth-value, a thesis that is an obvious consequence of his new view, and clarifies what he means by the ‘unsaturatedness’ of concepts, which is what distinguishes them from objects, from which he concludes that truth-values are objects. In the rest of this section I will comment briefly on the idea of unsaturatedness and indicate its role in motivating Frege’s conclusion. Frege’s claim that concepts are unsaturated (ungesättigt) was first made in his letter to Marty of 29 August 1882, and was connected with his view that judgements are logically prior to concepts. In discussing how ‘Christianity’ is a concept only in the sense in which it is used in the proposition ‘This way of life is Christianity’, Frege writes: 17. He recognizes, of course, that the introduction of truth-values does not exhaust what he had earlier had in mind in talking of ‘conceptual contents’, which is why he is led to introduce the notion of Sinn as well as that of Bedeutung. As he says in the preface to the Grundgesetze, what he had earlier called ‘content’ he has now “split … up into what I call thought and truth-value” (1893, p. X/1997, p. 198). Cf. the next note. The relationship between Frege’s early conception of content and his introduction of truth-values is also brought out by Frege’s remark in ‘On Sinn and Bedeutung’ that “By the truth-value of a sentence I understand the circumstance that it is true or false” (1892a, p. 34/1997, p. 157). This seems an odd thing to say, for the ‘circumstance’ that one sentence is true is surely different, in most cases, from the ‘circumstance’ that another sentence is true. What we may have here, though, is just a residue of his earlier talk of circumstances as conceptual contents. 18. This is Frege’s response to the objection that might immediately be raised to his introduction of truth-values as ‘contents’ — that there is surely far more to ‘content’ than just truth-value (cf. the previous note). As Frege puts it himself, “The objection here suggests itself that ‘2²  4’ and ‘2  1’ nevertheless tell us quite different things, express quite different thoughts” (1891, p. 13/1997, p. 138). The distinction between Sinn and Bedeutung is intended to deal with this objection, and is elaborated in greater detail in ‘On Sinn and Bedeutung’ (1892a).

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A concept is unsaturated in that it requires something to fall under it; hence it cannot exist on its own. That an individual falls under it is a judgeable content, and here the concept appears as a predicate and is always predicative. In this case, where the subject is an individual, the relation of subject to predicate is not a third thing added to the two, but it belongs to the content of the predicate, which is what makes the predicate unsaturated. Now I do not believe that concept formation can precede judgement, because this would presuppose the independent existence of concepts, but I think of a concept as having arisen by decomposition from a judgeable content. (1997, p. 81)

As noted above, that judgements are prior to concepts remained Frege’s view for the rest of his life, as did the associated view that concepts are unsaturated (or can be explained in terms of the idea of unsaturatedness), despite his later abandonment of the notion of content. Again, this latter view results from his use of function-argument analysis: expressions for (propositional) functions are formed by removing one or more names (just one in the case of a concept) from a proposition. Functional expressions thus have ‘gaps’ where the argument term or terms are to go, the ‘completion’ or ‘saturation’ of these gaps forming the proposition. At the level of expressions, talk of unsaturatedness makes good sense: it simply reflects what goes on in function-argument analysis. But are functions themselves unsaturated? Frege explicitly calls them such in introducing the idea of unsaturatedness in ‘Function and Concept’ (1891, p. 6/1997, p. 133; quoted above), but in his subsequent writing, even in ‘Function and Concept’ itself, he talks of functional expressions being unsaturated and tends to resist calling functions themselves unsaturated.19 This is surely right. Talk of concepts (and other functions) as unsaturated may have made sense in his early work, where concepts were seen as ‘extracted’ from judgeable contents, but once the notion of content is abandoned, and the Bedeutung of a proposition is regarded as a truth-value, such talk is rendered problematic, even understood metaphorically. It makes no sense at all to regard concepts as ‘extracted’ from truth-values. In his early work, then, despite his use of function-argument analysis, Frege was arguably still thinking of ‘contents’ in whole-part terms, in other words, as in some sense ‘composed’ out of concepts and objects. On this view, concepts might be regarded as a kind of entity which can only exist 19. There are exceptions — for example, when Frege is explaining his conception of a function in the Grundgesetze (1893, § 1/1997, pp. 211–12). But this clearly draws on his account in ‘Function and Concept’, being composed very shortly afterwards.

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when instantiated in an object, thus constituting a content. This might still be mysterious and problematic, but in his later work, such a model is simply inappropriate. Concepts are functions, understood as mappings from objects (in the case of first-level concepts) to one of the two truth-values.20 Perhaps mappings can themselves be characterized as ‘unsaturated’, though it does not seem the obvious word to use. Nevertheless, whether or not ‘unsaturated’ is the right word to use to characterize functions, it is clear that Frege saw a fundamental distinction between objects and functions, reflected in the corresponding forms of linguistic expressions. It is the absoluteness of this distinction that led him to conclude that truth-values are objects. Having shown how he has extended the notion of a function to include truth-values as values, he points out how analyzing propositions in function-argument terms also involves extending what can count as an admissible argument: “Not merely numbers, but objects in general” (1891, p. 17/1997, p. 140). Indeed, he has generalized the notion of a function to allow any object whatsoever as either argument or value (ibid.). He then goes on: When we have thus admitted objects without restriction as arguments and values of functions, the question arises what it is that we are here calling an object. I regard a regular definition as impossible, since we have here something too simple to admit of logical analysis. It is only possible to indicate what is meant [gemeint]. Here I can only say briefly: an object is anything that is not a function, so that an expression for it does not contain any empty place. A statement contains no empty place, and therefore we must take its Bedeutung as an object. But this Bedeutung is a truth-value. Thus the two truth-values are objects. (1891, p. 18/1997, p. 140)

Frege’s conclusion that truth-values are objects, then, follows from his claims that truth-values are the Bedeutungen of sentences and that the Bedeutungen of sentences are objects, not functions. The former claim emerged from taking truth-values as the values of functions in the case of propositions. The latter claim is based on his view that sentences are ‘complete’ or ‘saturated’ expressions, like proper names but unlike functional 20. In the case of higher-level functions, which take lower-level functions as arguments, there is a temptation to treat the lower-level functions accordingly as objects. In ‘The concept horse is a concept’, for example, it looks as if ‘the concept horse’ designates an object that is a first-level concept, which is being said to fall under the second-level concept ‘( ) is a concept’. But Frege notoriously denies that this shows that concepts are objects: this is his infamous paradox of the concept horse. But I shall not pursue this here. For discussion, see Wright (1998).

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expressions, which was seen as reflecting an absolute distinction at the corresponding level of Bedeutung between objects and functions. Underlying both these claims is thus his use of function-argument analysis. Thinking through the implications of analyzing propositions in function-argument terms led, via the notion of conceptual content, to the idea of truth-values as the values of the relevant functions; and extending the notion of a function to allow any object whatsoever as either argument or value of a function, and emphasizing the absolute distinction between object and function, yielded the conclusion that truth-values are objects. 4. Analysis and the introduction of objects How seriously or literally are we to take Frege’s claim that truth-values are objects? What metaphysical weight is to be placed on such a claim and what epistemological issues does it raise? It is clear that Frege had a very liberal understanding of what counts as an object. Anything that is the Bedeutung of a well-formed expression that contains no empty place, i.e., a proper name or sentence, counted as an object. This included not just the empirical objects of everyday life and science but also abstract objects such as numbers, extensions of concepts and truth-values. (Cf. 1893, § 2/1997, p. 213.) Frege admitted that such abstract objects were not ‘actual’ objects, that is, spatio-temporal objects, fully part of the causal network. But that did not make them any less ‘objective’.21 They still counted as objects. But how do we apprehend such objects? In the case of numbers, Frege’s answer (at the time of the Grundlagen) was clear. We apprehend numbers by grasping the sense of sentences in which number terms appear. This can be illustrated by returning to the Cantor-Hume Principle, which asserts the equivalence between the following (cf. p. [13] above]): (Na) The concept F is equinumerous to the concept G. (Nb) The number of Fs is identical with the number of Gs. Frege’s thought can be represented thus. We apprehend numbers by grasping the sense of number terms such as ‘the number of Fs’ (which is a ‘saturated’ expression, indicating that any referent is an object). We do 21. For the distinction between what is actual (wirklich) and what is objective (objectiv), the latter taken as including the former, see e.g. Frege (1884, § 26/1997, p. 96; 1893, p. XVIII/1997, pp. 204–5; 1918, pp. 76–7/1997, p. 343–5).

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this by grasping the sense of sentences such as (Nb) in which the number terms appear. This is where Frege’s context principle comes in: “The meaning of a word must be asked for in the context of a proposition, not in isolation” (1884, p. X/1997, p. 90). But what is the sense of (Nb)? By the Cantor-Hume Principle, (Nb) is equivalent to, i.e., has the same content as, (Na). So we grasp the sense of (Nb) by grasping the sense of (Na) and accepting the principle. And the point about (Na) is that it can be defined purely logically, since one-one correlation is involved. In offering a way of contextually defining numbers, then, the principle at the same time suggests an account of how we apprehend numbers. If all this is right, then our knowledge of logic is enough to explain our apprehension of numbers. Before commenting on the assumptions involved here, however, let us note that such a contextual definition can also be offered of ‘conceptual contents’. Recalling the criterion discussed in § 2 above, we can formulate the following analogue to (Na) and (Nb): (CCa) Proposition P is logically equivalent to (has the same possible consequences as) proposition Q. (CCb) The conceptual content of P is identical with the conceptual content of Q. Indeed, we can note that such a contextual definition can also be offered of ‘truth-values’. We have seen how truth-values came to replace conceptual contents as the values of (propositional) functions, on Frege’s account. There was thus a corresponding shift from a criterion based on logical equivalence (as understood at the time of the Begriffsschrift) to one based on what is essentially mere material equivalence: two propositions have the same truth-value if and only if they are materially equivalent. But to make the resultant principle even more obvious, let us talk of two propositions being ‘equi-truth-valent’ when they have the same truth-value. So now we have the following analogue to (Na) and (Nb): (TVa) Proposition P is equi-truth-valent to proposition Q. (TVb) The truth-value of P is identical with the truth-value of Q. The equivalence between (TVa) and (TVb) is trivial. (There may be debate about the Cantor-Hume Principle, but this is surely uncontroversial.) So if Frege’s strategy in the Grundlagen is right, then this would show that truth-values, like numbers, are objects, which can be apprehended by

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grasping the sense of the relevant sentences. That truth-values are objects was precisely the conclusion he arrived at in 1891, in ‘Function and Concept’. An argument for this conclusion — and a simpler one, apparently independent of function-argument analysis — was already available, it seems, in 1884. As noted above, however, even in the Grundlagen itself, Frege came to reject the contextual definition provided by the use of the CantorHume Principle, and offered an explicit definition instead (from which the Cantor-Hume Principle could then be derived). So does this suggest that he would have repudiated the argument just given? He would not have done so because he saw anything intrinsically wrong with the use of such principles as the Cantor-Hume Principle. For, as also alluded to above, in subsequently defining numbers as extensions of concepts, Frege implicitly appeals to an analogous principle. Indeed, he later made this appeal explicit in formulating Axiom V of the Grundgesetze, which asserts the equivalence between the following: (Va) The function F has the same value for each argument as the function G. (Vb) The value-range [Wertverlauf] of the function F is identical with the value-range of the function G. (Cf. 1893, §§ 3, 9/1997, pp. 213– 14.) Since concepts are functions whose values are truth-values, on Frege’s later view, (Va) and (Vb) yield the following as a special case: (Ca) The concept F applies to the same objects (generating the same true propositions) as the concept G. (Cb) The extension of the concept F is identical with the extension of the concept G. (Cf. e.g. 1891, pp. 15–16/1997, p. 139.) Far from repudiating the use of such principles, then, Frege came to sanctify their use by laying down a supposedly fundamental law of logic. More problematic, however, was the use of the context principle. Formulated as the second of the three fundamental principles Frege states that he adhered to in the Grundlagen (1884, p. X/1997, p. 90), the principle is never mentioned again in his later work (unlike the other two). The issue has been controversial, but what is clear is that the principle could not have been stated in its original form once the distinction between Sinn

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and Bedeutung had been drawn. In his early work (i.e., prior to 1891), Frege used the terms ‘content’ (‘Inhalt’), ‘Sinn’ and ‘Bedeutung’ more or less interchangeably. But once his early notion of content has been split into the notions of Sinn and Bedeutung, his strategy in the Grundlagen looks less convincing. For given that Frege allows that terms can have sense without Bedeutung, how can grasping the sense of a term constitute apprehending the object that is supposedly its Bedeutung? And how can grasp of the sense of a sentence such as (Na) constitute apprehension of the Bedeutung of a term that only appears in a different sentence, albeit one supposedly equivalent such as (Nb)? This suggests that, despite Frege’s strengthened endorsement of principles such as the Cantor-Hume Principle and Axiom V, there remains a problem with his understanding of them. The distinction between Sinn and Bedeutung only brings this problem into focus. On his early view, such principles embody sameness of content. On his later view, they clearly embody sameness of Bedeutung, but do they also embody sameness of sense? If Axiom V is to be a logical law, then it looks as if the answer must be ‘yes’, as Frege seems to indicate in some places.22 But he is never as explicit about this as one would like, and the thrust of his notion of sense suggests a negative answer. For is it not natural to say that (Va) and (Vb) present their common Bedeutung in different ways, in one case by using the notions of function and sameness of value and in the other case by using the notions of value-range and identity, and hence have different senses? However, if any such principle embodies sameness of sense, then it is surely that asserting the equivalence between (TVa) and (TVb). But if so, then is this not simply the result of a stipulation? For any equivalence relation holding between objects of a certain kind we can stipulate that there is something that they thereby have in common. We can stipulate that two propositions have the same ‘content’ iff they are logically equivalent, for example, or that they have the same ‘Bedeutung’ or ‘truthvalue’ iff they are materially equivalent. The relevant principle can then be regarded as embodying sameness of sense by definition, but in this case, it seems misleading to suggest that we are thereby apprehending or being introduced to objects of some new kind. The dilemma is clear. If such principles embody sameness of sense, then they cannot be seen as 22. See e.g. Frege (1891, pp. 10–11/1997, p. 136; 1893, § 3/1997, pp. 213–14 — and my editorial remark in fn. 26 on p. 213). For further discussion, see Beaney (1996, § 8.1; 2005).

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genuinely referring to objects that are already ‘out there’ (or explaining our apprehension of such objects). If, on the other hand, the relevant identity statement does indeed make such reference, then it would seem to have a different sense to the statement asserting the corresponding equivalence relation. Although Frege later abandoned the notion of content in favour of the notions of Sinn and Bedeutung, he did not fully think through the implications for his use of such principles as we have just been considering. We now call these principles ‘abstraction principles’. But it is significant that Frege did not himself call them this, despite recognizing Russell’s talk of ‘definition by abstraction’.23 For this term suggests that the relevant objects are indeed ‘abstracted’ out from the corresponding equivalence relation and hence not already ‘out there’. Frege seemed to think of such objects in a robustly realist way. In some notes written in 1906, for example, he remarks: The True and the False are to be regarded as objects, for both the sentence and its sense, the thought, are complete in character, not unsaturated. If, instead of the True and the False, I had discovered two chemical elements, this would have created a greater stir in the academic world. (1969, p. 211/1997, p. 297)

The implication here is that Frege saw himself as having ‘discovered’ two new logical objects in just the same sense as a chemist might discover two new elements. What tempted Frege to think in this way? There is no simple answer to this question, but the conception of analysis suggested by his use of function-argument analysis undoubtedly played a major role.24 As we have seen, central to function-argument analysis is the idea of alternative analyses. This looks plausible in the case of propositions such as those considered in § 1 above, where alternative function-argument analyses are clearly possible, and one can also see here how Frege thought that concept-formation can be explained. But Frege extends the idea to the case of abstraction principles. Here, it would seem, we have object-formation and not just 23. See Frege’s letter to Russell of 28 July 1902: Frege (1980, p. 141). For further discussion of the differences between Frege’s and Russell’s understanding of abstraction principles, see Beaney forthcoming; Levine forthcoming. 24. A fuller explanation would have to make reference, among other things, to nineteenthcentury geometrical methodology, which allowed the ready construction of imaginary points and other such ‘extension elements’. For a recent account of the influence of this on Frege, see Wilson (2005).

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concept-formation; but it is not clear that this is compatible with Frege’s realism. Abstraction principles can yield new ways of talking, but can their use be regarded as a form of analysis that identifies new objects? Of course, we now know that Frege’s understanding of the abstraction principle that is Axiom V of his Grundgesetze led to inconsistency. And this might be taken to show that talk of ‘discovering’ abstract objects is misleading. In ‘Der Gedanke’, written towards the end of his life, Frege no longer talks of the True and the False as objects. His concern, as he puts it himself, is with the Bedeutung of the word ‘true’, and he starts by considering whether ‘true’ denotes a property (cf. 1918, pp. 59–61/1997, pp. 326–8). It would be tempting to suggest that this shows that he came to have doubts about his conception of truth-values as objects — just as he did about numbers being logical objects (cf. 1969, pp. 288–9/1997, p. 369). But in his ‘Notes for Ludwig Darmstaedter’, written shortly afterwards (in 1919), we still find reference to ‘the True’ and ‘the False’, and he continued to think of the numbers as objects rather than concepts (cf. 1969, pp. 276–7/1997, pp. 365–7). The possibility of using abstraction principles to ‘analyze away’ rather than ‘analyze out’ objects seems never to have occurred to him.25

BIBLIOGRAPHY Baker, Gordon, 2001. ‘Function’ in Frege’s Begriffsschrift: Dissolving the Problem. British Journal for the History of Philosophy 9, 525–44. — 2005. Logical operators in Begriffsschrift. In: Beaney and Reck (2005), Vol. II, 69–84. Baker, Gordon P. and Hacker, Peter M. S., 2003. Functions in Begriffsschrift. Synthese 135, 273–97. Beaney, Michael, 1996. Frege: Making Sense, London: Duckworth — 2005. Sinn, Bedeutung and the Paradox of Analysis. In: Beaney and Reck (2005), Vol. IV, 288–310. 25. A talk based on the first draft of this paper was given at the conference on ‘Frege’s Conception of Truth’ in Santa Maria, Brazil, in December 2005. I am grateful to the participants for discussion, and to Dirk Greimann for detailed written comments. I would also like to acknowledge the support of the British Academy in awarding me an Overseas Conference Grant, and the Alexander von Humboldt Foundation and the Institut für Philosophie of the University of Jena during my tenure of a Humboldt Research Fellowship at Jena when the final version of this paper was completed.

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— forthcoming a. The Analytic Turn in Early Twentieth-Century Philosophy. In: Beaney (forthcoming b). — forthcoming b, ed. The Analytic Turn: Analysis in Early Analytic Philosophy and Phenomenology. London: Routledge. Beaney, Michael and Reck, Erich H., eds., 2005. Gottlob Frege: Critical Assessments, 4 vols., London: Routledge. Dummett, Michael, 1981. The Interpretation of Frege’s Philosophy, London: Duckworth. Frege, Gottlob, 1879. Begriffsschrift, Halle: L. Nebert; Preface and most of Part I (§§ 1–12) tr. in: Frege (1997), 47–78. — 1884. Die Grundlagen der Arithmetik, Breslau: W. Koebner; selections tr. in: Frege (1997), 84–129. — 1891. Function and Concept. Jena: H. Pohle; tr. in: Frege (1997), 130–48. — 1892a. On Sinn and Bedeutung. Zeitschrift für Philosophie und philosophische Kritik 100, 25–50; tr. in: Frege (1997), 151–71. — 1892b. On Concept and Object. Vierteljahrsschrift für wissenschaftliche Philosophie 16, 192–205; tr. in: Frege (1997), 181–93. — 1893. Grundgesetze der Arithmetik. Jena: H. Pohle, Vol. I (Vol. II 1903); selections tr. in: Frege (1997), 194–223. — 1918. Der Gedanke. Beiträge zur Philosophie des deutschen Idealismus 1, 58–77; tr. as ‘Thought’ in: Frege (1997), 325–45. — 1969. Nachgelassene Schriften. Ed. H. Hermes, F. Kambartel & F. Kaulbach, Hamburg: Felix Meiner; tr. as: Frege (1979). — 1976. Wissenschaftlicher Briefwechsel. Ed. G. Gabriel, H. Hermes, F. Kambartel, C. Thiel and A. Veraart, Hamburg: Felix Meiner; abr. and tr. as: Frege (1980). — 1979. Posthumous Writings. Tr. of Frege (1969) by P. Long and R. White, Oxford: Blackwell. — 1980. Philosophical and Mathematical Correspondence. Tr. of Frege (1976), ed. B. McGuinness, tr. by H. Kaal, Oxford: Blackwell. — 1997. The Frege Reader. Ed. with an introd. by M. Beaney, Oxford: Blackwell. Levine, James, 2002. Analysis and Decomposition in Frege and Russell. Philosophical Quarterly 52, 195–216; repr. in: Beaney and Reck (2005), Vol. IV, 392–413. — forthcoming. Analysis and Abstraction Principles in Russell and Frege. In: Beaney (forthcoming b). Reck, Erich H. and Beaney, Michael, 2005. Introduction: Frege’s Philosophy of Mathematics. In: Beaney and Reck (2005), Vol. III, 1–12. Wilson, Mark, 2005. Ghost World: A context for Frege’s context principle. In: Beaney and Reck (2005), Vol. III, 157–75.

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Wright, Crispin, 1998. Why Frege does not deserve his grain of salt: A note on the paradox of ‘the concept horse’ and the ascription of Bedeutungen to predicates. In: J. Brandl and P. Sullivan, eds. Grazer Philosophische Studien 55, New Essays on the Philosophy of Michael Dummett. Vienna: Rodopi, 239–63; repr. in: Beaney and Reck (2005), Vol. IV, 177–96.

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Grazer Philosophische Studien 75 (2007), 125–148.

DID FREGE REALLY CONSIDER TRUTH AS AN OBJECT? Dirk GREIMANN Universidade Federal de Santa Maria Summary It is commonly assumed that the conception of truth defended by Frege in his mature period is characterized by the view that truth is not the property denoted by the predicate ‘is true’, but the object named by true sentences. In the present paper, I wish to make plausible an alternative interpretation according to which Frege’s conception is characterized by the view that truth is what is expressed in natural language by the “form of the assertoric sentence”. So construed, truth is neither an object (like the True) nor a property (like the Bedeutung of the predicate ‘is true’) but something of a very special kind that belongs to the same logical category as the logical relations (like subsumption). The main argument justifying this interpretation is that Frege’s explication of truth does not hold of the True, but only of truth, considered as what is expressed by the form of the assertoric sentence.

Introduction What is truth? To determine this, we must previously answer the more fundamental question: What is the logical form of sentences containing the word ‘true’? With regard to the latter question, the competing theories of truth fall into two main categories: “conservative” and “revisionary” theories. Examples of the first type are the traditional conceptions of truth as correspondence, coherence, usefulness, etc. They start from the assumption that the logical category of the word ‘true’ agrees with its grammatical category. Since ‘true’ behaves grammatically like an adjective, they consider ‘true’ to be a logical predicate, that is, an expression that is actually used to ascribe a property. This analysis leads to the view that truth is the property denoted by the truth-predicate. Revisionary theories, on the other hand, claim that ‘true’ is a pseudopredicate in the sense that sentences containing the word ‘true’ have a

deep-structure in which ‘true’ does not really function as a predicate. According to the disquotation theory, for instance, the sentence ‘‘Snow is white’ is true’ has the deep-structure ‘Snow is white’, because the function of the word ‘true’ is not to ascribe a property, but to neutralize the quotation marks. This analysis implies that the talk of truth as a property is based on a grammatical misunderstanding. In his mature period, Frege defended a conception of truth that agrees with the revisionary approach on the assumption that the grammar of natural language is misleading with regard to the nature of truth. He argued that the adjectival occurrence of ‘true’ in sentences like ‘The thought that sea-water is salty is true’ suggests that truth is a property of thoughts. Actually, however, the relation of the thought to the True does not correspond to “the relation between subject and predicate”, but to the “relation between the Sinn of a sign and its Bedeutung”. From this he inferred that truth is not a property. Frege’s positive account of the nature of truth seems to be characterized by the view that truth is an object. For this reason, his conception of truth is commonly seen as a strange “naming theory of truth” according to which truth is the object named by true sentences. The basis of this theory consists of the doctrine that, from a logical point of view, a sentence is a special proper name, namely, a proper name of a truth-value. In what follows, my aim is to show that on closer examination Frege did not defend the naming theory, but an “assertion theory of truth” according to which truth is what is expressed by the “form of the assertoric sentence”. Its core is the thesis that an assertoric sentence like ‘Sea-water is salty’ has the deep structure Salty(sea-water)’, where ‘ ’ is a truth-operator whose counterpart in ‘ natural language is what Frege calls “the form of the assertoric sentence”. So construed, truth is neither an object nor a property but something else that belongs to a third logical category to which also the logical relations such as the “falling” of an object under a concept (subsumption) and the “standing” of objects in a relation belong. I do not deny that Frege conceived of the truth-values as objects; my point rather is that this does not also hold for truth, considered as what is expressed by the form of the assertoric sentence. The paper is structured as follows. In section 1, I briefly describe Frege’s system of logical categories. The task of section 2 is to show that, given the special role that Frege ascribes to the concept of truth in judgment and assertion, truth can neither be an object nor a concept. In section 3, it is argued that the naming theory of truth cannot be attributed to Frege.

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Finally, in section 4, the alternative interpretation is developed according to which Frege adopted the assertion theory of truth. 1. Frege’s logical categories According to the neo-Kantian logic defended by Frege’s contemporaries, to judge is to unite ideas. The unity of a judgment’s content is produced by the psychological act of uniting ideas, which is called “synthesis” by Kant. This approach is rejected by Frege on the ground that the truth of a judgeable content is independent of the act of judging.1 If the unity of the judgeable content was constituted by the act of judgment, he argues, then the existence and hence the truth of this content would depend on this act. In many cases, however, the content of a judgment was true already before the act of judgment took place. Thus, the judgeable content that snow is white was true already before it was grasped for the first time by a human being. In Frege’s own view, to judge is, not to unite ideas, but to acknowledge the truth of something whose unity and existence is independent of the act of judging, namely, a thought. He assumes that the unity of a thought is not constituted by the act of synthesis, but by a mechanism he calls “saturation”. This metaphor encapsulates the idea that the predicative component part of a judgment — the concept — unites itself with the non-predicative component part — the object — to form a judgeable content, without there being any psychological act constituting the unity. The logical category of a component part depends on the role it plays in judgment.2 Two components belong to the same category if they can be substituted one for another without dissolving the unity of the judgment, i.e., the connectedness of its parts. In the judgment that snow is white, for instance, the concept white can be substituted by other concepts, but not also by other objects, because an object combined with another object does not give a judgment, just as a proper name combined with another proper name does not give a sentence. The basic logical categories of traditional logic are subject and predicate. In Frege’s logic, these categories are substituted by the more general and flexible categories of function and object. A function is any entity that is 1. Cf. Frege (1990, p. 370f.) or Frege (1997, p. 354f.). 2. Frege himself does not use the term “logical category”.

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unsaturated and an object any entity that is saturated. A first-level function is a function taking objects as arguments, and a second-level function a function taking first-order functions as arguments. The values of functions are always saturated entities, that is, objects. Concepts and relations are considered as a species of functions. According to Frege’s first logical system, which is presented in Begriffsschrift (1879), concepts and relations are functions whose value is always a judgeable content, and, according to his second system, which is to be found in Grundgesetze (1893), they are functions whose value is always a truth-value. Intuitively, the role played by a concept in judgments may either be a predicative or a non-predicative one. Thus, in the judgment ‘Snow is white’, the concept white plays a predicative role, while in the judgment ‘The concept white is a color concept’, it obviously does not play such a role, because it is not predicated of anything. Contrary to this, Frege maintains that concepts are predicative by their very nature. This “essential predicativity” of concepts is explained by him in the posthumous writing “Logik in der Mathematik” from 1914 as follows:3 It is of the essence of the concept to be predicative. … So the sentence ‘x is a prime number’, does indeed contain the possibility of a statement [eine mögliche Aussage], but so long as no meaning [Bedeutung] is given to the letter ‘x’, we do not have an object about which anything is being said. Another way of putting this would be to say: we have a concept but we have no object subsumed under it. (Frege 1979, p. 214)4

Frege holds that the predicative role that is usually played by concepts in judgments is an essential feature of them in the sense that a concept can never be used to play the role of an element that is not used to predicate something of something. Since an object cannot be predicated of another object, it goes without saying that an object cannot play the role that concepts normally play. According to the thesis of the essential predicativity of concepts, the converse also holds: a concept can never play the role (or “occupy the position”) that is played by an object.5 When, for instance, we abstract from the objects 2 and 3 in the judgment that 2  3 by forming corresponding empty places in it, we gain the relation 3. This view is to be found also in Frege’s early writings as, for instance, in “Booles rechnende Logik und die Begriffsschrift” from (1880/81). Cf. Frege (1983, p. 19) or the translation in Frege (1979, pp. 17–18). 4. The German version of this citation is to be found in Frege (1983, p. 231). 5. Cf. Frege (1983, pp. 130, 167).

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of identity. Prima facie, it would be possible to put two concepts, say, the concepts white and green, into the empty places of this relation in order to form the judgment that the concept white is identical to the concept green. In this case, however, the concepts do not occur predicatively in the judgment because they are not predicated of any objects. According to the thesis of the essential predicativity of concepts, such an occurrence of a concept in a judgment is impossible because the concept plays by its very nature always a predicative role in judgment: it can even be defined, as Frege actually does, as “the predicate of a judgment”.6 This conception does not imply that it is in general impossible to predicate something of a concept. It implies only that when we want to predicate something of a concept, we must use this concept at the same time to predicate something of objects.7 Consider, for instance, the judgment expressed by ‘Snow is white’. We may consider this judgment as a judgment in which the second-order concept is a concept that applies to snow is predicated of the first-order concept white. In this case, the judgment predicates something of the concept white, but, at the same time, this concept is predicated of snow. An important consequence of the essential predicativity of concepts is that concepts cannot be designated by proper names. Since, for instance, in the sentence ‘The concept horse is a concept’ the singular term ‘the concept horse’ is not used predicatively, it cannot refer to a concept, because otherwise this concept would not have a predicative function in the judgment expressed by the sentence. Because of this symmetry it is possible, in Frege’s system, to characterize the logical categories of object and concept syntactically, as follows: an object is any entity that can be referred to by a proper name and a concept any entity that can be referred to by a predicate.8 Frege is well aware of the paradoxical consequence this conception has. They are illustrated by his “paradox of the concept horse”, as it is commonly called.9 The paradox is that a seemingly false sentence like ‘The concept horse is not a concept’, which has the form ‘The F is not an F’, must be true because the referent of the proper name ‘the concept horse’ must be an object, not a concept.10 6. See, for instance, the second footnote to “Booles rechnende Logik und die Begriffsschrift” in Frege (1983, p. 18). 7. Cf. Frege (1990, p. 174) or the English translation in Frege (1997, p. 189). 8. Cf. Frege (1990, p. 134) or the English translation in Frege (1997, p. 140). 9. Cf. Frege (1990, p. 170) or the English translation in Frege (1997, p. 185). 10. In his middle period, Frege assumed that definite descriptions like ‘the concept horse’

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The difference between a concept and a (dyadic) relation is that the latter has two empty places that must be filled in order to get saturated. However, according to Frege, there is a very special relation that does not have these properties, namely, the relation of saturation itself. Natural language suggests that saturation is the relation denoted by the copula ‘is’ in sentences like ‘Snow is white’ and ‘2 is prime’. Here, ‘is’ seems to denote a relation with two empty places: one that must be filled with an object and another that must be filled with a concept. In Frege’s view, on the other hand, saturation is not a relation in the ordinary sense nor an object but the mechanism by means of which objects and such relations are glued together in judgment. In contrast to objects and ordinary relations, this mechanism is not an entity at all. In the posthumous writing “Über Schoenflies: Die logischen Paradoxien der Mengenlehre” from 1906, Frege explains his view as follows: In the sentence ‘Two is a prime’ we find a certain relation: that of subsumption. We may also say the object falls under the concept prime, but if we do so, we must forget the imprecision of linguistic expression we have just mentioned [the imprecision that the proper name ‘the concept prime’ hides the predicative nature of the concept prime, D.G.]. This also creates the impression that the relation of subsumption is a third element supervenient upon the object and the concept. This isn’t the case: the unsaturatedness of the concept brings it about that the object, in effecting the saturation, engages immediately with the concept, without need of any special cement. Object and concept are fundamentally made for each other, and in subsumption we have their fundamental union. (Frege 1979, p. 178)11

The relation that Frege calls “subsumption” is a special case of saturation. According to the system in Begriffsschrift, this relation holds between an object and a concept if the result of the concept’s saturation by the object is a fact, and, according to the system in Grundgesetze, if the result is the True. In both systems, the relation of saturation is expressed, not by a corresponding predicate, but by the parentheses together with its empty places.12 When, for instance, we fill the empty place of the predicate ‘F( )’ with the name ‘A’, then the resulting expression ‘F(A)’ expresses the result denote “a special kind of object”, whereas, in his last period, he became convinced that these terms do not have any denotation at all. Cf. Frege (1983, p. 289) or the English translation Frege (1979, p. 269). 11. The German version of this citation is in Frege (1983, p. 193). 12. Cf. Frege (1983, pp. 131–132).

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of the saturation of the concept F by the object A.13 Note that because of the very special role that this relation plays in judgment it cannot be substituted by ordinary relations. While the role of ordinary relations is to form the predicative part of judgments, the role of saturation is to connect this part with the non-predicative parts. Therefore, saturation, subsumption and kindred relations like the standing of two objects in a relation and the falling of a first-level concept under a secondlevel concept form a logical category sui generis: they are, anachronistically speaking, not genuine relations, but “non-relational ties” whose sole task is to tie the predicative and the non-predicative parts of judgments together. Let us call these very specials relations “logical relations”. They form the subject of the logic of “primary propositions”, that is, the subject of predicate logic.14 In contrast to genuine relations, logical relations are not entities at all. For this reason, the relation of subsumption must be distinguished from the (genuine) relation denoted by the predicate ‘x falls under y’. Whereas the latter is an unsaturated entity, the former is neither an unsaturated nor a saturated entity, but “no-thing”: there is no ontological category to which this very special relation belongs. Unfortunately, Frege does not give a positive account of what the logical relations are. In particular, he does not describe in more detail how the mechanism of saturation works. The reason might be that the notion of a non-relational tie cannot really be made intelligible, for principled reasons. This is, at least, what the current discussion about this notion suggests.15 2. The special role of truth in judgment and assertion With regard to the role of the concept of truth in judgment and assertion, the most striking feature of Frege’s conception in his first period is that every sentence of his formal language contains the predicate ‘is a fact’. In § 3 of Begriffsschrift (1879), he describes this language as follows:

13. In natural language, saturation is expressed by the syntactic operation of concatenation. Thus, the sentence ‘Snow is white’ expresses in virtue of the concatenation of the proper name ‘snow’ with the predicate ‘is white’ in this order the result of the saturation of the concept white by snow. 14. Cf. Frege (1983, p. 53) or the English translation in Frege (1979, p. 47). 15. For an overview over the problems this notion is facing, see Campbell (1990, p. 14–15).

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Imagine a language in which the proposition ‘Archimedes was killed at the capture of Syracuse’ is expressed in the following way: ‘The violent death of Archimedes at the capture of Syracuse is a fact’. Even here, if one wants, subject and predicate can be distinguished, but the subject contains the whole content, and the predicate serves only to present it as a judgment. Such a language would have only one predicate for all judgments, namely, ‘is a fact’. […] Our Begriffsschrift is its common predicate for all judgments. is such a language and the symbol (Frege 1997, p. 54).

It is widely assumed that Frege’s motive for paraphrasing normal sentences in this way derives from the erroneous conviction, later corrected by him, that in order to assert something we need a truth-predicate. It is, however, possible to interpret Frege in a much more charitable way according to which his motive for paraphrasing sentences in the way indicated above derives from his goal to construct a logically transparent language whose sentences have a syntactic structure that reveals the logical structure of the judgments they express.16 The basic structure of a judgment, according to Frege, is the structure R(G), where G is the act of the mere grasping of a thought and R the act of recognizing its truth. When, for instance, a scientist asks himself whether neutrinos lack mass, he already grasps the thought that neutrinos lack mass, but, in order to judge that neutrinos lack mass, he must also recognize the truth of this thought. In Frege’s view, the structure of judgments is hidden by the surface grammar of natural language because there is no word or group of words that corresponds to the act of recognizing a thought as true. In particular, the surface grammar of natural language hides the fact that it is possible to grasp a thought without judging whether it is true or false. For, by uttering an assertoric sentence like ‘Snow is white’, we do not only express the thought that snow is white, but at the same time assert this thought as true. Here, the linguistic act of the mere expression of a thought corresponds to the cognitive act of the mere grasping of a thought and the act of asserting a thought as true corresponds to the act of judging a thought to be true.17 16. This interpretation is elaborated in Greimann (2000, pp. 215–228) and in Greimann (2003, Chap. 3). 17. In “Logik”, Frege writes: “In an assertoric sentence two different kinds of thing are usually intimately bound up with one another: the thought expressed and the assertion of its truth. And this is why these are often not clearly distinguished. However, one can express a thought without at the same time putting it forward as true.” (Frege 1997, p. 239). The German version of this citation is to be found in Frege (1980, p. 150).

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In a logically transparent language, these linguistic acts must be separated in order to make transparent that R(G) is the basic structure of judgments. To achieve this, it is necessary to construe a sort of expressions that express a thought without asserting it as true. Given the semantics of Begriffsschrift, the simplest way to achieve this is to nominalize sentences in the way indicated by Frege in the passage cited above. Thus, the nominal phrase ‘The violent death of Archimedes at the capture of Syracuse’ expresses the same judgeable content as the sentence ‘Archimedes was killed at the capture of Syracuse’, but, in contrast to this sentence, it does not put this content forward as true. Consequently, the sentence ‘The violent death of Archimedes at the capture of Syracuse is a fact’ is logically transparent, because its syntactic structure makes explicit that the judgment it expresses consists of two different acts: the mere grasping (expressing) of the thought that Archimedes was killed at the capture of Syracuse and the recognition (assertion) of its truth. Syntactically, the speech act of assertion is associated with what Frege calls “the form of the assertoric sentence”, which is the bundle of syntactic properties in which assertoric sentences differ from interrogative and imperative ones. The primary expressive function of the assertoric form is to indicate the type of the speech act that is (normally) performed by uttering sentences of this form. Thus, the assertoric form of the sentence ‘Archimedes was killed at the capture of Syracuse’ indicates that the utterance of this sentence is meant as an assertion, and the interrogative form of the sentence ‘Was Archimedes killed at the capture of Syracuse?’ that its utterance is meant as a question. Since, on Frege’s analysis, to assert that p involves the act of putting the thought that p forward as true, the assertoric form is regarded by him also as a truth-operator of natural language, that is, a device of expressing that something is true: “… it is really by using the form of the assertoric sentence that we state truth [womit wir die Wahrheit aussagen]”, he writes in the fragment on “Logik” from 1879, “and to do this we do not need the word ‘true’” (Frege 1983, p. 140).18 In the system of Grundgesetze, Frege represents the assertoric form by a special sign, the judgment-stroke, which is an illocutionary act indicating device by means of which thoughts are asserted as true. It seems to be probable that this conception is anticipated by the Begriffsschrift. For, ac’ serves only to “present something as a cording to § 3, the predicate ‘ ’ serves as an illocutionary act indicating judgment”. This means that ‘ 18. The German original is in Frege (1997, pp. 228–229).

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device whose counterpart in natural language is the form of the assertoric sentence. Moreover, in the posthumous writing “Boole’s logische Formelsprache und meine Begriffsschrift”, written in 1882, Frege explicitly says that he makes “a distinction between judgment and possible content of judgment, reserving the first word for cases where such contents are put forward as true [als wahr hingestellt wird]” (Frege 1979, p. 47).19 The judgment-stroke is accordingly introduced by him as a truth-operator by means of which a possible content of a judgment is asserted as true: Now in order to put a content forward as true, I make use of a small vertical stroke, the judgment-stroke, as in 3²  9 whereby the truth of the equation is asserted, whereas in 3²  9 no judgment has been made. Hence since the judgment-stroke is lacking, we can 3²  4 without saying anything untrue. (Frege 1979, even write down p. 51)20

It would be natural to think that to judge or assert something as true is a special case of the ascription of a property to an object. On this assumption, to judge a given content as true is to subsume it under the concept of truth. However, according to Frege’s analysis, the use of the concept of truth in judgment and assertion is a constituent of the act of subsumption, not a special case of it. In order to subsume the number 2 under the concept prime, it does not suffice merely to express that 2 is prime; it is necessary, in addition, to judge this as true. This analysis is suggested in by § 10 of Begriffsschrift, where Frege introduces the logical relations of the falling of an object under a concept (subsumption) and the standing of two objects in a relation into his system. The corresponding passage reads: )(A) can be read: ‘A has the property )’. <(A,B) may be translated as ‘B stands in the <-relation to A’ or as ‘B is a result of an application of the procedure < to the object A’. (Frege 1997, p. 69). 19. The German version of this passage is to be found in Frege (1983, p. 54). 20. For the German version of this citation, see Frege (1983, p. 54).

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According to this analysis, the ascription of a property F to an object A has the structure R(F(A)), where R is the act of recognizing (or putting forward) the predication F(A) as true. Prima facie, this analysis appears to be wrong because sentence pairs like (1a) 2 has the property of being prime. (1b) The primality of 2 is a fact. express different thoughts. Thus, (1b) implies the existence of the judgeable content that 2 is prime, while (1a) does not have this consequence. From the viewpoint of Begriffsschrift, however, this objection must be repudiated because concepts are construed as functions whose values are judgeable contents. The truth of (1a) implies that the function x has the property of being prime has a value for the number 2 as argument, and this value is the judgeable content that 2 is prime. To say of the number 2 that it is prime is thus the same as to say of the thought that 2 is prime that it is true. Hence, (1a) and (1b) may indeed be regarded as paraphrases. From the point of view of the system in Grundgesetze, the semantics of Begriffsschrift involves a confusion of Sinn and Bedeutung, because the nominalization ‘the primality of 2’ designates the Sinn of the sentence ‘2 has the property of being prime’, not its Bedeutung. Since the nominalization and the sentence differ with regard to their Bedeutung, they cannot express the same Sinn. As a consequence of this, a nominalization like ‘the primality of 2’ cannot be used to separate the mere expressing of a thought from the assertion of its truth. This separation is accordingly achieved in the system of Grundgesetze in a different way. According to the description in Funktion und Begriff (1891) and the first paragraphs of Grundgesetze, 2 3  5’ as ‘The truth-value of the identity of 2 3 we must read ‘ and 5 is the True’, where the judgment-stroke indicates that the truthvalue designated by ‘2 3  5’ is the True. As Frege explicitly notes, the expression ‘2 3  5’, as it is used in his formal language, “simply designates a truth-value without saying which of the two it is” (§ 5), whereas 2 3  5’ asserts that this value is the True. This becomes clear also ‘ from a passage from Funktion und Begriff in which Frege explains the use of expressions like ‘5  4’ in his formal language as follows: If we write down an equation or inequality, e.g. 5 
4, we ordinarily wish at the same time to express a judgment; in our example, we want to assert that 5 is greater than 4. According to the view I am here presenting, ‘5 4’

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and ‘1
3  5’ just give us expressions for truth-values, without making any assertion. […] We need thus a special sign in order to be able to assert something. To this end I make use of a vertical stroke at the left end of the horizontal, so that, e.g., by writing 2
3  5 we assert that 2
3  5. Thus we are not just writing down a truth-value, as in 2
3  5, but also at the same time saying that it is the True. (Frege 1997, p. 142)21

Since the English expression ‘the truth-value of 2 3’s being identical to 5’ has exactly the semantic and pragmatic properties that Frege ascribes to ‘2 3  5’, considered as an expression of the formal language of Grundgesetze, the former can be regarded as an English translation of the latter. It might be objected that pairs of expressions like (2a) the truth-value of 2 3’s being identical to 5. (2b) The sum of 2 and 3 is identical to 5. cannot express the same sense, because (2b) is a sentence expressing a thought while (2a) is a singular term whose sense is a “mode of presentation” of a truth-value. This objection, however, is unjustified because according to the truth-conditional semantics sketched in § 32 of Grundgesetze, pairs of expressions like (2a) and (2b) do express the same sense. This is a consequence of the identification of thoughts with truth-conditions: since the conditions under which the Bedeutung of (2a) is the True are identical with the conditions under which the Bedeutung of (2b) is the True, (2a) and (2b) do express the same sense. Hence, the singular term (2a) can actually be used to express the thought that the sum of 2 and 3 is identical to 5 without asserting it as true. This is, in nuce, Frege’s solution of the problem of separating predication and assertion in Grundgesetze.22 By “assertoric force” Frege obviously means the truth-claim that is characteristic for assertions. Whereas the interrogative sentence ‘Is the sum of 2 and 3 identical to 5?’ leaves open whether the thought expressed is true or false, the assertoric sentence ‘The sum of 2 and 3 is identical to 5’ 21. The German version of this citation is reprinted in Frege (1990, pp. 136–137). 22. For more details, see Greimann (2000).

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expresses that this thought is true. To be sure, there are some passages in Frege’s writings according to which to assert is to express the speaker’s inner recognition of something as true. In “Der Gedanke” (1918), for instance, he characterizes “assertion” as the “manifestation” of the act of judgment.23 This characterization suggests that, by asserting ‘Snow is white’, the speaker communicates to the hearer that he holds this predication to be true, and not that it is true. However, from a letter to Jourdain it becomes clear that Frege does not construe assertion in this psychological way, as the mere communication of a belief. He answers the question posed by Jourdain ) as psychological” as follows: “When whether he “regards assertion ( I assert something as true, I do not want to speak of myself, of an occurrence in my soul”.24 This means, when I assert that p, my claim is not that I am holding the thought that p as true, but that this thought is true. It can, moreover, be shown that Frege is committed to rejecting the psychological conception of assertion because it is incompatible with his criticism of psychologism. Suppose that, by asserting that snow is white, the speaker merely communicates his belief that snow is white, without asserting this as true. In this case, the assertoric force of his utterance does not consist in the claim that the thought expressed is true, but in the claim that the speaker believes this thought to be true. As a consequence, another speaker could assert that snow is not white without contradicting the first speaker. For, the claim made by the second speaker is only that he does not believe something that the first speaker believes. The psychologistic conception of assertion hence implies that a scientific debate is impossible, because it rules out that one speaker denies what another asserts. This implication, however, is one of the main targets in Frege’s criticism of psychologism.25 We saw that Frege construes in his first period the logical relations such as the falling of an object under a concept as special cases of truth. In his mature period, Frege sticks to this approach. In § 4 of Grundgesetze, he explains the falling of an object under a concept and the standing of two objects in a relation in terms of truth, as follows: We say that the object * stands in the relation <([,]) to the object ' if <(*,') is the True; just as we say that the object ' falls under the concept )([) if )(') is the True. (Frege 1997, pp. 214–215) 23. Cf. Frege (1990, p. 346) or the English translation in Frege (1997, p. 329). 24. Cf. Frege (1976, pp. 126–7), my translation. 25. See, for instance, the preface of Grundgesetze.

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Note that Frege’s reduction of subsumption to truth is the reversal of Tarski’s reduction of truth to subsumption (or satisfaction). From the Tarskian point of view, the truth of a proposition is a special case of the falling of objects under concepts or, more precisely, of the satisfaction of propositional functions by sequences of objects. Contrary to this, Frege explains subsumption in terms of truth, not vice versa: to say of an object ' that it falls under the propositional function )([) is to say that the truth-value of )(') is the True. The subsumption of objects under concepts and relations is accordingly expressed in Frege’s logical systems in terms of truth, by means of ‘|’. Thus, according to § 4 and § 5 of Grundgesetze, )(')’, we are saying that the truth-value designated by by writing ‘ )(')’ is the True and hence that ' falls under )([). ‘ According to this approach, the role of the concept of truth in judgment and assertion is a very peculiar one: unlike ordinary properties and relations, truth does not constitute a predicative part of a judgeable content; rather, its function is to determine the logical relation in which the predicative and the non-predicative parts of a judgeable content are supposed to stand. Thus, in the simplest case of an elementary judgment of the form )(')’, the function of the ascription of truth, as it is represented ‘ by the judgment-stroke, is to determine the logical relation in which the object ' and the concept )([) are supposed to stand: whether or not ' falls under )([). This analysis has a venerable tradition. It goes back to Kant’s doctrine in the Kritik der reinen Vernunft that truth is a “modality” of judgment whose use has a “very peculiar function”, because it “does not make a contribution to the judgment’s content”, but determines “the value of the copula” (Kant 1781, B 100–101). It follows from this that truth is indefinable in the sense that it cannot be reduced to ordinary properties. The reason is that no such property can play the role that is played by truth in judgment, just as no ordinary relation can play the role of saturation. In the posthumous writing “Logik” from 1897, Frege presents the following argument for the indefinability of truth: Now it would be futile to employ a definition in order to make it clearer what is to be understood by ‘true’. If, for example, we wished to say ‘an idea is true if it agrees with reality’ nothing would have been achieved, since in order to apply this definition we should have to decide whether some idea or other did agree with reality, in other words: whether it is true that the idea agrees with reality. Thus we should have to presuppose the very thing that is being defined. The same would hold of any definition of the form ‘A is true

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if and only if it has such-and-such properties or stands in such-and-such a relation to such-and-such a thing’. In each case in hand it would always come back to the question whether it is true that A has such-and-such properties, or stands in such-and-such relation to such-and-such a thing. Truth is obviously something so primitive and simple that it is not possible to reduce it to anything still simpler. (Frege 1997, p. 228)26

Given the present interpretation of Frege’s conception of truth, this argument must be reconstructed as follows. In order to define truth, we have to give an explanation of the form ‘A is true if and only if A falls under the concept C or stands in the relation R to such-and-such things’. Since circular definitions are illegitimate, the explanation must not presuppose that it is already known what truth is. Obviously, an explanation of the form in question presupposes that it is already known what it means that A falls under C or that A stands in the relation R to such-and-such things. Since, however, these logical relations are special cases of truth, the explanation given is circular: it presupposes a prior grasp of what is to be explained. This argument implies that truth is something that cannot be properly called a property. If the logical relations are special cases of truth, then truth must belong to the same logical category as these relations. In particular, truth cannot be identified with the property denoted by ‘is true’, just as the relation of subsumption cannot be identified with the relation denoted by ‘x falls under y’. 3. The naming theory of truth The secondary literature generally proceeds from the assumption that Frege determines in his mature phase the logical category of truth on the basis of a syntactical analysis of sentences according to which sentences form a species of singular terms. On this assumption, the logical category of truth is determined by Frege in his mature period on the basis of the following argument: 27

26. I have slightly modified this translation by inserting one missing sentence. The German text is to be found in Frege (1983, p. 140). 27. The locus classicus of this interpretation is Burge (1986).

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1. Sentences are a species of proper names. 2. A true sentence is a proper name of truth and a false sentence a proper name of falsehood. 3. The bearer of a proper name is always an object, never a property. 4. Hence, truth and falsehood are objects, not properties. Prima facie, this interpretation of Frege’s conception of truth can be confirmed by means of the following citation from “Einleitung in die Logik”: The True and the False are to be regarded as objects, for both the sentence and its sense, the thought, are complete in character, not unsaturated. (…) If we say ‘the thought is true’, we seem to be ascribing truth to the thought as a property. (…) But here we are misled by language. We don’t have the relation of an object to a property, but that of the sense of a sign to its meaning [Bedeutung]. (Frege 1979, p. 194) 28

There can, of course, be no doubt that Frege construes the truth-values as objects. But this does not imply that he considers also truth and falsity as objects. On the contrary, there is some evidence that Frege wants to distinguish between truth and the True. In “Der Gedanke”, he states the following paradox about the cognitive function of the concept of truth. On the one hand, this concept seems to play an essential role in acquiring knowledge because the transition from the mere hypothetical assumption that p to the acknowledgement of its truth is a crucial step in acquiring the knowledge that p. But, on the other hand, this concept seems to be completely redundant because the sense of the word ‘true’ does not make any essential contribution to the senses of the sentences in which it occurs: the sentence ‘It is true that snow is white’ says nothing more than the simpler sentence ‘Snow is white’. From this he concludes that “the Bedeutung of the word ‘true’ seems to be altogether sui generis” and that it might be that we are dealing here “with something which cannot be properly called a property in the ordinary sense” (Frege 1997, p. 328). Then, however, he does not go on to introduce the True, but to explain that truth is expressed, in natural language, not by the truthpredicate, but by the form of the assertoric sentence.29 This suggests that Frege wishes to identify truth with what is expressed by the assertoric form. 28. The German version of this citation is in Frege (1983, p. 211). 29. The truth-values do not occur in “Der Gedanke” at all.

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In any case, it can be shown that Frege is committed to distinguish truth from the True because the True does not fulfill his explication of truth. This explication is almost exclusively negative in its character: it tells us what truth is not (“correspondence with facts”, “being acknowledged as true”), but it does not tell us what truth positively is. The main component of his explication accordingly is the thesis of the indefinability of truth. The main positive component is the elucidation that “stating truth is always included in stating anything whatever”. Both components do not apply to truth, considered as the True, but only to truth, considered as what is expressed by the assertoric form: [i] The True is treated in Frege’s system, not as a primitive object, but as a defined one. Since the German expression ‘das Wahre’ (‘the True’) means ‘that what is true’, it would be natural to define the True as the class of all true truth-bearers. Actually, however, Frege identifies the True with the x. extension of the function x’ is a functional sign that yields a complex name The Horizontal ‘ of the True when it is applied to a name of the True and a complex name of the False when it is applied to a name whose denotation is not the True. 2 3  5’ denotes the True, because ‘3 2  5’ denotes the Thus, ‘ 3 2’ denotes the False, because 3 2 is not the True. True, and ‘ The English counterpart of the Horizontal is the functional expression ‘the truth-value of the identity of x with the True’.30 If, as in Frege’s system, sentences are treated as proper names, the Horizontal may be regarded as a truth-predicate because it satisfies the following variant of Tarski’s truth-scheme: p is the True iff p. In § 10 of Grundgesetze, Frege reduces the function of identity, by means of the following definition:

x to the relation

x iff x  (x  x). Since ‘x  x’ is always the True, ‘x  (x  x)’ is the True if and only if x is 30. In Greimann (2003a) the role of the Horizontal in the system of Grundgesetze is described in detail.

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the True. This definition is equivalent to the following definition of the truth-operator ‘it is true that p’ in terms of ‘p iff q’: It is true that p iff (p iff (p iff p)). ‘p iff (p iff p)’ is true if and only if ‘p’ and ‘(p iff p)’ have the same truthvalue. Since ‘(p iff p)’ is always true, ‘p iff (p iff p)’ is true if and only ‘p’ is true. Hence, ‘It is true that p’ and ‘p iff (p iff p)’ have the same truthconditions. In order to overcome the referential indeterminacy of value-course terms in his system, Frege identifies in § 10 of Grundgesetze the True with the value-course of the function — x. This identification, however, clearly shows that the True is definable in Frege’s system, namely as follows: The True  (

).

Since the function x coincides with the concept of being identical to the True, this definition amounts to the identification of the True with the extension of the concept of being identical to the True.31 [ii] In order to explain the intended meaning of the primitive terms of his system, Frege uses the method of “elucidation”. In contrast to a definition, an elucidation does not aim at a reduction of the concept to be explained, but only at a characterization of this concept in order to distinguish it from other concepts with which it can easily be confused. In “Logik”, directly after having reached the conclusion that truth is indefinable, Frege gives the following elucidation of his notion of truth: Consequently we have no alternative but to bring out the peculiarity of our predicate by comparing it with others. What, in the first place, distinguishes it from all other predicates is that stating it is always included in stating anything whatever [dass es immer mit ausgesagt wird, wenn irgend etwas ausgesagt wird]. If I assert that the sum of 2 and 3 is 5, then I thereby assert that it is true that 2 and 3 make 5. […] Therefore it is really by using the form of the assertoric sentence that we state truth [womit wir die Wahrheit aussagen], and to do this we do not need the word ‘true’. Indeed, we can say that even where we use the 31. Since the True is the only object falling under this concept, this identification implies that the True is identical with its own unit class.

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locution ‘it is true that …’ the essential thing is really the form of the assertoric sentence.32

It is clear that the True, being an object, cannot be stated (ausgesagt) at all. Hence, this elucidation of truth does not apply to the True, either. If, on the other, truth is considered as what is expressed by the assertoric form, it immediately follows that “stating truth is always included in stating anything whatever”. For, according to this approach, the statement about an object A that it has the property F has the structure R(F(A)), where R is the act of putting the predication F(A) forward as true.33 4. The assertion theory of truth The “assertion theory of truth”, as I would like to call it, is the doctrine that truth is what is expressed in natural language by the form of the assertoric sentence.34 Regarded from the point of view of this approach, Frege’s claim that truth is not a property appears in an entirely new light: its point is not that truth is an object, but that the ascription of truth contained in the acts of judging or asserting something as true must not be regarded as the ascription of a property. In the remainder of this paper, this is explained in more detail. If we consider the form of the assertoric sentence as the primary truth operator of natural language, we must distinguish between two fundamentally different uses of the concept of truth: a predicative one, which consists in the predication of the word ‘true’, and an assertoric (or “modal”) one that consists in the use of the form of the assertoric sentence. In ‘Is it true that snow is white?’, for instance, we have a predicative use, and in ‘Snow is white’, an assertoric one. The predicative use consists in uniting the sense of the truth-predicate with the sense of the sentences to which it is applied. Since pairs of sentences like ‘Is it true that snow is white?’ 32. For the German version of this citation, see Frege (1983, p. 140). The translation is partly taken from Frege (1997, pp. 228–229) and partly my own. 33. One might think that Frege derives this peculiarity of truth from his thesis that the sense of ‘true’ is redundant. If sentence pairs of the form ‘p’ and ‘It is true that p’ always express the same thought, then the sense of ‘true’ is contained in every thought. This interpretation, however, does not fit in with the thesis that truth is expressed by the form of the assertoric sentence. This form does not express any sense, but provides predications with assertoric force, that is, the truth-claim that is characteristic for assertoric utterances. 34. In Greimann (2000) and Greimann (2005) this theory is described in more detail.

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and ‘Is snow white?’ express the same sense, this use is redundant, that is, it can be eliminated without any significant loss of expressive power. The function of the assertoric use is to determine the relation of the thought expressed to reality. Thus, by asserting that snow is white, we are not merely expressing the thought that snow is white, but at the same time are putting it forward as true. This means, in Frege’s terminology, that the assertoric use determines the relation in which the thought is supposed to stand to the realm of Bedeutung. Since, in natural language, the assertoric use of the concept of truth does not correspond to any sign or phrase, it provokes a confusion of the assertoric use with the predicative one. This seems to be Frege’s point in the following passage from the posthumous writing “Meine grundlegenden logischen Einsichten” from 1915: … the word ‘true’ seems to make the impossible possible: it allows what corresponds to the assertoric force to assume the form of a contribution to the thought. … ‘true’ only makes an abortive attempt to indicate the essence of logic, since what logic really is concerned with is not contained in the word ‘true’ at all but in the assertoric force with which a sentence is uttered. (Frege 1997, p. 323) 35

If we take the word ‘true’ to be the primary truth-operator of natural language, that is, the effective means of presenting a thought as true, then we are forced to assume that the truth-claim made by asserting the thought that p as true is constituted by the application of the sense of ‘true’ to the thought that p. It would, indeed, be natural to reduce the assertoric use of the concept of truth to the predicative one as follows: (RED) To assert the thought that p as true is to (merely) express the thought that it is true that p. To judge the thought that p as true is to (merely) grasp the thought that it is true that p. This reduction, however, is clearly faulty. For, it is possible to merely express (or grasp) the thought that it is true that p without determining the relation between the thought that p and reality. An example of this is the question ‘Is it true that snow is white?’, by means of which we express the thought that it is true that snow is white without determining its relation to reality. Consequently, (RED) leads into either a vicious circle or an infinite regress. 35. The German version of this citation is in Frege (1983, p. 272).

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This argument is to be found in the following passage from “Über Sinn und Bedeutung”:36 One might be tempted to regard the relation of the thought to the True not as that of sense to Bedeutung, but rather as that of subject to predicate. One can, indeed, say: ‘The thought that 5 is a prime number is true’. But closer examination shows that nothing more has been said than in the simple sentence ‘5 is a prime number’. The truth claim arises in each case from the form of the assertoric sentence, and when the latter lacks its usual force, e.g., in the mouth of an actor upon the stage, even the sentence ‘The thought that 5 is a prime number is true’ contains only a thought, and indeed the same thought as the simple ‘5 is a prime number’. It follows that the relation of the thought to the True may not be compared with that of subject to predicate. Subject and predicate (understood in the logical sense) are just elements of thought; they stand on the same level for knowledge. By combining subject and predicate, one reaches only a thought, never passes from a sense to its Bedeutung, never from a thought to its truth-value. One moves at the same level but never advances from one level to the next. (Frege 1997, p. 158)37

By the “relation of the thought to the True” Frege means the relation into which the thought and reality are brought by means of the acts of judgment and assertion, i.e., the assertoric use of the concept of truth. Since, in natural language, there is no word or group of words representing the assertoric use of truth, but only the truth-predicate, the grammar of this language suggests that we determine the “relation of the thought to the True” by predicating truth of the thought. However, by means of this predication we do not determine the thought’s relation to reality, but merely combine the sense of the predicate ‘is true’ with the sense of the sentence ‘Snow is white’. This act amounts to nothing more than the mere expressing of the thought that it is true that snow I white. But this maneuver is completely redundant, for two reasons. First, by combining the sense of ‘Snow is white’ with the sense of ‘true’, we do not add anything essential to the former sense, because the senses expressed by sentence pairs like ‘Snow is white’ and ‘The thought that snow is white is true’ are identical. Second, by combining the sense of ‘Snow is white’ with the sense of ‘true’, we do not determine the relation of the thought to the True, because the mere expressing of the thought that it is true that snow is white leaves 36. Compare also the parallel argumentation in Frege (1983, pp. 251–252). 37. For the German version of this citation, see Frege (1990, p. 150).

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open whether this thought is true or false. We remain at the level of sense and do not advance to the level of reality. Analogously, in order to recognize that p, it is not sufficient to merely grasp the thought that p; it is, in addition, necessary to form the belief that p, i.e., to judge that p as true. For this reason, the assertoric use of the concept of truth is also a necessary condition for recognition and knowledge. In the case of these cognitive acts, the assertoric use can also not be reduced to the predicative use. This is made clear by Frege in the following passage from “Der Gedanke”: All the same it is something worth thinking about that we cannot recognize a property of a thing without at the same time finding the thought this thing has this property to be true. So with every property of a thing there is tied up a property of a thought, namely truth. It is also worth noticing that the sentence ‘I smell the scent of violets’ has just the same content as the sentence ‘It is true that I smell the scent of violets’. So it seems, then, that nothing is added to the thought by my ascribing to it the property of truth. And yet isn’t it a great success when, after long hesitation and laborious research, the scientist can finally say ‘What I have conjectured is true’? The Bedeutung of the word ‘true’ seems to be altogether sui generis. May we not be dealing 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 truth were a property, until some more appropriate way of speaking is found. (Frege 1997, pp. 328–329)38

The more appropriate way of speaking that Frege uses in the remainder of “Der Gedanke” is not to talk about truth considered as an object, but to talk about truth considered as what is expressed by the form of the assertoric sentence. His thesis that truth is not a property must accordingly be reconstructed as follows. For Frege, a property (or concept) is the “predicate of a possible judgment”. Truth is, in his view, what is expressed by the form of the assertoric sentence. This form, however, does not represent the predicate of a judgment, but determines the logical relation between subject and predicate. Thus, the form ‘S is P’, which is the common form of elementary assertoric sentences like ‘Snow is white’ and ‘Socrates is wise’, expresses that S really falls under P, in contrast to the interrogative form ‘Is S P?’, by means of which we express that S is P without expressing that S really falls under P. 38. The German version of this citation is to be found in Frege (1990, p. 345). For a detailed reconstruction of this passage, see Greimann (2004).

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Note that the falling of S under P cannot also be expressed by an ordinary predicate, say, ‘x falls under y’. The latter denotes a genuine relation, not a non-relational tie. It would therefore be a category mistake if one wanted to define the falling of S under P by means of an explanation of the form ‘S falls under P if and only if S has such-and-such properties’. Analogously, truth, considered as what is expressed by the form of the assertoric sentence, cannot be expressed by an ordinary predicate like, for instance, the predicate ‘is true’. Consequently, it would be a category mistake to define truth by means of an explanation of the form ‘S is true if and only if S such-and-such properties’.39 For, if the falling of an object under a concept is a special case of truth, then truth belongs to the same logical category as saturation. Just as saturation is not a relation in the ordinary sense, so too truth is not a property in the ordinary sense. In particular, it would be a category mistake to identify truth with the Bedeutung of the predicate ‘is true’, because the latter is an unsaturated entity whereas the former is no entity at all.40

REFERENCES Burge, Tylor, 1986. Frege on Truth. In L. Haaparanta and J. Hintikka, eds. Frege Synthesized. Dordrecht: Reidel Publishing Company, 97–154. Campbell, Keith, 1990. Abstract Particulars, Oxford: Basil Blackwell. Frege, Gottlob, 1879. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Reprinted in: G. Frege, Begriffsschrift und andere Aufsätze, Hildesheim: Olms, second edition, 1988. — 1893. Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet. Vol. I. Reprint: Darmstadt: Wissenschaftliche Buchgesellschaft, second edition, 1962. — 1976. Wissenschaftlicher Briefwechsel. Ed. by G. Gabriel, H. Hermes, F. Kambartel, Ch. Thiel and A. Veraart. Hamburg: Felix Meiner Verlag. — 1979, Posthumous Writings. Transl. by P. Long and R. White. Oxford: Basil Blackwell. 39. In Sluga (1999, pp. 36–38) and Sluga (2002, pp. 86f.) this category mistake is explained in a slightly different way. The main difference is that Sluga does not derive the indefinability thesis from Frege’s thesis that truth is expressed by the form of the assertoric sentence, but from the illegitimacy of categorial predicates like ‘is a concept’ and ‘is an object’. 40. I am indebted to Michael Beaney, Erich Reck and Marco Ruffino for their very helpful comments on an earlier draft of this paper. I am also grateful to the CNPq (Brazil) for supporting my research on Frege’s conception of truth with a grant.

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— 1983. Nachgelassene Schriften. Ed. by H. Hermes, F. Kambartel and F. Kaulbach. 2nd revised ed. Hamburg: Felix Meiner Verlag. — 1990. Kleine Schriften. Ed. by I. Angelelli, second edition, Hildesheim: Olms. — 1997. The Frege Reader. Ed. by M. Beaney. Oxford: Blackwell. Greimann, Dirk, 2000. The Judgment-Stroke as a Truth-Operator: A New Interpretation of the Logical Form of Sentences in Frege’s Scientific Language. Erkenntnis 52, 213–238. — 2003. Freges Konzeption der Wahrheit, Hildesheim, Zürich, New York: Olms. — 2003a. Frege’s Horizontal and the Liar Paradox. In: M. Ruffino, ed. Logic, Truth and Arithmetic. Essays on Gottlob Frege. Special edition of Manuscrito, Campinas: CLE-Unicamp, 359–387. — 2004. Frege’s Puzzle about the Cognitive Function of Truth. Inquiry 47, 425– 442. — 2005. Frege’s Understanding of Truth. In: M. Beaney and E. Reck, eds. Gottlob Frege: Critical Assessments of Leading Philosophers, Vol. II, Frege’s Philosophy of Logic. London: Routledge, 295–314. Kant, Immanuel, 1781. Kritik der reinen Vernunft. Ed. by R. Schmidt, Meiner: Hamburg, 1990. Sluga, Hans, 1999. Truth before Tarski. In: J. Woleński and E. Köhler, eds. Alfred Tarski and the Vienna Circle. Austro-Polish Connections in Logical Empiricism. Dordrecht: Kluwer, 27–41. — 2002. Frege on the Indefinability of Truth. In: E. Reck, ed. From Frege to Wittgenstein. Oxford: Oxford University Press, 75–95.

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Grazer Philosophische Studien 75 (2007), 149–173.

FREGE ON TRUTH, JUDGMENT, AND OBJECTIVITY Erich H. RECK University of California at Riverside Summary In Frege’s writings, the notions of truth, judgment, and objectivity are all prominent and important. This paper explores the close connections between them, together with their ties to further cognate notions, such as those of thought, assertion, inference, logical law, and reason. It is argued that, according to Frege, these notions can only be understood properly together, in their inter-relations. Along the way, interpretations of some especially cryptic Fregean remarks, about objectivity, laws of truth, and reason, are offered, and seemingly opposed “realist” and “idealist” strands in his position reconciled.

I. In Frege’s writings, the three notions mentioned in the title of this paper — truth, judgment, and objectivity — are all prominent and important. They are also closely related to each other, as is made explicit at various places. In “On Sinn and Bedeutung”, Frege relates the first two as follows: “Judgments can be regarded as advances from a thought to a truth value” (Frege 1997, p. 159); at other places, including the late article “Thought”, he also characterizes judging as “the acknowledgement of the truth of a thought” (ibid., p. 329). Relating the second and third notions, he remarks in The Foundations of Arithmetic: “What is objective … is what is subject to laws, what can be conceived and judged, what is expressible in words” (Frege 1994, p. 35). As these initial quotations already indicate, it is not just truth, judgment, and objectivity that are connected for Frege, but also several other notions, including those of thought, law, conceivability, and expressibility; and among the relevant laws, those of logic are especially important. Thus, in Foundations Frege advises us “always to separate sharply the psychological from the logical, the subjective from the objective” (Frege 1994, p. x);

and in “Thought” he adds: “I assign to logic the task of discovering the laws of truth. … The meaning of the word ‘true’ is spelled out in the laws of truth.” (Frege 1997, p. 326) Occasionally the notion of reason is thrown into the mix as well: It is in this way that I understand objective to mean what is independent of our sensations, intuitions, and imagination, and of all construction of mental pictures out of memories of earlier sensations, but not what is independent of reason. For what are things independent of reason? To answer that would be as much as to judge without judging, or to wash the fur without wetting it. (Frege 1994, p. 36)

The purpose of this paper is to explain, or better to elucidate, all of these notions as understood by Frege, including providing interpretations of the passages just quoted, cryptic as some of them are.1 It will become apparent that, according to Frege, a proper elucidation of them will require exactly paying close attention to their inter-relations. Crucial in this connection, since centrally related to all the others, is the notion of judgment. Its centrality has been pointed out before, in particular by Thomas Ricketts.2 My discussion will build on some of Ricketts’ insights and arguments, not just concerning the notion of judgment, but also those of truth and objectivity. On that basis, I will attempt to clarify further some relatively neglected aspects of Frege’s position, including especially cryptic remarks about objectivity, laws of truth, and reason. II. It will help to begin by reminding ourselves, briefly, of the broader context in which Frege brings up the notions mentioned. Throughout his work, Frege’s main interest is in the foundations of mathematics, and especially in the foundations of arithmetic. What he intends to investigate in this connection, clearly and in depth, are the following issues: What are the fundamental concepts and principles of arithmetic; what does their ulti1. This paper complements Reck (1997) and (2000/2005); see also Reck (forthcoming). With respects to my considerable debts to other Frege scholars, especially Thomas Ricketts, see the following footnotes. 2. See Ricketts (1986) and (1996), as well as the summary of Ricketts’ interpretation in Kremer (2000). A similar emphasis on judgment in connection with Frege can be found in Sullivan (2005).

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mate justification consist in; and thus, what does arithmetic’s content and objectivity amount to? Considering such questions leads straight away to Frege’s logicism, his claim that arithmetic is reducible to logic. To establish that claim, he develops a new logic, much more powerful than what was available previously; and he attempts to reconstruct arithmetic within that framework. Along the way, he reflects on the nature of logic, both to combat widespread misconceptions concerning it and to defend his novel logical system. The main kind of misconceptions combated by Frege consists of a group of psychologistic views about mathematics and logic, or more generally of empiricist and naturalistic views, which he finds in the literature of his time.3 Frege’s initial attack on such views occurs in The Foundations of Arithmetic, at first focused on the case of mathematics. The attack gets extended and sharpened in subsequent writings, from Basic Laws of Arithmetic to the late article “Thought”, now with the focus more on logic. Overall, Frege locates the core of the problem in a confused understanding of the nature of judgment, an understanding that, as he tries to make evident, is incoherent and ultimately self-refuting.4 We have arrived at the main negative reason why the notion of judgment is central for Frege. Before we can get clear about the foundations of logic and mathematics, what needs to be rooted out first, according to him, are various misguided accounts of the nature of judgment, as they can be found in the literature of his time. Typically, such accounts involve psychologistic appeals to “sensation, intuition, imagination, construction of mental pictures out of memories and earlier sensations” (as quoted above). More generally, they appeal to naturalistically conceived and empirically accessible aspects of cognition, i.e., general features of thinking conceived of as a mental occurrence or even as a process in the brain. In itself it is not illegitimate, of course, to study thinking in a naturalistic way. Frege’s point is that doing so tends to mislead in connection with what he is ultimately concerned with — the foundations of logic and mathematics, thus their content and objectivity — especially if it is done in the forms criticized by him. It is precisely in order to counteract such tendencies that Frege admonishes us, as well as himself, “always to separate sharply 3. Frege’s opposition to such views was first emphasized and put into historical context in Sluga (1980). 4. In Ricketts’ words, Frege opposes “a confused admixture of psychology and logic”, rooted in “a naturalistic empiricist view of cognition”, that “collapses into subjective idealism” (Ricketts 1986, p. 121).

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the psychological from the logical, the subjective from the objective” (as quoted above). Now, if we are not to conceive of the foundations of logic and mathematics in such problematic terms, how are we to do so instead? Answering that question is the main positive challenge for Frege. III. Below I will elaborate on how Frege’s response to the challenge just mentioned leads him to his sustained reflections on close relations between the notions of judgment, truth, objectivity, assertion, inference, logical law, etc. But before doing that, let me bring up another issue, one that is unavoidable in this connection. I am referring to Frege’s alleged platonism, often conceived of precisely as an explanation of the content and the objectivity of logic and mathematics. Turning to this issue first will be useful both to make clear the inherent limitations of such an explanation and to prepare my later discussion of its relation to an alternative. What philosophers usually mean by “platonism”, in our context, is a basic realism concerning abstract objects. In Frege’s writings, two kinds of abstract objects come up centrally: classes (or value ranges more generally), with numbers conceived of as a special case; and thoughts (or senses more generally). In The Foundations of Arithmetic, it is the nature of numbers that takes center stage. Against psychologistic attempts to conceive of numbers as mental entities or, more generally, to account for the content of arithmetic by appeal to mental states or processes, Frege insists that numbers are not “psychological and subjective”, but “logical and objective” (as quoted above). He also characterizes numbers as “independent” and “selfsubsistent objects” (Frege 1994, p. 72), both in the sense of being different from concepts and of not needing a bearer for their existence, as mental phenomena do. As such, they are the things referred to, and attributed properties to, in objectively true or false arithmetic statements.5 In Frege’s later writings, from Basic Laws of Arithmetic to “Thought”, his reaction against psychologistic trends typically involves an appeal to the objective status of thoughts, conceived of as the contents of judgments, including logical and mathematical judgments. In order to distance himself from problematic psychologistic views in this connection — which, in his view, render such contents hopelessly subjective — Frege goes so far as 5. See Reck (1997) and (2000/2005) for extensive quotations.

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talking about a “third realm” in which objective thoughts reside (Frege 1997, p. 337). This third realm is presented as parallel to the “first realm”, the spatio-temporal world of physical objects and processes, and to the “second realm”, the world, or worlds, of mental pictures, subjective ideas, etc., as occurring in individual people’s minds.6 Both Frege’s characterization of numbers as “independent, self-subsisting objects” and his location of thoughts in a “third realm” feed into interpretations of him as a strong and perhaps crude platonist — a “metaphysical platonist”, as I propose to call it. Along those lines, it is assumed that Frege’s response to psychologistic views is founded on a primarily and essentially ontological construal of the subjective-objective distinction, and more specifically, on the postulation of abstract entities in an objectbased metaphysics. That postulation is then taken to form the basis of an explanation of the content and objectivity of logical and mathematical judgments.7 It is also often seen as immediately problematic as a position in itself. The main problem is familiar: How could we ever have access to a realm of abstract objects, as postulated here, especially if any causal contact to them is ruled out, as it seems to be by Frege? IV. What is crucial with respect to a metaphysical-platonist interpretation of Frege, as just sketched, is the following: We start with the postulation of certain abstract entities, conceived of independently of how we make judgments. It is such entities that are first and foremost seen as objec6. See again Reck (1997) and (2000/2005) for further discussion. While Frege does not say so explicitly, I assume that numbers, classes, and value-ranges in general, plus all other abstract entities countenanced by him (truth values, concepts, functions in general), are also inhabitants of the “third realm”. An alternative interpretation of Frege’s “three realms” talk, suggested to me by Danielle Macbeth, would be the following: Take the first realm to contain all objective entities to which we can refer (physical as well as abstract objects, also functions, including concepts); the second realm to contain everything subjective; and the third realm to contain thoughts and other senses, understood as objective entities distinct from both objects and functions (as “modes of presentation”, i.e., ways in which entities in the other two realms can be given to us). While the question which of these two interpretations is correct seems significant for understanding Frege’s notion of sense, I do not see that it affects the discussion in the present paper much. 7. I owe the phrases “ontological construal of the objective-subjective distinction” and “object-based metaphysics” to Ricketts (1986). Compare Reck (1997) and (2000/2005) in which a further clarification of the position is attempted, by focusing on the order of explanation inherent in “metaphysical platonism”.

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tive (entities both in the first and third realms, while those in the second realm are subjective). This is, in a subsequent step, supposed to lead to an explanation of the objectivity of corresponding judgments. A tempting way to arrive at the latter explanation may be in terms of the notion of correspondence, in the sense of adjudicating arithmetic statements or thoughts against facts involving the initially postulated abstract entities. In any case, the objectivity of judgments, including arithmetic and logical judgments, is treated as secondary — as derivable, or explainable, in terms of more basic notions. Several formulations Frege uses, especially in the context of expressing his strong opposition to psychologistic views, seem indeed to support an interpretation of him as a metaphysical platonist. However, there are some immediate problems with such an interpretation. Besides saddling him, rather uncharitably, with a heavy-handed and objectionable view, these problems arise from what he says about three of our main notions: truth, objectivity, and reason. Concerning each, Frege makes remarks that appear to be in direct conflict with a metaphysical-platonist interpretation; or more cautiously, if we assume it, the remarks appear to lead to an internal tension in his views. We just saw that, if we take the notion of objectivity to apply primarily to entities, then an explanation of the objectivity of corresponding judgments needs to be added. And what may suggest itself in connection with the latter — especially if the postulation of numbers, classes, etc. is the starting point — is an appeal to correspondence. In fact, if we take such an appeal seriously, the notion of truth may seem to be explainable at the same time, in the form of a correspondence account of objective truth (whatever the more specific details). But if so, then a first interpretive problem is the following: As is well known, Frege explicitly rejects an explanation of truth as correspondence at various points in his writings. Even stronger, in “Thought” he rejects any attempt to reduce truth to other, more basic notions. His argument is, in a nutshell, that all such attempts will be circular; while trying to explain truth, they will already presuppose it.8 Now, perhaps there is no conflict, after all, between Frege’s explicit rejection of a correspondence account of truth and a metaphysical-platonist reading of him; since perhaps an explanation of the objectivity 8. See the discussion of Frege’s “regress argument” in Ricketts (1986) and (1995). In the present paper, I put aside Ricketts’ stronger claim that Frege’s views about truth rule out any kind of metalogic. Compare Tappenden (1997) for criticism and an alternative perspective.

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of judgments, even along platonist or realist lines, need not be tied to such correspondence. But in that case, an alternative explanation seems called for, especially in the case of logical and mathematical judgments. What could such an explanation look like? Perhaps we need to start with the postulation of thoughts as abstract objects, rather than with that of numbers and classes? (More on that idea below.) Then again, if we look at what Frege himself says about objectivity, in connection with logic and mathematics as well as more generally, it seems different from anything one would expect along metaphysical-platonist lines. As quoted initially, Frege characterizes what is objective as “what is subject to laws, what can be conceived and judged, what is expressible in words” (Frege 1994, p. 35). In passages such as these, he connects the notion of objectivity neither with that of correspondence nor with any other object-based notions. Instead, the notions of law, judgment, conceivability, and expressibility are brought up as crucial. A little later in Foundations, the following passage can be found: My explanation [of number in terms of logic] lifts the matter onto a new plane; it is no longer a question of what is subjectively possible, but of what is objectively definite. For in fact, that one proposition follows from certain others is something objective. (ibid., p. 93)

Here again, what is objective is not explained in terms of what one would expect from a metaphysical-platonist perspective. Rather, objectivity is connected with the notion of logical consequence or inference, thus with judgments according to logical laws. The conflict or tension that should be apparent now, between Frege’s remarks about truth and objectivity, on the one hand, and a metaphysical-platonist interpretation, on the other, becomes even more pronounced if we bring in some related remarks about reason. Remember this passage from Foundations (quoted early on): It is in this way that I understand objective to mean what is independent of our sensations, intuitions, and imagination, and of all constructions of mental pictures out of memories of earlier sensations, but not what is independent of reason. For what are things independent of reason? To answer that would be as much as to judge without judging, or to wash the fur without wetting it. (ibid., p. 36)

Later on in the same work, Frege comes back to the notion of reason briefly, as follows: “In arithmetic we are not concerned with objects which we

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come to know as something alien from without through the medium of the senses, but with objects given directly to our reason and, as its nearest kin, utterly transparent to it”. He adds: “And yet, or rather for that very reason, these objects are not subjective fantasies. There is nothing more objective than the laws of arithmetic (ibid., p. 115). Now, neither of these two passages is immediately transparent; both call for interpretation. But it should be noted that Frege connects objectivity again with the notions of judgment and law in them; and all three are presented as closely tied to the notion of reason. V. As argued so far, from the perspective of a metaphysical-platonist reading of Frege the following should puzzle us: his rejection of any reductive account of truth, including a correspondence account; remarks in which he associates the notion of objectivity closely, not with object-based notions, but with those of judgment and law; and corresponding remarks about reason. As noted earlier, perhaps it is possible to argue that, as a platonist or realist, one does not have to subscribe to a correspondence account of truth. Still, why does Frege relate objectivity so closely to judgment and law? Even more, what about his remarks on reason, which seem to fly in the face of a metaphysical-platonist reading — should we simply put them aside, as rhetorical flourishes that are not to be taken seriously, or perhaps as an early aberration in Foundations, never repeated in Frege’s later, more mature writings? Well, not if there is a plausible alternative. Instead of interpreting Frege as a metaphysical platonist — as someone who starts with the postulation of abstract entities, in an object-based metaphysics — the alternative is to attribute a judgment-based metaphysics to him.9 What that means is, first of all, to recognize the notion of judgment as central and primary, not the notion of object. Second, it amounts to taking more seriously than so far the close relationships between the notions of judgment, truth, and objectivity, as well as their ties to the notions of thought, assertion, inference, logic, and reason. In fact, crucial for this alternative interpretation of Frege is the claim that these notions 9. The phrase “judgment-based metaphysics” is again from Ricketts (1986). In Reck (1997), I talk about “contextual platonism” as opposed to “metaphysical platonism”; see also again Reck (2000/2005).

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can only be clarified, or elucidated, together, as opposed to being reduced to more primitive notions. First again to truth. In the article “Thought”, right after the passage in which Frege presents his argument that truth cannot be explained in terms of any more basic notion, he remarks: “So it seems likely that the content of the word ‘true’ is sui generis and indefinable” (Frege 1997, p. 327). This does not mean, however, that we cannot say anything further about truth. Frege himself goes on to talk about its relation to the notion of judgment; more specifically, this is one of the places where he talks about judging as “the acknowledgment of the truth of a thought” (ibid., p. 329). Looking at the latter remark out of context, it may be mistaken as a Fregean reductive explanation, or definition, of judgment. Doing so would presuppose that the notion of truth can be understood prior to, or independently of, the notion of judgment. In contrast, one may see Frege’s remark, not as a definition or reduction, but as an elucidation, i.e., as a clarification in which the two notions are related to each other.10 Let me clarify further the notion of elucidation as it is employed here. It is presupposed in such an elucidation that we already have some understanding of the notions involved, even if only a partial or implicit understanding, perhaps also an understanding that is easily misrepresented. What the elucidation provides, then, is an articulation of that understanding, by relating the relevant notions to each other in an explicit, particular, and hopefully illuminating way. As such, it involves a kind of circle — not a vicious circle, but a hermeneutic one — which distinguishes it from more linear explanations, reductions, or definitions. This can be illustrated by means of the case just mentioned. Actually, the elucidation here involves not just truth and judgment, but also a third notion: thought. Concerning all three notions, what we are told has three aspects or “directions”: First, a thought, in Frege’s sense, is (to be understood as) that which we acknowledge as true in a judgment. Second, a judgment is (to be understood as) the acknowledgement of the truth of a thought. Third, truth is (to be understood as) what we acknowledge about a thought in a judgment. Taken together, these three statements form an elucidatory circle, or here a triangle. Saying that truth is what we acknowledge about a thought in a judgment may still seem obscure. The following additional elucidations, in terms of 10. Ricketts, whom I again follow here, call Frege’s remark “equally elucidatory of judgment and truth” (Ricketts 1986, p. 131).

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further cognate notions, may help: Truth is the aim, or goal, of a judgment, while a thought is its content. More specifically, we can distinguish the act of judging, which has truth as its goal, from its particular content, the thought. We can also distinguish between thinking, as the mere grasping of a thought, and judging that the thought is true. And we can recognize assertion as the explicit manifestation or expression of a judgment (verbally, in writing, etc.). In Frege’s own words, again from “Thought”: We can distinguish: (1) the grasp of a thought — thinking, (2) the acknowledgement of the truth of a thought — the act of judgment, (3) the manifestation of this judgment — assertion. (Frege 1997, p. 329)

VI. In the previous section, I started to take seriously the passages in which Frege relates the notions of truth, judgment, thought, and assertion closely to each other. We still need to do the same with several further notions, especially those of inference, logic, and logical law, before then coming back, in the next section, to those of reason and objectivity. The ways in which inference, logic, and logical law are related to the notions discussed so far come to the fore when we recognize that thoughts, as the contents of judgments, often stand in logical relations to each other. In particular, two thoughts can exclude each other; one thought can imply another; and a thought can be a generalization of another. This much is, once again, understood implicitly when we understand the corresponding thoughts. What logic allows us to do is to make that understanding explicit, by using the notions of negation, conditional, and universal quantification — exactly the three basic notions in Frege’s logical system — and by formulating general laws concerning the latter.11 The notion of logic at work here is that of a field that deals with inference. But how exactly should the latter notion be understood? Staying 11. Compare the discussion of the explicative role of logic in Brandom (1994). I assume here a standard interpretation of Frege’s logic, including his conception of quantification. For an interesting alternative, see (Macbeth 2005). Along Macbeth’s lines, some of my remarks about Frege’s views on inference, logic, and logical laws would have to be reformulated. But my overall point seems unaffected.

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in line with our earlier suggestions concerning Frege, we can conceive of inference as a kind of judgment. What we do in an inference is this: We judge one thought to be true on the basis of judging other thoughts to be true.12 Logical laws — as made explicit in Frege’s logical writings, starting with Begriffsschrift — are then the general principles governing this activity, i.e., this kind of judging. Insofar as logical laws are laws of judgment and insofar as truth is the goal of judgment, a close connection between logic and truth then manifests itself. And it is precisely such considerations that lead Frege to saying: logical laws are “laws of truth”; and even: “the meaning of ‘true’ is spelled out in the laws of truth” (as quoted above). While the general ties between logic and truth should now be apparent, these last two quotations from Frege still need further clarification. We noted earlier that for him “the content of the word ‘true’ is sui generis and indefinable”. If so, then we cannot take logical laws to provide any definition of truth. Still, these laws — as alleged “laws of truth” — make explicit something crucial. What is it they make explicit? Again, truth is what we aim at in judging and inferring; or put slightly differently, truth is their norm. Now, insofar as we cannot define truth — insofar as we cannot reduce it to anything more basic, anything on which we have an independent handle — we cannot give “external standards” for truth. What we can do, instead, is to articulate “internal standards” for judgment and inference, i.e., basic principles concerning them, namely the logical laws; and that makes these laws internal standards for truth as well. Let me reformulate this last point slightly, thereby illuminating another aspect of Frege’s views. While logical laws do have a descriptive content for Frege — as he conceives of them, they are themselves truths — they also play a normative role. Namely, they prescribe how we are to judge, or how we are to aim at truth. Moreover, unlike the laws of special sciences such as geometry or mechanics, they do so in a very general way. In the posthumously published piece “Logic”, Frege puts it this way: 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 the utmost generality for all thinking, whatever its subject matter. … Consequently, we can also say: logic is the science of the most general laws of truth. (Frege 1997, p. 128) 12. Here I again follow Ricketts closely; see especially Ricketts (1986, pp. 135 ff.).

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To paraphrase, any scientific law provides a standard for judging correctly, as it demands that we judge in accordance with it. What is peculiar about logical laws is that they provide the most general such standards; and this is the sense in which they, especially or exclusively, deserve the title “laws of truth”. To round off this discussion of the inter-relations between truth, judgment, logic, and inference, let me come back to a notion already treated briefly above: assertion. Recall that what we do in an assertion is to make a judgment manifest. Now, this statement should again be seen as elucidatory of both judgment and assertion. That is to say, we can understand an assertion as the making manifest of a judgment; but we can also understand a judgment as what can be made manifest in an assertion. Put slightly differently, we understand what a judgment is insofar as, or to the extent to which, we understand what an assertion is. We also understand what truth is insofar as, or to the extent to which, we understand what a judgment is, as noted above. Combining these two points, we get a direct connection between assertion and truth. Moreover, the laws of logic, by being laws of judgment, are laws of truth. But then we can conceive of the laws of logic as laws of assertion as well. Indeed, occasionally Frege goes so far as saying: “[W]hat 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” (Frege 1997, p. 323). VII. At this point, we are in a good position to return to Frege’s striking, but rather cryptic remarks about objectivity and reason. Remember especially this passage: It is in this way that I understand objective to mean what is independent of our sensations, intuitions, and imagination, and of all construction of mental pictures out of memories of earlier sensations, but not what is independent of reason. For what are things independent of reason? To answer that would be as much as to judge without judging, or to wash the fur without wetting it. (Frege 1994, p. 36)

We noted early on in this paper why Frege distances himself from an appeal to “sensation, intuition, and imagination, …”, namely to oppose psychologistic views. Logic and arithmetic should not be seen as dependent on what

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the “psychologistic logicians” appeal to, since that would, in particular, undermine their objectivity, and thus misrepresent them in a fundamental way. Saying this much explains the first half of the passage just quoted. But what about the rest — why does Frege accept, indeed emphasize, that objectivity, including the objectivity of logic and arithmetic, is not independent of “reason”? Given our discussion so far, Frege’s reference to judging in the last sentence of the passage can point us in the right direction. One more conceptual relationship needs to be elucidated, however: that between reason and judgment. Frege himself is not very helpful in this connection, as he is largely silent on the issue. But consider the following: What is it we do in reasoning, i.e., what kind of activity is it? Well, both according to traditional logic and according to Frege, in reasoning we apply concepts (for Frege, also higher-level concepts) to make determinations in judgments and inferences. Indeed, reasoning amounts to doing these things in a systematic, law-governed way. What, then, is reason? It is the competence for reasoning, i.e., the normative ability for applying concepts in judgments and inferences, in a systematic, law-governed way.13 Returning to another of our initial quotations, not just the connection between reason and judgment, but also those between reason and objectivity and between judgment and objectivity fall into place now. Remember, once again, that for Frege “what is objective … is what is subject to laws, what can be conceived and judged, what is expressible in words” (Frege 1994, p. 35). While his rejection of “sensation, intuition, and imagination …” amounts to characterizing objectivity negatively, what we have here is Frege’s main way of characterizing it positively. The part about expressibility can be seen as a reference to the notion of assertion, itself closely tied to that of judgment; and the part about conceivability and judgeability, together with the reference to law-governedness, can be seen as references to reasoning, as just explained. Note that, along such lines, it is exactly right to say that being objective does not amount to being “independent of reason”; and to assume otherwise would be an attempt to “judge without judging, or to wash the fur without wetting it”. This is so because objectivity amounts to the possibility of making judgment (“what can be judged”), in terms of concepts (“what can be conceived”), 13. Using Kantian terminology one could talk about the corresponding „faculty“, understood precisely as the normative ability for doing these things. The ability is normative in the sense that it involves doing things correctly or incorrectly relative to a certain goal, in this case truth.

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as expressible in assertions (“what is expressible in words”), and in a lawgoverned way (“what is subject to law”). I have tried to clarify how Frege ties the notion of objectivity primarily to that of judgment (and some closely related notions), not to that of object.14 Our earlier slogan that Frege’s is a judgment-based metaphysics, not an object-based metaphysics, should thus have become clearer. Yet, how are we to think about the objectivity of various entities, including numbers, along such lines? Frege certainly wants to say that numbers, too, are objective, not just arithmetic judgments; similarly for physical objects. Along present lines, the suggestion is the following: Here we are dealing with a secondary and derivative use of the notion of objectivity. That is to say, the objectivity of entities is to be understood in terms of the objectivity of the corresponding judgments. And again, the latter has to do with the systematic law-governedness of those judgments, within the general framework provided by logical laws. This fits together well with the following (already quoted) Fregean remark: My explanation [of number in terms of logic] lifts the matter onto a new plane; it is no longer a question of what is subjectively possible, but of what is objectively definite. For in fact, that one proposition follows from certain others is something objective. (Frege 1994, p. 93)

Note, in addition, that what is special about numbers and arithmetic is this: In their case all we need to appeal to for ensuring “objective definiteness” is logical laws, while in the case of other sciences (geometry, mechanics, etc.) further laws play a role as well (basic laws of those fields). To highlight what is crucial about the interpretation we have arrived at, the following slogan may help: “Don’t ask what numbers are except by asking how we reason about them!” And as Frege tries to establish, we reason about them purely logically. I take this to be the core of Frege’s logicism.15

14. As Ricketts puts this point suggestively: “[Frege] begins philosophizing from a conception of objectivity that is internal to judgment making” (Ricketts 1995, p. 140). In Sullivan (2005) a related kind of “internalism” is attributed to Frege, although the author disagrees with Ricketts’ discussion of truth. 15. See again Reck (1997) and (2000/2005) for further discussion; compare also Tait (1986) which influenced me strongly, in this and other respects.

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VIII. Central to my interpretation is the claim that, according to Frege, objectivity is to be understood primarily in connection with the notion of judgment, not in connection with the notion of object. Similarly, truth is to be conceived of as closely related to judgment (and some cognate notions), not in an independent way, especially not in a reductive way. In the past few sections, the focus was on establishing the following: If we accept such claims, we can make sense of several of Frege’s cryptic and otherwise puzzling remarks about objectivity; likewise for his even more cryptic remarks about reason. Now I want to add some further reflections on the interpretation overall. A main remaining issue concerns the sense, if there is any, in which we are left with a “platonist” or “realist” position. Indeed it may appear, at this point, that what is attributed to Frege is an idealist position, as opposed to a realist one. More specifically, my emphasis on Frege’s remarks about reason, coupled with making the notion of judgment central, may prompt such a challenge. After all, aren’t these exactly the notions at the core of traditional idealist views, including Kant’s transcendental idealism? At some level, I want to say that this challenge works with a distinction between “realism” and “idealism” that is too simple and unreflective, i.e., in serious need of clarification and refinement if it is to advance the discussion. But to make such a response plausible at all, we first need to get clearer about the proposed interpretation in one crucial respect. For that purpose, let me distinguish four possible reactions to my focus on the notions of judgment, reason, etc. in connection with Frege. First, someone may question, and ultimately reject, that this gets at a significant aspect of Frege’s position at all, insisting instead on a metaphysical-platonist or realist reading of him, based on his remarks about “self-subsistent objects”, the “third realm”, etc. Second, someone may acknowledge that the discussion has brought to the fore a significant idealist strand in Frege’s thinking, but then argue that what this really shows is that there is a tension in his views, perhaps even an incoherency, as the remarks about “self-subsistence” etc. obviously pull in a realist direction.16 Third, one may go to the other extreme and deny the significance of the platonist16. I was confronted with the second response at the conference where this paper was originally presented. Thanks to Oswaldo Chateaubriand and Michael Beaney for pressing me on this issue. The first response seems fairly widespread, especially among critics of Frege; compare again Reck (1997) and (2000/2005).

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or realist-sounding remarks in Frege, by either seriously deflating them or essentially explaining them away, so that all we are left with is the idealist side. Fourth and finally, one may attempt to reconcile the “two strands” in Frege’s writings, arguing that, if understood properly, the “realist” and the “idealist” sides are not opposed to each other, so that neither needs to be explained away. It may appear that it is the third of these options that has been adopted in the present paper. But that would be a misunderstanding; it is actually the fourth option — probably the hardest one to make plausible, both as an interpretation of Frege and as a position in itself — that is intended. As this is an important point, let me put aside options one and two from now on.17 Instead I want to address the question of how, or in which sense, my proposal amounts to adopting option four above, not option three. In other words, I want to make clearer how Frege’s remarks about judgment, reason, etc., as interpreted in the previous sections, can be seen as reconcilable with his platonist- or realist-sounding remarks, in such a way that the latter are not simply explained away. IX. So far, two aspects of Frege’s remarks about objectivity have been emphasized. On the negative side, these remarks are meant as an emphatic rejection of, in particular, psychologistic views about logic and arithmetic. On the positive side, objectivity is to be understood primarily in connection with the notion of judgment, not the notion of object. Insisting on the latter does, no doubt, lead to an interpretation — some will want to say a re-interpretation — of certain Fregean claims about mathematical objects, especially numbers. In particular, it leads to making sense of the objectivity of numbers by way of the objectivity of arithmetic judgments.18 Yet this move is not meant to deflate completely, or to reveal as having no content, Frege’s corresponding platonist- or realist-sounding remarks. Instead, they 17. Like Reck (1997) and (2000/2005), this paper is not meant to provide decisive refutations of options one and two. However, the more interpretive option four can be made plausible, the more this will speak against both of them, as well as against option three; or so I hope. 18. To avoid a possible misunderstanding, it is not denied here that the notion of objectivity as applied to numbers has an “ontological” sense for Frege. What is denied, instead, is that an object-based explanation lies at its bottom; rather, judgment-based notions take priority. Compare again Ricketts (1986), Reck (1997), and Reck (2000/2005), especially the appeal to explanatory priority in the latter two.

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are seen as amounting to something substantive and interesting. It is just that what they amount to — what the significance is of saying that numbers are “independent”, “self-subsistent objects” etc. — is not taken to be self-evident, but in need of interpretation, indeed an interpretation that allows for a defense of them. If this is the intention, is it really what we have arrived at? As a first step in responding to this question, note the following: If explaining the objectivity of arithmetic objects in terms of the objectivity of corresponding judgments were coupled with certain views about the latter — e.g., with a psychologistic understanding of such judgments or, perhaps, with an account that bases them ultimately on conventions (as suggested by some twentieth-century philosophers) — this would indeed lead to a rather deflationary reading of Frege. But we do not need to adopt such views about arithmetic judgments. In fact, it is clear that Frege himself rejects them, explicitly those of a psychologistic character, more implicitly also those of a conventionalist character. Turning this line of thought around: If we want to retain real substance to Frege’s platonist- or realist-sounding statements about numbers, along the lines suggested in this paper, we better not start with views about arithmetic judgments that are too “thin”. Frege’s logicist project has the goal of reducing arithmetic to logic. In particular, arithmetic judgments are to be reduced to logical judgments. More than that, everything is to be grounded in basic logical laws in the end (and corresponding definitions). As noted earlier, for Frege the laws of logic are different from, say, the laws of geometry or of mechanics in one important respect: they guide inquiry more generally. At the same time, they share with geometric and mechanical laws that they are supposed to be truths. Furthermore, they are truths investigated and established systematically in a science — the science of logic. The latter point is crucial now. Remember, once again: My explanation [of number in terms of logic] lifts the matter onto a new plane; it is no longer a question of what is subjectively possible, but of what is objectively definite. For in fact, that one proposition follows from certain others is something objective. (Frege 1994, p. 93)

The suggestion is this, then: It is by keeping in mind, and taking seriously, that for Frege arithmetic judgments are grounded in logical laws and that those laws are established in a systematic science — logic, conceived of as such — that we can avoid working with an understanding of such judgments that is not “thick” enough.

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This suggestion can be reinforced by coming back to a Fregean view already mentioned, although only briefly: the view that logical laws are truths. At various points in his writings, Frege emphasizes that “being true” should not be confused with “being held as true”. As he puts it early on in Basic Laws of Arithmetic: Being true is quite different from being held as true, whether by one, or by many, or by all, and is in no way to be reduced to it. There is no contradiction in something being true which is held by everyone as false. (Frege 1997, p. 202)

For Frege, this point explicitly applies to logic. Thus, even what is taken to be a logical falsehood, “by one, many, or all”, may actually be true; and conversely, what is taken to be a logical truth at some point in time may later reveal itself as false (or at least as not true).19 But then genuine logical laws, as well as all truths that follow from them, are clearly different in status from conventions, not to speak of mere subjective convictions. X. What the previous section has made explicit, but what was already implicit earlier, is that for present purposes a lot hangs on Frege’s views about logical laws and their status. In particular, if we want to retain a sense that Frege’s platonist- or realist-sounding remarks amount to something substantive, as opposed to just explaining them away, we have to take seriously his claim that logical laws are truths, indeed truths established in a systematic science. Can anything further be said in this connection? There is a temptation, at this very point, to attribute to Frege once more a version of metaphysical platonism, now not by starting with numbers and other mathematical objects, but by focusing on logical laws and the objectivity of thoughts. To see how this fits into our earlier discussion, let us go back to the notion of reason, as appealed to by Frege. Given the conclusions we just reached, reason — our capacity for systematic reasoning — cannot be based on a merely subjective or conventional stipulation of laws, especially logical laws. A different understanding of its founda19. Compare Frege’s reaction to Russell’s antinomy. Frege took it to show that Basic Law V, which he and others had taken to be true, is not true after all (but without truth value, as some apparent names in it lack reference). Indeed, Frege had already made room for such a possibility earlier; see Frege (1997, p. 195).

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tions is needed. But what is available as an alternative? In search of one, we may be tempted to think of reason as grounded in some kind of grasp of a logical structure “out there”, in particular a structure built into the “third realm” of thoughts. In other words, don’t we have to assume, on pain of falling back into subjectivism or conventionalism, that in reasoning we let ourselves be informed by an abstract, independent structure of thoughts; and how could the latter be understood except in a metaphysical-platonist sense?20 Should such a position be attributed to Frege, though? I think it shouldn’t, for several reasons. Most crucially, this position involves again a notion of objectivity that is object-based, not judgment-based. The relevant objects are now thoughts, together with the system of logical relations between them, while in our earlier discussion it was numbers and other mathematical objects; but their objective existence is again simply postulated, as something basic and primitive. Yet is it really possible to make sense of objective, interrelated thoughts prior to and independently of the notions of judgment, assertion, logical law, reasoning, etc.? Also, if thoughts are conceived of in this way it seems problematic, once more, that we have access to them. And if we read Frege along such lines, his “idealist” remarks have to be explained away again; or it has to be acknowledged that there is a tension in his views, perhaps even an inconsistency. Someone may respond that these are all Frege’s problems, not ours. That is to say, perhaps such a position is really what his platonist- and realist-sounding remarks amount to. It is just that the position is hard to make sense of, even harder to defend, and seemingly in conflict with other remarks he makes. Then again, interpretive charity usually demands of us to resist such a conclusion, at least if there is an alternative.

20. See Burge (1992, p. 645, fn. 16): “Frege sees the whole logical structure, not just objects, in a Platonic fashion.” Compare also Hart (1992), in which a reading of Frege is sketched that is similar to mine, exactly up to the point where a “robustly platonic” position on logic and reason is attributed to Frege. Hart is dismissive of Frege in this respect, while Burge endorses his “Platonic” perspective. Concerning the latter — and as suggested to me by Michael Beaney — I should add that in more recent work Burge seems to play down that aspect and move closer to the reading of Frege suggested in the present paper; see the introduction to Burge (2005). Compare also footnote 23 below.

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XI. There is a dilemma at this point that can be described as follows: It seems that more needs to be said about Frege’s views on the status of logical laws, and with them about his understanding of the notions of objectivity, reason, etc., if we want to substantiate and further defend the line of interpretation suggested above. In doing so, we don’t want to fall into various kinds of subjectivism, including a crude form of idealism, the versions of psychologism explicitly rejected by Frege, and the simple conventionalism mentioned above. We also don’t want to fall into metaphysical platonism or realism, neither concerning numbers nor concerning thoughts and the structure of logical relationships between them. It is like a high wire act. How can we avoid falling down on either side? What does staying on the wire even amount to? A subtle response to this dilemma is contained in Thomas Ricketts’ work. As he writes: “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” (Ricketts 1986, p. 72). To prevent a possible misunderstanding right away, I do not interpret Ricketts as saying in such passages that, for Frege, objectivity simply amounts, or can be reduced, to some aspects of our linguistic practices, where the latter are seen as basically conventional. Otherwise we would be back to a kind of subjectivism already rejected. Rather, the idea is this: What we have reached when we think of objectivity as grounded in logical laws, and what is as such built into our linguistic practices, is itself “rock bottom”. In connection with commenting on “the sources of Frege’s conception of objectivity and logic” (ibid., p. 68), Ricketts elaborates on this point as follows: “Frege’s primary given is our awareness of obvious implications and contradictions” (ibid., p. 73). In other words, it is such implications and contradictions that are basic and primitive. There are two things we can do in connection with them: First, we can formulate logical laws in which we make them explicit. In doing so, we provide “laws of truth”, in the sense of internal standards for truth, judgment, assertion, etc. Second, we can relate the notions of truth, judgment, objectivity, though, assertion, logical law, and reason explicitly to each other, as we did above. But that is all — these notions can only be elucidated in terms of inter-relating them, as opposed to reducing them to something else. Likewise, the logical laws cannot be grounded in anything deeper. And why is that? As

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Frege indicates in the case of truth, any attempt at a further explanation or grounding would already presuppose what it is supposed to provide. As should be obvious by now, I am very sympathetic to this general perspective on Frege. However, I think that Ricketts’ remarks, as considered so far, are not entirely satisfactory, or not quite the final word. For one thing, simply calling something basic and primitive seems unsatisfactory. One may wonder, then, whether the resulting position is really less mysterious than the metaphysical-platonist position rejected above. Also, is it really impossible to say anything further in this connection? Maybe we cannot “dig deeper”; but perhaps we can elaborate things further in some other ways, moving more “laterally”? Actually, Frege himself has a few further remarks to offer concerning the justification of logical laws. While these remarks are again rather cryptic, we should consider them (as Ricketts does, too). At a couple of points in his writings, including The Foundations of Arithmetic, Frege simply states that logical laws “neither need nor admit of proof ” (Frege 1994, p. 4). In Basic Laws of Arithmetic, he is slightly more explicit: 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. (Frege 1997, p. 204)

Frege goes on to explain that any appeal to what we are forced to “by our nature and external circumstances” in connection with logical laws would merely concern their “being held as true”, not their “being true”; it thus wouldn’t, indeed it couldn’t, account for “with what right we acknowledge [them] to be true” (ibid.). That is to say, no such psychological or naturalistic considerations can provide a justification for the logical laws, in the sense Frege is concerned with. And as the passage above indicates, within logic only reductive justification is available, which doesn’t help either when we reach the bottom level of such reductions, namely basic logical laws. Does this mean that no justification of basic logical laws is possible at all? More generally, is any further inquiry concerning which logical laws to adopt impossible? Let us not jump to conclusions here. The following two points seem clear enough: First, Frege rejects problematic psychologistic or more broadly naturalist appeals in this context. Second, in the case of basic logical laws any further justification, if there was one, could not proceed reductively. Now, is that the end of the matter? Ricketts’

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remarks, as quoted above, suggest so. He seems to conclude that the second point here, about the limits of reductive justification, rules out inferential justification of basic logical laws in general; and as he adds: “Frege has almost nothing to say about non-inferential justification” (Ricketts 1986, p. 74). I want to close this paper by challenging Ricketts’ conclusion or, put more positively, by suggesting an alternative involving inferential, but non-reductive justification. XII. My concluding suggestion starts not from anything specific Frege writes about this issue, as he indeed says little, not only about non-inferential justification, but also about justification in the relevant non-reductive sense.21 Instead, it starts from what he does more generally, or from what is implicit in his procedure. What follows is thus somewhat speculative, as I am aware. Consider again Frege’s overall project. His goal is to reduce arithmetic to logic; and to do so he introduces a new logical system — first in Begriffsschrift and then, in a revise and expanded form, in Basic Laws of Arithmetic — so as to explore what is derivable in that system. Now, the fact that Frege modifies his logical system from Begriffsschrift to Basic Laws is already noteworthy. Beyond that, observe what happens after he is informed of Russell’s antinomy: He fundamentally questions, and ultimately rejects, one of his logical laws, Basic Law V. Moreover, this is done because of inferential considerations, namely the realization that, against the background of the other laws, a contradiction can be inferred from it. But if so, aren’t we dealing with a case of justification for a basic logical law that goes beyond what was considered in the previous section, both in terms of rejected naturalistic appeals and reductive inferences? To be sure, it is a case of negative justification; still, it seems significant. If this is correct, the question arises what a similar kind of positive justification might amount to, if there can be one at all. Even in that respect, 21. I am putting aside Frege’s remarks in the Introduction to Basic Laws about a “logical madman”, i.e., someone who rejects a basic logical law. I hope to be able to pick up on this issue — in connection with Ludwig Wittgenstein’s related discussions in his later works — in a future publication, but I have to leave it aside here for lack of space. (Ricketts has some interesting comments about those remarks.) As pointed out to me by Dirk Greimann, there are also some little known remarks in “Compound Thoughts” on how and why we accept logical laws (Frege 1977, pp. 55–77). Those, too, I have to leave aside here.

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there are some hints in Frege. At points he seems to suggest, more implicitly than explicitly, that the successful completion of his whole project would have constituted such a positive case — the working out of a comprehensive logical system, together with a systematic, conclusive demonstration that arithmetic can be derived (from suitable definitions) within this system, and perhaps also other parts of mathematics. This leads to an extension and strengthening of my interpretation so far. Part of it was that, for Frege, reasoning involves the systematic, law-governed use of concepts in judgments and inferences. What is emphasized now is the systematic aspect, as exemplified in scientific inquiry, including logical inquiry. And the additional observation is that, for Frege, such inquiry involves more than the piece-by-piece derivation of one thought from a few others. It also brings with it more global, holistic desiderata, including the following: the commitment to coherence and consistency; the goal of elaborating and extending our earlier, pre-scientific understanding in a systematic form, as refined in successive accounts; and corresponding virtues such as fruitfulness, both with respect to basic laws and definitions.22 What I am suggesting, then, is not that basic logical laws can be justified in some reductive or extrinsic sense, such as an appeal to a logical structure “out there” or to what we are forced to “by our nature and external circumstances”. Rather, a further kind of non-reductive, intrinsic justification seems possible, both in a negative and a positive form. Indeed, Frege’s de facto commitment to such justification can be seen as another aspect of his appeal to reason, as manifested in systematic, scientific inquiry.23 How far this kind of justification can lead us, especially in the positive direction, is another question. The history of logic and the foundations of mathematics in the twentieth century indicates that no quick solutions to all relevant questions can be expected, in some cases perhaps no definite solutions at all. Nevertheless, this aspect of rational inquiry should not be ignored, including for understanding Frege’s views.24 22. Concerning the issue of fruitfulness, see the eye-opening discussion in Tappenden (1995). With respect to the importance of the notion of system in Frege, or of systematic scientific inquiry, I am indebted to conversations with Norma Goethe; compare Goethe (2006). Here the present paper also connects directly with Reck (forthcoming). 23. Compare the rich discussion of Frege’s (largely implicit) views about rational inquiry in Burge (1998), parts of which seem compatible with my suggestion. However, the compatible parts would have to be divorced from the metaphysical-platonist aspect; see fn. 20. Similarly for Jeshion (2001), although the specific account of „self-evidence“ presented in it seems psychologistic at bottom, thus less congenial. 24. I am grateful to Dirk Greimann for organizing the conference Frege on Truth, Federal

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REFERENCES Beaney, Michael & Reck, Erich, eds., 2005. Gottlob Frege: Critical Assessments of Leading Philosophers, Volumes I-IV. London: Routledge. Brandom, Robert, 1994. Making It Explicit. Reasoning, Representation, and Discursive Commitment. Cambridge, Mass.: Harvard University Press. Burge, Tyler, 1992. Frege on Knowing the Third Realm. Mind 101, 633–649. Reprinted in: Burge (2005), 299–316. — 1998. Frege on Knowing the Foundations. Mind 107, 305–347. Reprinted in: Burge (2005), 317–355. Also in: Beaney & Reck (2005), Vol. II, 317–357. — 2005. Truth, Thought, Reason: Essays on Frege. Oxford University Press. Frege, Gottlob, 1977. Logical Investigations. P. Geach and R. Stoothoff, eds. and trans. New Haven: Yale University Press. — 1994. The Foundations of Arithmetic. J. L. Austin, trans., Evanston: Northwestern University Press, 1994. Originally published, in German, as: Die Grundlagen der Arithmetik. Halle: Koeber, 1884. — 1997. The Frege Reader. M. Beaney, ed. and trans. Oxford: Blackwell. Goethe, Norma B., 2006. Frege on Understanding Mathematical Truth and the Science of Logic. In: B. Löwe, V. Peckhaus & T. Räsch, eds. Foundations of the Formal Sciences IV. London: College Publications, 27–50. Hart, W. D., 1992. Frege and Carnap on Structure, Logic, and Objectivity. In: D. Bell & W. Vossenkuhl, eds. Wissenschaft und Subjektivität/Science and Subjectivity. Berlin: Akademie Verlag, 169–184. Jeshion, Robin, 2001. Frege’s Notion of Self-Evidence. Mind 110, 937–976. Reprinted in: Beaney & Reck (2005), Vol. II, 358–398. Kremer, Michael, 2000. Judgment and Truth in Frege. Journal of the History of Philosophy 38, 549–581. Reprinted in: Beaney & Reck (2005), Vol. I, 375–408. Macbeth, Danielle, 2005. Frege’s Logic. Cambridge, Mass.: Harvard University Press. Reck, Erich H., 1997. Frege’s Influence on Wittgenstein: Reversing Metaphysics via the Context Principle. In: W. W. Tait, ed. Early Analytic Philosophy: Frege, Russell, Wittgenstein. Chicago: Open Court, 123–185. Reprinted, in abridged form, in: Beaney & Reck (2005), Vol. I, 241–289. University of Santa Maria, Brazil, December 2005; to my audience at that conference, including Oswaldo Chateaubriand, Norma Goethe, Marco Ruffino, and Hans Sluga, for feedback on a first version of this paper; and to Michael Beaney, Dirk Greimann, and Danielle Macbeth for comments on later drafts. As I did not follow all of their suggestions, the remaining weaknesses and mistakes should, as usual, be attributed to me.

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— 2000/2005. Frege on Numbers: Beyond the Platonist Picture. The Harvard Review of Philosophy 13, 2005, 25–40. Originally published, in German, as “Freges Platonismus im Kontext”, in: G. Gabriel & U. Dathe, eds. Gottlob Frege: Werk und Wirkung. Paderborn: Mentis, 2000, 71–89. — forthcoming. Frege-Russell Numbers: Analysis or Explication? In: M. Beaney, ed. The Analytic Turn: Conceptions of Analysis in Early Analytic Philosophy and Phenomenology. London: Routledge. Ricketts, Thomas, 1986. Objectivity and Objecthood: Frege’s Metaphysics of Judgment. In: L. Haaparanta & J. Hintikka, eds. Frege Synthesized. Dordrecht: Reidel, 65–95. Reprinted in: Beaney & Reck (2005), Vol. I, 313–339. — 1996. Frege on Logic and Truth. Proceedings of the Aristotelian Society Supplement 70, 121–140. Reprinted in: Beaney & Reck (2005), Vol. II, 231–247. Sluga, Hans, 1980. Gottlob Frege. London: Routledge. Sullivan, Peter, 2005. Metaperspectives and Internalism in Frege. In: Beaney & Reck (2005), Vol. II, 85–105. Tait, W. W., 1986: Truth and Proof: The Platonism of Mathematics. Synthese 69, 341–370. Reprinted in: The Provenance of Pure Reason: Essays in the Philosophy of Mathematics and Its History. Oxford University Press, 61–88. Tappenden, Jamie, 1995: Extending Knowledge and ‘Fruitful Concepts’: Fregean Themes in the Foundations of Mathematics. Nous 29, 427–467. Reprinted in: Beaney & Reck (2005), Vol. III, 67–114. — 1997: Metatheory and Mathematical Practice. Philosophical Topics 25, 213–264. Reprinted, in revised form, in: Beaney & Reck (2005), Vol. II, 190–228.

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Grazer Philosophische Studien 75 (2007), 175–197.

EVIDENCE, JUDGMENT AND TRUTH Verena MAYER Universität München Summary Although Frege was eager to theoretically eliminate the judging subject from logic and mathematics, his system is permeated with notions that refer to subjective mental processes, such as grasping a thought, assuming, judging, and value. His semantic system depends on such notions, but since Frege in general shuns explaining them, his central conception of judgment and truth remains dark. In this paper it is proposed to fill out the gaps in Frege’s explanations with the help of Husserl’s phenomenological descriptions, especially those of the sixth Logical Investigation. This leads to a comparison between Frege’s notion of judgment and Husserl’s “Evidenz”, and finally also to a phenomenological classification of Frege’s remarks on truth.

1. Introduction Frege’s anti-psychologism is well known. When he designed his semantics and, in this context also wrote about truth, he was careful to extinguish wherever possible any essential involvement of a living subject that means or feels something, or judges, thinks or has mental images of any kind. Thus, for Frege, linguistic signs have a meaning and a reference quite for themselves, thoughts and truths exist independently of their being “grasped” by a person, objects fall under concepts independently of whether someone judges them to do so or not, and the correct definition of a term is rather discovered than developed by interested persons and can lie hidden forever. We know how Frege dealt with the seemingly indispensably subjective deictic expressions: they are to be substituted by objective indices which give the sentences in which they occur an “eternal” meaning (cf. Frege (1918)).1 This extinction of living subjects is last but not least due to the special goal Frege pursued: to provide mathematics 1. In what follows translations of Frege’s texts are mostly from Beaney (1997).

with a firm foundation that repels every attempt of psychologising it, an attempt which would weaken mathematical rigour and make mathematics self-refuting in the end. The remedy of psychologism Frege saw in a strict deductive derivation of mathematics from the concepts and laws of logic alone. Frege thereby adopted the Kantian conception of logic as the pure form of any possible judgment which can be expressed and made rigorous by a conceptual notation. His notation, a Leibnizian lingua universalis, is a mechanical tool that allows us to manipulate interpreted symbols in such a way that from true judgments only true judgments follow. At first sight it seems that any contribution of a subject is quite useless and definitely not desirable in this context. However, Frege could not totally dispense with subjective elements, some of them being rather implicit, others being quite essential, as he reluctantly admitted. Generally, his lingua characteristica is distinguished from a mere calculus by being a semantically interpreted language. This entails that the truth of the premises must not only be assumed but must be acknowledged by a judging person. Thus, Frege’s system necessarily comprises several subjectivist features.2 The first of these is the characterization of the sense of an expression as “a way in which the object is given”. One is immediately tempted to ask: To whom? And in which way? Frege avoids the problem by adding that the sense itself is objective and only “grasped” by the subject, this process of grasping being a topic for psychology or epistemology and of no interest to the logicist project as such. From this explanation follows what Dummett called the “linkage problem”, the question how objective abstract “thoughts” can be represented at all by individual and subjective thinking (cf. Dummett (1981, p. 681) and Dummett (1993, chap. 10)). Secondly, in connection with this notion of sense, identity statements forced Frege to introduce some kind of epistemology. The semantic difference between A  A and A  B cannot be accounted for except in terms of the “cognitive value” of the second type of statement. However, a value is something that is evidently in need of a subject to whom it is valuable. Actually, it is remarkable that Frege uses the term “value” on various occasions within his allegedly strictly objectivist semantics (see also the term “truth value”), in spite of the fact that values in general seem to presuppose feeling and thinking subjects. Thirdly, in connection with his considerations on value Frege character2. For further psychological elements, see Sokolowski (1987) and Mohanty (1982, chapter 4).

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izes logic as a normative science, however, normativity seems necessarily connected with the pragmatic concerns of living people. Husserl, to the contrary, was quite suspicious towards normativism, exactly because he suspected it of implying psychologism.3 A fourth subjectivist feature can be seen in Frege’s use of the notion assertion (“Behauptung”). The “assertoric force” with which a sentence is uttered differentiates what a scientist says from the words of an actor. Scientific assertoric force consists of the speaker’s purpose to aim for truth, while the actor merely pretends to do so. Assertion is surely a mental process, which Frege discerns clearly from the process of merely grasping a thought. Moreover, assertion is psychologically distinct from judging, which may count as a fifth subjective element in Frege’s logic. Judgment is the only psychological element that is explicitly incorporated — in terms of the so called “judgment stroke” — into the conceptual notation, and which expresses, as Frege declared, “the inner acknowledgement” of the truth of a sentence (cf. Frege (1983, p. 150)). Frege was quite explicit that the judgment stroke is a necessary ingredient of his formal language. Thus, as an answer to Jourdain’s question, “whether you now regard assertion as merely psychological?” Frege wrote that something essential would be missing if one would omit the judgment stroke from the premises of a conclusion.4 Frege was also explicit as to the psychological nature of grasping, asserting and judging, and even in turn used this as an argument against the necessity of a thorough analysis of the processes involved. He says: Just because [grasping and judging] is a mental (seelisches) event, we do not have to care about it. It is enough that we can grasp thoughts and acknowledge their truth; how this might happen is another question,

a question, he says in a footnote, whose difficulty has not been fully apprehended as yet (Frege 1983, p. 157). In sum, there are several essential aspects of the lingua characteristica that reach deeply into the mental acts of the people using the device. Frege did not deny this fact, he simply declares it not to be of concern for logic and mathematics, which is why its detailed analysis can readily be set aside. However, some of these aspects do not simply wait for the work of psychologists to become explained; they presumably cannot be explained in any satisfying sense. This is true especially for the terms judgment and 3. See Mohanty (1982). 4. “Wenn man bei der Darstellung eines Schlusses in meiner Begriffsschrift die Urteilsstriche bei den Prämissensätzen wegliesse, fehlte etwas Wesentliches” (Frege 1976, letter XXI/12).

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truth. To judge means to acknowledge the truth of a thought, or, as Frege occasionally declares, to make a “transition from a thought to a truth value” (Frege 1976, letter XIX/1; Beaney 1997, p. 150); however, it seems to him that no further clarification of the process is possible. Judgment, after all, is a basic concept of logic. This basic notion, however, entails a second, non-psychological notion, which must also be left undefined. Necessarily, if one judges, one acknowledges the truth of a proposition, and thus one must at least implicitly know what the truth of a thought consists in. Yet for Frege there seems to be no reasonable explanation of the concept of truth. For whatever criterion of truth we would accept, in any special case we have to ask if it is true that the criterion can be applied, and “so we should be going round in a circle” (Frege 1918, p. 60; Beaney 1997, p. 327). Thus, the connection between the psychological process of judging and the objective concept of truth cannot be elucidated either. Frege therefore leaves the two notions of judgment and truth basically unexplained. Given that logicism today is no scientific option, while Fregean semantics is widely used even for the analysis of natural language, it seems that this attitude is no longer acceptable. The linkage problem cannot be neglected. As Eva Picardi puts it: “The psychological ‘I’ has probably no essential role to play in logic, but a philosophical account of the contents of knowledge, belief and desire which disregards the circumstance that they must be thinkable by a human agent is seriously defective” (Picardi 1997, p. 165). 2. Husserl and Frege Husserl’s phenomenology aims at describing the essential objective structure of consciousness, and as such is explicitly contrasted with any empirical psychological investigation of individuals. There are also many other systematic features that Frege’s system shares with Husserl’s. Thus, Husserl and Frege both started with a clarification of the foundation of arithmetic, both emphasized the objective character of meanings, and both distinguished between two semantic levels, i.e. sense (noema) and reference (the intended object).5 Moreover, both exhibited a similar attitude towards language — natural language being some kind of covering of thought 5. On the relationship mostly with reference to the notions of sense and noema, see Føllesdal (1958), Mohanty (1982), Sokolowski (1987), and Bernet (1996).

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which could be removed and ameliorated by an ideal language — and Frege as well as Husserl was interested in logic not (so much) as a rational calculus but as a means of promoting the knowledge of truths.6 It was the decisive attitude with which Frege rejected every attempt to look into subjective processes and to connect epistemology with logic that has for a long time hindered a productive comparison between the two systems. Thus, the development of phenomenology was quite separated from that of logicism, and Husserl and Frege were ancestors to quite different philosophical schools. Yet it could be shown that this development is at least partially based on a misunderstanding on the part of analytic philosophy: the implicit presupposition that there is no possibility to bridge the gap between the psychological and the realm of objective thoughts, or Frege’s “Vorstellung” and the content of sentences. Husserl, however, shared Frege’s reservations with respect to psychological explanations of logical relationships without abandoning an explanation. In the Prolegomena he argued in a quite Fregean way against psychologism, for instance in the following passage: Undoubtedly our knowledge of logical laws, considered as an act of mind, presupposes an experience of individuals, has its basis in concrete intuition. But one should not confuse the psychological ‘presuppositions’ and ‘bases’ of the knowledge of a law, with the logical presuppositions, the grounds and premises, of that law: we should also, therefore, not confuse psychological dependence (i.e. dependence of origin) with logical demonstration and justification. […] All knowledge ‘begins’ with experience, but it does not therefore ‘arise’ from experience. (Husserl 2001, p. 42)7

Most clearly he points to the equivocation inherent in many terms central to logic, for instance “truth” and “concept”, which have a psychological as well as a logical meaning. While to hold something for true or to think 6. See on the latter point Mohanty (1982, p. 87ff.). 7. In what follows the translations are, wherever possible, from Husserl (2001). In his Philosophy of Arithmetic Husserl was not so decisive as to the difference between a logical and a psychological explanation. His interest there was mainly focussed at the constitution of the concept of number within consciousness, however, this was not meant to describe individual mental ideas or processes, but an objective constitutive structure. Therefore, Husserl is not content with Frege’s extensional definition of number via the term “Gleichzahligkeit” because it cannot explain how the intension of the term “number” comes about. On the other hand, Frege is not interested in the latter project, and generally tends to identify concepts with their extensions. To him the question how senses are constituted — which would be a Fregean translation of the phenomenological project — is of no bearing on their use in logic.

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about a concept are mental facts, the truth of a sentence and the falling of an object under a concept are in themselves quite independent of psychological events. In their phenomenological sense such terms signify ideal entities that are not to be understood in terms of a generalisation of mental occurrences, but are “ideal singulars” or “genuine species”. However, Husserl is not content with simply stating the equivocation and restricting logic to the domain of ideal entities. Instead he demands to clearly grasp what the ideal is, both intrinsically and in its relation to the real, how this ideal stands to the real, how it can be immanent in it and so come to knowledge. (Husserl 2001, p. 64)

In other words, the task Husserl sets himself is exactly the task Frege left unsolved, without Husserl, on the other hand, falling prey to the psychologistic prejudices Frege deplored. It therefore seems worthwhile to ask how Husserl explained the mysterious epistemic functions, which Frege labelled under the terms “grasping a thought” and “judgment”, the latter of which is quite deficiently described as “a transition from a thought to a truth value”, while the former is mostly left unexplained. 3. Husserl and Frege on grasping a thought The idea that a thought, an ideal objective entity, is merely “grasped” — in contrast to, for instance, being produced — by a thinker, is one of the seemingly rather bizarre aspects of Frege’s system. There seems to be no reasonable elucidation of the idea throughout Frege’s published works, apart from some negative explanations in his article “The Thought”. There, he says that “we are not owners of thoughts as we are owners of ideas. We do not have a thought as we have, say, a sense impression, but we also do not see a thought as we see, say, a star” (Frege 1918, p. 74; Beaney 1997, p. 341). In the same line lies the negative comparison of grasping a thought with sense perception, which has provoked quite different interpretations.8 Only in a letter to Husserl from the year 1906 does Frege give a kind of positive explanation. Husserl had sent him a critical review of Anton Marty’s “Über subjektlose Sätze und das Verhältnis der Grammatik zu Logik und Psychologie”9, which provoked Frege to remark that logicians make many distinctions with respect to judgments that to him seem to 8. See Malzkorn (2001). 9. For references see Frege (1976, XIX/3).

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be immaterial, while on the other hand they do not make many distinctions which he regards as important (cf. Frege (1976, XIX/3); Beaney (1997, p. 301)). Among the differences he refers to, there is one that must have sparked Husserl’s special interest. It concerns the ideal identity of different “tokens” of a judgment (or, for Frege, a sentence), which lies, according to Frege, in the thought grasped or expressed. Yet, how do we know that two sentences express the same thought, or that the sentence token uttered by someone else means the same as my own sentence token? In other words: how do we grasp the identical thought that is meant by different speakers, in different languages and wordings, and at different times? Here Frege, instead of simply referring to an unexplainable mental function, declares equipollence, i.e. logical equivalence, to be the criterion of identity.10 In particular, from our knowledge that certain equivalence relations hold, we “abstract” a so-called “Normalsatz”, which contains only those semantic aspects that are of interest for the logical evaluation of the sentence. This is reminiscent of Frege’s idea that we have to “separate a thought from its trappings” to arrive at the logically pure thought. “Seeing that the same thought can be worded in different ways”, Frege once said, we learn better to distinguish the verbal husk from the kernel with which, in any given language, it appears to be originally bound up. This is how the differences between languages can facilitate our grasp of what is logical. (Frege 1983, p. 154; Beaney 1997, p. 243)

Husserl’s reply to Frege’s letter, in which he referred to the notion of equipollence, is lost; however, we already know from his Logical Investigations that his account is quite parallel to Frege’s. Husserl uses a notion similar to Frege’s “thought”: the “matter” or “interpretative sense” (Auffassungssinn) of an objectifying act (which can, but does not have to be a meaningful sentence).11 Husserl describes the criterion for the identity of matter in the following way: 10. “Man muss sich in der Logik dazu entschließen, äquipollente Sätze als nur der Form nach verschieden zu betrachten” (Frege 1976, XIX/3). 11. The “Auffassungssinn” is, as Frege’s sense, the way-of-being-given of an object, however, Husserl makes it clear that there are more types of givenness of which the sense is only one: 1) the quality or “speech act”, i.e. if we are directed to an object in terms of an assertion, a question, a wish and so on; 2) the form of being given, i.e. only as a sign or only as an intuition or both, i.e. in a mixed way; 3) the sense or matter or meaning by which the object is given; 4) the content or representation (“Vorstellung”) involved, i.e. as a sign or an intuition. Thus, this red object can be given by way of a question (“Is this thing red?”) or an assertion (“This is a

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[…] essential identity can be equivalently defined if we say: Two presentations are in essence the same, if exactly the same statements, and no others, can be made on the basis of either regarding the presented thing […]. The same holds in regard to other species of acts. Two judgments are essentially the same judgment when (in virtue of their content alone) everything that the one judgment tells us of the state of affairs judged would also be told us by the other, and nothing more is told by either. Their truth value is identical […]. (Husserl 2001, p. 237)

Husserl adds that they can be otherwise quite different, for instance the “intuitive fullness and vividness” of the sensuous contents of one of them might be increased or decreased without the identity of matter of the act being changed. For Frege, too, equipollence serves to separate subjective aspects of being given, for instance feelings or a certain “illumination”. Thus, the considerations of both Husserl and Frege concerning judgment and truth have a comparable background: it is the thought or the matter of a judgment (in contrast to its intuitive fullness) which is true or false12, and which can be grasped via its different expressions by looking at their inferential relations, thus separating it from the subjective content and point of view. Husserl (1984/1, V § 22ff.) describes the phenomenological processes involved in much more detail, especially when he asks what the matter of a judgment is in relation to the acts which express or contain it. He, too, makes it clear that the objective sense or matter of a sentence is not the “constant psychic character of meaning” (Husserl 2001, I § 31), but an ideal entity which is the species or type of those acts of meaning. Grasping a thought, then, consists in recognizing the type of what one means in a concrete situation, a recognition which can in reflection be reconstructed by looking at what Frege calls “the whole system of equipollent sentences”. Thus, Husserl and Frege both do explicate the process of “grasping” a thought in terms of an intuitive abstraction which can be reconstructed in terms of a rational cognitive activity. However, for Husserl the grasping is intimately tied to the complex mental acts it “begins” with, while Frege red thing”); it can be given through a sign, an intuition or a phantasm or in a mixed way, the latter, for example when I read of a red thing and imagine it, or when I see a thing which is red on the front and use this redness as a sign for the redness of its backside too; it can be given by description (“This thing has the colour of a dark rose”); or the thing can be given by this or that sign or representation. (Husserl 1984/2, § 27) 12. Husserl, however, remarks that this is only a restricted (if common) way of considering truth, and that we can also speak of parts of sentences or of representations like “this green house” as being true if there is a corresponding intuition.

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reduces it to a simple logical procedure which results in a “Normalsatz”. Once the “Normalsatz” is abstracted, the whole process of grasping is no longer of concern for logic. Thus, again, the phenomenological background explanations of Frege and Husserl are quite similar, however, while Husserl goes into descriptive details, Frege skips the question in favour of the logical relations of the thoughts thus grasped. 4. Judgment in Frege and Husserl The logical relations between thoughts are indeed Frege’s main interest. In a letter to Dingler he once said that in mathematics the most important case of relationship is the one between a conditional sentence and a conclusion which is drawn from it. This is a fortiori true for logicism as a meta-mathematical account that tries to deduce mathematics from logic (cf. Frege 1976, Letter IX/4). It is no coincidence that in On Sense and Reference he discusses at length linguistic configurations of this type, which in natural language might be expressed even by one single sentence. From this point of view it also seems natural that the basic sentential connective of the “Begriffsschrift” is the conditional. Therefore, when Frege says that a judgment consists in a transition from a thought to a truth value, he mostly refers to a transition from certain premises to a conclusion, i.e. to a purely logical relation. In his first paper on logic (Frege 1983, pp. 1ff.) he also speaks of the “justifying reasons” logic does consider, and contrasts them with two other possible “transitions”, only to exclude the latter from the task of the logician. The first of these other ways (of relating sense to truth) occurs, when we acknowledge the truth of a sentence by recognizing the natural cause of the fact. For instance, we can accept that it is true that a certain person suffers from rabies because s/he has been bitten by an infected dog. However, for the latter to be a justifying reason, it must be given in terms of a quantified conditional (a natural law), which can be formally related to the conclusion. Only then is the relation between cause and effect again of logical concern. Secondly, we can acknowledge the truth of a sentence because there is some direct sensual evidence. This is a relationship that seemingly — at least if we take Frege’s general remarks on truth seriously — cannot be elucidated in any perspicuous way at all. We could add as a third type of transition the intuitive acceptance of logical axioms. Naturally, logic cannot dispense with these ways of judgmental relationships, since some premises must be given and

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some logical rules must be accepted before any conclusion can be drawn. This is true especially since Frege’s conceptual notation was meant to be not a formal calculus but an interpreted scientific language that must have a certain input from sense perception. However, even while logical “Gedankengefüge” depend on the existence of judgments that are justified “in other ways”, those ways must be excluded from logical investigation. Rather laconically, Frege says, “here begins the task of epistemology”, only to drop the subject for the rest of his discussion.13 Thus, it seems that the definition of judgment as a “transition from a thought to a truth value” refers mainly to the logical transition via inference. If we therefore “inwardly acknowledge” the truth of a thought in a Fregean way, we are in possession of logical reasons, but we do not, for instance, acknowledge the truth because we have a perception that confirms our hypothesis. Or at least, when the latter is the case, the logician does not have to explain it. This seems to be exactly the place where any elucidation of the process of judging must go beyond Frege, and turn to accounts that give the judgments of perception a central place, as Husserl does especially in the sixth Logical Investigation. According to Husserl, every true judgment must eventually be founded in an act of self-evidence, which in turn consists in an intuition of either a sensual or a categorical kind.14 Self-evidence for Husserl is not a “feeling of certainty” that accompanies certain cognitions; thus, it is not a subjective psychological event. Using the terminology of cognitive science it could rather be described as an “internal matching” between the intended and the represented object, or, in Husserl’s terms, as a relation of fulfilment or congruence between a meaningful predicative act and an act of intuition (“Anschauung”) or “givenness”. Acts of perception, in particular, have an intricate structure; they do not, for instance, simply predicate a property of a non-propositionally given object. Instead, they unite a thought of that object, which predicates something from it, with an intuition (or series of intuitions) that “represents” exactly the state of affairs as it was meant in the thought. For to be a case of self-evidence, however, not only must the perception confirm the thought, but the congruence between the two must be acknowledged in a further act of reflection, thus a judgment of perception proper consists of three interwoven 13. He returns to the topic when he briefly mentions the justification of logical laws, and then again says: “If [their justification] is not founded in other truths, logic must not care about it” (Frege 1983, p. 6). 14. For a more detailed analysis of Husserl’s concept of knowledge, see Willard (1995).

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acts: the act of judgment that means the object, the act of intuition that “gives” it, and the reflecting act of acknowledging the fulfilment of the former by the latter. Husserl tries to prove the phenomenological soundness of this account by discussing linguistic acts that contain indexicals. From a phenomenological point of view deictic expressions seem to refer directly to the thing which is meant within the context, however, the fact that we understand sentences containing such expressions without an adequate perception at hand, shows that there is a mediating act between interpreting the sentence and the perception. The mediating act corresponds roughly to Frege’s “sense”, and the already mentioned “Auffassungssinn” or matter of a judgment. Husserl concludes that also in judgments of perception we have to strictly distinguish the perception from the meaning of the judgment in such a way that the meaning presents what is intended while the perception fulfils or satisfies this meaning. Both, the act of perceiving and the act of meaning the perceived are intimately related to each other “in a relationship of congruence and a unity of fulfilment”, so that normally in perception we experience them as being one and the same. This is perhaps the reason why it seems impossible to analyse the relationship or to “define” it. However, this static unity can also become “floating”, for instance when an anticipation is immediately realized or when a hypothesis is confirmed. In such cases the dynamic process that leads from the meaning to the object given in perception as that which was meant, can be experienced. Husserl speaks of a “transitional experience” in which “the intentional essence of the act of intuition gets more or less fitted into the semantic essence of the act of expression” (Husserl 1984/2, p. 32; Husserl 2001, p. 296). Only in such cases do we properly speak of knowledge. Thus, knowledge is not a simple act of grasping either a fact or a thought, but the result of a conscious experience of congruence; it needs not only the intentional relation towards an object, but the intention of becoming fulfilled by an appropriate intuition. This complex description could work as an explication of Frege’s unexplained transition from a thought to a fact. 5. Intuitions and truth value Self-evidence is a relation that holds between the matter of an act and the corresponding intuition, while a Fregean judgment is a relation between

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the sense and the reference of a thought. The two accounts therefore could be schematised as follows:15

(1) (2) (3)

Husserl judgment16 expresses matter of a predicative act/thought is (reflectively) fulfilled by intuition

Frege sentence expresses objective thought/proposition is inwardly judged to have truth value

It is commonly believed that Husserl’s matter and Frege’s sense are roughly the same kind of entities or are at least comparable over the respective systems. Moreover, not only is self-evidence a reflective act, also the Fregean judgment has the character of being conscious, since it is definitely distinguished from the mere grasping of a thought, and consists in the “inner acknowledgment” of truth. Thus, the most striking difference between the two accounts lies at level (3), where an intuition is opposed to a truth value. It is here that the semantic conceptions of both thinkers are normally seen to fall apart.17 If this were true, the transition from a thought to a truth-value could not be illuminated by Husserl’s self-evidence, and the linkage problem would arise again. However, there are some possibilities to assimilate the two notions. First of all, Husserl’s epistemological semantics is of an externalist structure. In an intuition, we have direct access to the thing, which is intended in an act of consciousness, and which — provided that the thought is true — is immediately given with the properties meant. (Thus, the intuited, given object is in fact a “Sachverhalt”, which could be formalised with help of the function-argument-structure Frege proposed.) In an appendix to the fifth Logical Investigation, Husserl most clearly argues against the “picture-theory” of representation that is doomed to end in an infinite regress of the picture relating to an object via another picture; instead we must realize 15. Frege’s schema has famously been compared with Husserl’s in a quite different way, such that to Frege’s “expression” corresponds Husserl’s “act”, to the Fregean “sense” the “noema”, and to the “reference” the “object” (cf. Føllesdal (1969)). That such a parallel holds has been denied by Sokolowski (1987). 16. Husserl understands the term judgment — as usual in contemporary logic — as an intentional meaningful predicative sentence, much like Frege’s “Satz”; the latter is normally not meant to include questions or meaningless sentences but only assertions. 17. See for example Simons (1995).

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that a transcendent object is not present to consciousness merely because a content rather similar to it simply somehow is in consciousness — a supposition which, fully thought out, reduces to utter nonsense — but that all relation to an object is part and parcel of the phenomenological essence of consciousness […]. (Husserl 1984/1, p. 423; Husserl 2001, p. 239f.)

Thus, what we know by self-evidence is the fact itself, albeit the fact meant in a certain way. This is reminiscent of Frege’s argument in “The Thought”, where he says that truth cannot consist in a relation between a picture and a fact (cf. Frege (1918, p. 59)). Perhaps Frege’s difficulties with any definition of the concept of truth thus could be traced to his implicit externalist account, which is mostly, and somehow misleadingly, expressed in his repeated arguments against psychologism. Husserl, in turn, makes it clear that to speak of recognizing objects (or to judge that a certain sentence is true) and of self-evidence in the explicated externalist sense means essentially the same. Thus, it seems that the two accounts of judgment describe a similar procedure, only from slightly different points of view. Again, in Husserl’s wording: What is characteristic about this unity of knowing [in a state of self-evidence, V.M.], is now shown up by the dynamic relationship before us. In it there is at first the meaning-intention, quite on its own: then the corresponding intuition comes to join it. At the same time we have the phenomenological unity which is now stamped as a consciousness of fulfilment. Talk about recognizing objects, and talk about fulfilling a meaning-intention, therefore express the same fact, merely from differing standpoints. The former adopts the standpoint of the object meant, while the latter has the two acts as its foci of interest. (Husserl 1984/2 § 8; Husserl 2001, p. 296)

If this is true, it remains to be explained why Frege speaks of a truth value where Husserl introduces a “consciousness of fulfilment”, an intuition, or an “object immediately given”, terms that essentially relate the object to the judging subject. At first sight a Fregean truth value is an abstract object, which is the result of “saturating” a function with an argument — a process that remains entirely on the level of reference, and has nothing of a “consciousness” in it. Generally Frege tends to place truth solely within the realm of reference, thus entirely adopting the standpoint of the object meant. However, it seems striking that Frege (by extending the mathematical notion of a functional value) invented of all things the term “truth value”, thus giving the abstract object a place within philosophical value theory, which is certainly not subject-indepen-

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dent.18 It therefore seems that Frege did not think of the formal object, “the True”, as existing without people that value it, that is, without a cognitive structure that constitutes it in terms of a value. As is well known, Frege’s explanations of his notion of a truth value are sparse, and it is generally assumed that the truth values “the True” and “the False” are arbitrary abstract objects that serve to rigidify the semantic scheme in a way especially adapted to logical purposes (cf. Mayer (2003)). However, as Frege insists, to say that a thought “has a truth value” does neither mean that it has a “real” property, nor that it refers to an arbitrary object, but only that it is either true or false. Thus, talk of a thought having a truth value is a formalised, abbreviated way of saying that it is true or false. Moreover, if we look at the semantic schema Frege exhibits in his letter to Husserl, it seems clear that the “Bedeutung” of a sentence must incorporate the “Bedeutungen” of the parts of the thought, which are concepts and objects. Therefore, a truth value must somehow “consist” of concepts and objects of first and possibly also second order. Last but not least, if a true thought is a fact, and a fact consists in the fulfilment of a concept by an object, the natural explanation of “truth value” would be “fact”. However, there are several reasons why this is not the correct reconstruction. For Frege, a fact may be considered as an instantiation of truth, but not as truth itself, and what is judged is not only an a in relation to an unsaturated concept f( ), but that f(a) is the True. This is just one of many possible ways to express the same relationship which is already implicitly contained in the predicative form of sentences: that the object falls under the concept; that f(a) is the case; that f(a) is true; or simply f(a). It is this relationship, the falling of an object under a concept, which is acknowledged by a judgment. This subsumption is not a single fact, an instance of f(a), but so to speak the abstract idea at which, among others, scientific research is focussed, and which could be abstracted from the set of all instantiations in a similar way as the concept of number can be abstracted from the class of all one-to-one related sets. The abstract object, then, is a value in the sense that it is dependent on a special interest of persons, and thus is intimately related to the process of judging itself. Frege himself did not try to define the truth value in this way because the definition obviously would presuppose subsumption itself.

18. That this is not a coincidence is shown by Frege’s repeated reference to the three basic values true, good and beautiful. See for instance Beaney (1997, p. 227).

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6. Truth The difference between the respective systems of Husserl and Frege lies not only in the fact that Husserl is occupied with a thorough analysis of the processes involved, while Frege is not interested in these processes as such, but rigidifies them with the help of an abstract notion to make his formal language work. The essential distinction is rather to be seen in the fact that Husserl, because of his overall project, i.e. the structural description of consciousness, never leaves the phenomenological standpoint, and thus never arrives at a concept of truth that would explicate the idea of objective subsumption — for him such an idea would always have to be translated into talk about acts, eidetic types and intentional objects. Frege, to the contrary, officially knows only one sense of truth: the objective truth of thoughts expressed by sentences that consists in the falling of the object under the concept, which is a special case of saturation. However, when he tries to explicate this notion, he gives several different characterizations, for instance, that truth is redundant, indefinable, the aim of science etc. Not all of these explanations are compatible with each other; for instance, the definition of the True in § 10 of Grundgesetze seems to contradict his repeated claim that truth is indefinable, and, in turn, the aim of science is surely not a formal object. Husserl, on the other hand, is more susceptive to the ambiguities of philosophical notions such as judgment or idea. As he once claimed, with such notions we sometimes mean psychological events, sometimes abstract entities, thus that theoretical reflection can be misled by usage. In the same way the notion of truth reflects quite different ways to conceive of the relation between meaning and fact. Therefore, Husserl distinguishes carefully between several truth concepts, which in a way are all abstracted from the essential structure of self-evidential acts described. In what follows we will try to loosely relate them to the different characterizations of truth Frege gives throughout his writings. Before going into details, however, a quite straightforward comparison must be accounted for that Frege himself hints at in his first letter to Husserl. There he refers to Husserl’s first book, the Philosophy of Arithmetic, which is still formulated in a rather psychological terminology. Frege gives the following schema which is meant to reconstruct Husserl’s notion of a concept (Begriff):

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Concept word p Sense of the concept word (sense) p Object falling under the concept, thus indicating that for Husserl judgment is a saturation relation (“fulfilment”) between a mental concept and a real object. In contrast, as he says, for him the step from concept to object “takes place on the same level”, and thus is a relation within the objective realm. This is, why he has difficulties to clearly discern a concept from its extension (Begriffsumfang) as is shown in his Foundations of Arithmetic. However, for Husserl at the time of the Logical Investigations (and possibly already in 1891), it makes no sense to distinguish between entities of a mental and entities of a real world except in terms of their phenomenological structure; rather, this is exactly the psychological preoccupation that phenomenology wants to overcome. Fulfilment, therefore, is to be translated into the relation between meaning and “the given”, which is the world as it is immediately present to consciousness. Thus, for Husserl, Frege is mistaken in believing that the thinking and perceiving subject can be extinguished from a logical language, as already the many “psychological” ingredients of his system have shown. Frege only tries to describe “from the standpoint of the object” what must be shown to be act-relative from the standpoint of the thinker or perceiver. Husserl’s various notions of truth that are meant to explicate the ambiguity inherent in the term result from this. Even if these are gained from acts of self-evidence, it must be stressed that Husserl’s truth concepts are objective notions. For Husserl, the objectivity of truth is simply an analytic entailment of the term. Thus, it literally makes no sense to speak of something as being true “relative to” a speaker or a community or make truth dependent on the actual act of judging. Here, the respective arguments of both, Frege and Husserl, are again quite similar. For instance, both make it clear that to explicitly deny the objectivity of truth is paradoxical or self-refuting. Thus Frege says in his Logik from 1897: If anyone tried to contradict the statement that what is true is true independently of our recognizing it as such, he would by his very assertion contradict what he had asserted; he would be in a similar position to the Cretan who said that all Cretans are liars.” (Frege 1983, p. 144; Beaney 1997, p. 232f.)

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Husserl gives various arguments to the same conclusion, for instance in the first Logical Investigation: An assertion, e.g., whose content quarrels with the principles whose roots lie in the sense of truth as such, is self-cancelling. For to assert is to maintain the truth of this or that content. (Husserl 1984/2, p. 123; Husserl 2001, p. 51)

This is a claim that may also be taken to show that Husserl shares Frege’s “redundancy-thesis”. The various notions of truth that spell out the idea of fulfilment therefore must not be interpreted in a subjectivist sense; in contrast, as Husserl warns us in § 39 of the sixth Logical Investigation, the idea that what is self-evident for one person could be an absurdity for another, makes no sense. This, of course, follows already from his externalist account of knowledge.19 Generally Husserl conceives of truth in the classical sense of an adequatio rei et intellectu, where the adequatio can be more or less perfect; however, the “regulative idea” of truth is that of a perfect congruence: Where a presentative intention has achieved its last fulfilment, the genuine adequatio rei et intellectus has been brought about. The object is actually ‘present’ or ‘given’, and present as just what we have intended it; no partial intention remains implicit and still lacking fulfilment. (Husserl 1984/2 § 37; Husserl 2001, p. 328f.)

The several notions of truth Husserl describes stem from the fact that such an experience of congruence is itself an act of consciousness which exhibits, as other acts do, different phenomenological aspects. Thus, it has an “intentional essence” as well as an “objective correlate”, that is, an act of self-evidence instantiates certain types or ideas. The first notion of truth is meant to refer to the objective correlate of the experience of self-evidence or congruence itself. Since such an act says that the perceived is the same as what is meant, it is an act of identification, which belongs to what Husserl calls “positing” or “objectifying acts”, i.e. acts that are about something 19. For the following, see Husserl (1984/2, § 39). A thorough presentation and discussion of the text is given in Tugendhat (1967, especially § 5). In my own reconstruction of the text I skip many difficulties that Tugendhat detects, especially Husserl’s purported lack of a distinction between object and “Sachverhalt”. I take it that often when Husserl prefers to speak of an object, he does not mean an object simpliciter, but an object meant by a concept (recalling Kant’s “Anschauungen ohne Begriffe sind blind”) that imposes on it a “Sachverhaltsstruktur”. Husserl himself makes it clear that his use of “truth” comprises also nominal acts, thus that a simple naming can be true. This use, of course, has no parallel in Frege’s system.

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that is assumed to be the case. In contrast, a mere reverie or fantasy is not an objectifying act. Truth in the very sense of the objective correlate of positing acts comes very close to Frege’s realistic truth concept, however, always under the restriction described. In this context Husserl warns us not to confuse this notion of truth with the correlate of the copula “is”, which would simply consist in the being of a Sachverhalt, or, if generalized, in the class of all facts. Since Husserl — despite his praise of Frege’s article “Funktion und Begriff” — generally sticks to the elder analysis of judgments in terms of subject and predicate related by the copula, he thereby comes as close as possible to Frege’s concept of subsumption. He thus reminds us that a fact, the class of facts or even the abstract idea of a fact is not the adequate explication of the concept of truth. For the correlate of self-evidence is not only being but congruence of being with meaning, or, as Husserl says, “being in the sense of truth” (Husserl 1984/2, § 39). The notion of truth thus includes an implicit reference to somebody who judges, who is interested in the fact given, and to whom the fact is valuable. Truth in this sense, in contrast to any real property attributed with help of the copula, cannot be expressed, but is always only “experienced”; thus it cannot coincide with the meaning of the copula — a remark that is again reminiscent of Frege’s “redundancy thesis”, even if the latter is formulated with respect to sentences, not to acts and experiences. Secondly, truth is the idea, essence or species of the act of self-evidence itself; it is, so to speak the “type” of all acts of congruence of res and intellectus in contrast to the mere “token” of the singular act. The type of an act should not be confused with what all singular acts have in common, but is an ideal entity which is instantiated by a singular act. On the surface there seems to be no Fregean explanation of truth that comprises this idea. In contrast, Frege explicitly denies that the concept of correspondence can serve to elucidate the notion of truth. However, when he says that “all sciences have truth as their goal” (Frege 1983, p. 139; Beaney 1997, p. 227), he might be understood as meaning something like this: the general process of the confirmation of a hypothesis or of what is meant in a certain scientific way, which is, in a sense, the idea behind all self-evidential acts. Thirdly, we speak of truth as that which is really given in contrast to that which is merely intended. The given is what makes the intention true, which has a certain “fullness” of determinations that the sense lacks; it is

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the truthmaker20 “in an indefinite relation to an intention that is to make true or fulfil adequately” (Husserl 1984/2 § 39, Husserl 2001, p. 333). This seems to be a more popular use of the word, as it appears, for example in sentences like: “She is a true friend”, i.e. she exemplifies and exhibits everything that is only meant with the term “friend”. Frege excludes this meaning from the logical sense of “true” when he says that the word cannot be applied to something material (cf. Frege (1983, p. 140); Beaney (1997, p. 229)). However, facts of course are the truthmakers of sentences, thus, with respect to entities of a “Sachverhaltsstruktur” we are again close to what Frege refers to with his realistic notion. Fourthly, we can speak of truth as a relation between sense and reference, which Husserl calls the “accurateness” (Richtigkeit) of the sense: the sense correctly says what is the case. Frege would not have accepted a truth concept that comprises the idea of accuracy, since he does not allow for degrees of truth21, as Husserl naturally does. For even if self-evidence in the end consists of perfect congruence, most of our cognitions are not of this kind, without them therefore being false. For instance, to say of an apple, whose surface is mostly red, that it is red, is normally correct, if not literally true. Self-evidence exactly is a dynamic process because it allows for the gradual sharpening of congruence in the course of a temporarily extended perception. Thus, a phenomenological analysis of truth on the basis of congruence does not yield an unambiguous analogue to Frege’s notion of truth, however, it can illuminate the phenomenological foundation of Frege’s explanations. The most striking point of departure is the idea of correspondence which Frege rejects in “The Thought”, a rejection which might even be implicitly directed against the Logical Investigations. Frege has, especially, three objections. The first is the well-known circularity objection that Kant (1902f., Vol. IX, p. 50) had already formulated in the Jäsche lecture on logic: if we would stipulate that truth consists in a congruence between thought and reality, or representation and represented object, we would have to decide in a concrete case if the congruence holds, and thus the definition would be circular. This is generally not believed to be a good argument, since it lacks an explanation why a definition could not spell out its own formal structure. It would need quite elaborate Tractarian semantics to prove that “what can be shown, cannot be said” (Wittgenstein 20. For a discussion of the idea of a truthmaker in Husserl and his contemporaries, see Rojzczak and Smith (2002). 21. See Frege (1918).

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1986, §§ 4.121, 4.1212). Be that as it may, Husserl’s reflective recognition of the congruence of meaning and fact exhibits a basic “intuitive faculty” (a faculty of judgment in the Kantian sense) behind the notion of truth, which presumably must be available to the speaker before a truth definition can be understood. Frege’s second argument is relevant only for Husserl’s fourth notion of truth in terms of accurateness. Frege here refers to a (somewhat doubtful) analytic implication of the word “true”, when he says, “What is only half true, is untrue” (Frege 1918, p. 60; Beaney 1997, p. 327). This is perhaps the price Frege pays for proposing a subject-independent truth concept: the falling of an object under a concept either takes place or not; there is no third possibility. Also, it shows that Frege has a rigidified notion of truth in mind that does not allow for the laxness common to natural language. Husserl, in contrast, argues from the structure of our acts which perfectly allows for partial truth. However, also for Husserl, to be able to make true judgments at all, we must possess an ideal notion of truth, that is, the idea of perfect congruence, which serves at least as a kind of regulative idea in the Kantian sense. Frege’s third argument again is easily rejected by Husserl: Congruence, says Frege, can only take place where the congruent things are of the same nature, which is not the case with the correspondence thesis: res and intellectus are categorically different entities. However, self-evidence in the sense of Husserl is a relation between two acts of consciousness, such that no contradiction is involved. Also, the second aspect of the argument, namely that if the congruent acts are of the same category, we would not speak of congruence (Übereinstimmung) but of identity, is not relevant for Husserl: intention gives the object as it is meant, while in perception it is self-given (selbstgegeben), thus, the same matter is given in different ways. Since it is the way of being given that counts, “congruence” is the correct term. 7. Conclusion Frege conceived of truth as a property of senses, i.e. of subject-independent entities that are somehow real, even if they are not thought of. Truth is not a correspondence between senses and facts, but true thoughts are facts: a thought is true if the function-argument relation that it “means” really takes place, or, in other terms, if the value is the True. Since thoughts as well as facts are objective entities, there is in principle no place in logic and

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semantics for a thinking person, and this is, why psychologism is erroneous. On the other hand, the thinking subject is necessarily the author of such activities as using a language and understanding it, drawing logical inferences, or judging a thought to be true. Moreover, such activities are essential for the task Frege sets himself: to create a formal language that could serve as a means for developing and proving scientific sentences. A subtask of this programme was logicism. Thus, Frege had to incorporate “psychological” processes into his lingua characteristica, without, on the other hand, being able to explain how they are linked with its essential aspects, i.e. sense, reference, and truth. Husserl, on the other hand, conceived of truth as something that is based on (or founded in) acts of thinking persons, taking place within the realm of consciousness. Thus it seems at first sight that his conception can have nothing in common with Frege’s, even if many similarities between the two systems can be found. However, Husserl disposed of the psychologistic muddle by distinguishing between tokens and types (“species”) of acts and their contents. A concrete idea of a thinking person is an instantiation of an objective type that can have properties or aspects not thought of by the thinker. Thus, Husserl can entertain similar arguments against subjectivism in logic as Frege, without at the same time neglecting the fact that thoughts are thought of by people with various and complex intentions. He can therefore yield an explanation of truth that is not open to Frege, and that reduces truth basically to an experience of congruence. It has been shown that both conceptions do not lie as far apart as it seems. In Frege’s account the thinker essentially lurks in the notion of a judgment, of which Frege cannot give any reasonable explanation. However, having once accepted that by judging something to be true the thinking person has a reflective experience (or “inner acknowledgement”) of the truth of a thought it instantiates (or “has grasped”), the concept of truth cannot remain unaffected. From a phenomenological standpoint the class of all facts (or, in Husserl’s terms, the objective correlate of the copula) cannot be identified with the notion of truth, since truth is the essence of selfevidence, that is, of the experience of congruence between what is meant (the sense) and what is given (the reference). Even from the standpoint of Frege it is hard to see how there can be a notion of truth independent of the judging person who sees that subsumption takes place. Thus phenomenology can throw some light on Frege’s much restricted logical concept of truth.

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REFERENCES Beaney, Michael, 1997. The Frege Reader. Oxford: Blackwell. Bernet, Rudolf, Kern, Iso and Marbach, Eduard, 1996. Edmund Husserl. Darstellung seines Denkens, 2nd ed. Hamburg: Felix Meiner. Dummett, Michael A. E., 1981. Frege. Philosophy of language, 2nd ed. London: Duckworth. — 1988. Ursprünge der analytischen Philosophie. Frankfurt/Main: Suhrkamp. English version: 1993. Origins of Analytical Philosophy. Cambridge, Mass.: Harvard University Press. Føllesdal, Dagfinn, 1958. Husserl und Frege. Ein Beitrag zur Beleuchtung der Entstehung der phänomenologischen Philosophie. Oslo: Aschehoug. English translation: Husserl and Frege. A contribution to elucidating the origins of phenomenological philosophy. 1994. In: Leila Haaparanta, ed. Mind, meaning and mathematics. Essays on the philosophical views of Husserl and Frege. Dordrecht: Kluwer, 3–47. — 1969. Husserl’s notion of noema. The Journal of Philosophy 66, 680–687. Frege, Gottlob, 1918. Der Gedanke. Eine logische Untersuchung. Beiträge zur Philosophie des deutschen Idealismus II, 58–77. — 1962. Grundgesetze der Arithmetik, vol. I and II, 2nd ed. Darmstadt: Wissenschaftliche Buchgesellschaft. — 1964. Begriffsschrift und andere Aufsätze, 2nd ed. I. Angelelli, Hildesheim: Olms. — 1976. Wissenschaftlicher Briefwechsel. G. Gabriel et al., eds. Hamburg: Felix Meiner. — 1983. Nachgelassene Schriften. H. Hermes, F. Kambartel and F. Kaulbach, eds. 2nd revised ed. Hamburg: Felix Meiner. Husserl, Edmund, 1975. Logische Untersuchungen. Erster Band. Prolegomena zur reinen Logik. E. Holenstein, ed. Husserliana XVIII. The Hague: Martinus Nijhoff. — 1984. Logische Untersuchungen. Zweiter Band. Untersuchungen zur Phänomenologie und Theorie der Erkenntnis. U. Panzer, ed. Husserliana XIX/ 1 & 2. The Hague: Martinus Nijhoff. — 2001. The Shorter Logical Investigations. London, New York: Routledge. Kant, Immanuel, 1902. Logik. In: Kants Werke. Akademieausgabe. Berlin, Vol. IX. Malzkorn, Wolfgang, 2001. How Do We Grasp a Thought, Mr. Frege? In: A. Newen, U. Nortmann and R. Stuhlmann-Laeisz, eds. Building on Frege. New essays on sense, content and concepts. Stanford: CSLI Publications, 35–51.

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Mayer, Verena, 2003. Wahrheitswerte und Wahrheitsbegriff. In: D. Greimann, ed. Das Wahre und das Falsche. Studien zu Freges Auffassung von Wahrheit, Hildesheim: Georg Olms, 181–202. Mohanty, Jitendra Nath, 1982. Husserl and Frege. Bloomington: Indiana University Press. Picardi, Eva, 1997. Sigwart, Husserl and Frege on Truth and Logic, or Is Psychologism Still a Threat? The European Journal of Philosophy 5, 162–182. Rojszczak, Artur and Smith, Barry, 2003. Truthmakers, Truthbearers and the Objectivity of Truth. In: J. Hintikka et al., eds. Philosophy and Logic, In Search of the Polish Tradition. Dordrecht: Kluwer, 229 – 268. Simons, Peter, 1995. Meaning and Language. In: B. Smith and D.W. Smith, eds. The Cambridge Companion to Husserl. Cambridge: Cambridge University Press, 108–137. Sokolowski, Robert, 1987. Husserl and Frege, The Journal of Philosophy 84, 521–528. Tugendhat, Ernst, 1967. Der Wahrheitsbegriff bei Husserl und Heidegger. Berlin: Walter de Gruyter. Willard, Dallas, 1995. Knowledge. In: B. Smith and D.W. Smith, eds. The Cambridge Companion to Husserl. Cambridge: Cambridge University Press, 138– 167. Wittgenstein, Ludwig, 1986. Tractatus Logico-philosophicus. Frankfurt/Main: Suhrkamp.

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Grazer Philosophische Studien 75 (2007), 199–215.

THE TRUTH OF THOUGHTS: VARIATIONS ON FREGEAN THEMES Oswaldo CHATEAUBRIAND Pontifícia Universidade Católica do Rio de Janeiro/CNPq Summary In this paper I present an abstract theory of senses, thoughts, and truth, inspired by ideas of Frege. “Inspired” because for the most part I shall not pretend to interpret Frege in a literal sense, but, rather, develop some of his ideas in ways that seem to me to preserve important aspects of them. Senses are characterized as identifying properties; i.e., roughly, as properties that apply, in virtue of their logical structure, to exactly one thing, if they apply to anything at all. When Frege’s analysis of sentences in terms of function and arguments is combined with his analysis of quantification as higher-order predication, all sentences (formal and informal) can be analyzed in various ways as a function (predicate) applied to one or more arguments. This allows for an abstract characterization of thoughts as senses that combine other senses in a uniform way, and whose truth derives from their instantiation by corresponding items of reality.

1. Predication Although Frege (1879, p. 12) is adamant in rejecting the traditional subject-predicate analysis of sentences, it is quite clear that his work contains an important generalization of this distinction. This is due, on the one hand, to his analysis of sentences in terms of function and arguments, and, on the other hand, to his analysis of quantification as higher-order predication. Taken together these two ideas make it possible, and natural, to treat every sentence as a predication. Evidently, as Frege points out (1879, pp. 22–23), a sentence of the form aRb, such as (1) Plato taught Aristotle, can be analyzed as a subject-predicate sentence in many different ways.

For instance: (1a) (1b) (1c) (1d)

[x taught y] (Plato, Aristotle) [x taught Aristotle] (Plato) [Plato taught x] (Aristotle) [Plato Z Aristotle] (taught).

In this notation, derived from model theory, the initial part within brackets is the predicate (function), with the arguments listed within parentheses at the end.1 From the possibility of these different analyses, Frege concludes that it is not correct to talk about the subject-predicate analysis of a sentence. The same distinctions can be made for any sentence, including the sentences of natural languages and the formal sentences of logic built up from atomic sentences by means of quantifiers and connectives. For example, a sentence of the form (2) x(Ax o Bx) can be analyzed in subject-predicate form in any of the following ways: (2a) (2b) (2c) (2d)

[x(Zx o Wx)] (A, B) [x(Zx o Bx)] (A) [x(Ax o Zx)] (B) [xZx] (Ax o Bx)

In fact, Frege’s own notation already contains such an analysis, essentially corresponding to (2a). My notation generalizes Frege’s by allowing many different “amalgamations”. It is unfortunate that these distinctions, so clearly stated by Frege, were largely lost in the further notational development of logic. So much so, that it is quite common to hear that quantified sentences are not of subject-predicate form. In fact, the talk of ‘open sentence’, which assimilates predicates to sentences, completely obscures the predicative nature of sentences. Consider the example (3) Fido is a black dog. 1. I use the notation informally, but the interpretation should be clear in each case.

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The natural way to analyze (3) is as the predication (4) [x is a black dog] (Fido), which, together with the further analysis of the predicate as a conjunctive predicate, becomes (5) [x is black š x is a dog] (Fido). But when we use the standard open sentence analysis it becomes (6) Fido is black š Fido is a dog, which is no longer a predication, but rather a conjunction of the two sentences (7) Fido is black and (8) Fido is a dog. As another example, consider the sentence (9) Nobody can jump more than 10 feet up in the air. The idea that this may be analyzed as a subject-predicate sentence is sometimes ridiculed by asking whether the subject is “nobody”2. But, in a Fregean analysis, (9) is a quantified sentence whose logical subject is (the first-order predicate) ‘x is a person who can jump more than 10 feet up in the air’, and whose predicate is (the second-order predicate consisting of ) the negation of an existential quantification; thus (10) [™xZx] (x is a person who can jump more than 10 feet up in the air).3 2. E.g., by Denyer (1991, p. 13). 3. In Grundlagen (1884, § 21) and elsewhere Frege distinguishes the “grammatical” and the “logical” subject of a given sentence. The grammatical subject of (9) is “nobody”, but the

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My first variation on a Fregean theme, therefore, is to treat every sentence of natural language and of logic as a predication, with the proviso that there are many different ways in which we can distinguish predicate and arguments. In a specific linguistic or logical context it may be more natural, or more appropriate, to use one rather than another analysis, and the results need not be equivalent — in any of various possible senses of equivalence. To illustrate this point consider the sentence (11) Einstein plays violin like Sherlock Holmes, which can be analyzed in two different ways as (11a) [x plays violin like y] (Einstein, Sherlock Holmes) and (11b) [x plays violin like Sherlock Holmes] (Einstein). According to Frege, (11a) is neither true nor false, because it contains as argument a non-denoting name4 — and this is another Fregean theme that I shall adopt. On the other hand, in (11b) the name ‘Sherlock Holmes’ is part of the predicate, rather than an argument, and following up on an idea of Quine (1939, p. 198) we can say that this predicate is well defined, although it contains the name ‘Sherlock Holmes’, which does not have a denotation. Given Conan Doyle’s descriptions in his novels, we have reasonable identity conditions for asserting or denying the predicate ‘x plays violin like Sherlock Holmes’ of a specific individual, say, Einstein. Therefore, while (11a) is neither true nor false, (11b) may be either true or false. A somewhat similar issue arises in connection with negation. Just as under the usual open sentence analysis the conjunctive predication (5) becomes a sentential conjunction, under the open sentence analysis a negative predication becomes a sentential negation. And in certain cases this also affects the truth-value of sentences. Thus, consider the standard (predicate) negation of (11) logical subject is the concept denoted by the grammatical predicate ‘is a person who can jump more than 10 feet up in the air’. 4. See, e.g. Frege (1892, pp. 32–33).

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(11™P) Einstein does not play violin like Sherlock Holmes, and the predicative interpretations (11a™P) [™(x plays violin like y)] (Einstein, Sherlock Holmes) and (11b™P) [™(x plays violin like Sherlock Holmes)] (Einstein). Given the interpretations that I gave before, (11a™P) is neither true nor false, for the same reason as (11a), namely, because the name ‘Sherlock Holmes’ does not denote. And given that (11b) has a truth-value, (11b™P) will have a truth-value, too. With the open sentence interpretation, on the other hand, the negation of (11) is normally interpreted as (11™S) It is not the case that Einstein plays violin like Sherlock Holmes, which is false if (11) is either true or truth-valueless, and is true otherwise. Thus, in contrast to (11a™P), (11a™S) ™([x plays violin like y] (Einstein, Sherlock Holmes)) is false, while (11b™S) ™([x plays violin like Sherlock Holmes] (Einstein)) will have the same truth-value as (11b™P). 2. Definite descriptions In my book Logical Forms (2001, chapter 3), I proposed a theory of definite descriptions that combines ideas of Frege and of Russell. I shall not present that theory in detail here; it suffices to explain its main feature, the sharp distinction between descriptive terms and descriptive predicates.5 5. An account of the main ideas is also given in Chateaubriand (2002).

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A descriptive term is a singular term of the form ‘the F’, and I agree with Frege that such terms denote if there is a unique thing that is F, and do not denote otherwise. A descriptive predicate is a predicate of the form ‘is the F ’; and my view, which is also suggested by some formulations used by Russell, is that such predicates apply to an object if that object is an F, and is the only thing that is an F. I agree with Frege’s analysis of definite descriptions insofar as they occur in subject position. Thus, (12)

the present King of France is bald,

is neither true nor false, because the descriptive term ‘the present King of France’ does not denote. I also agree with those formulations of Russell’s6 in which he analyzes the sentence (13)

Scott is the author of Waverley

as (13P) Scott is an author of Waverley šx(x is an author of Waverley o x  Scott). On the other hand, I disagree both with Russell’s analysis of (12) as (12R) x(x is presently King of France š y(y is presently King of France o y  x) š x is bald), and with Frege’s analysis of (13) as (13F) Scott  the author of Waverley. I also disagree with Russell’s frequent use of (13F), interpreting it in the manner of (12R) by means of existential quantification as

6. See, e.g. Russell (1905, p. 427), (1911, p. 151), (1912, p. 53). For a detailed analysis of Russell’s formulations in “On Denoting”, see Chateaubriand (2005b).

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(13R) x(x is an author of Waverley š y(y is an author of Waverley o y  x) š Scott  x). This distinction between descriptive terms and descriptive predicates will be the basis for my interpretation of Frege’s notion of sense. But before I come to that I need to make a few remarks about predicates and properties. 3. Predicates and properties I take predicates to be characterized by applicability conditions, in the sense that for any legitimate predicate there are conditions associated with it that specify in which circumstances the predicate applies, and in which circumstances the predicate does not apply. In particular, in order to introduce a predicate into the language (natural or formal) it is necessary to specify in some way its conditions of applicability. Evidently, these conditions are not always specified explicitly, or precisely, but there must be some specification. One way in which we can specify the applicability conditions of a predicate is by saying that it applies to things of a certain kind, as given by a sample. This is essentially what Plato does when he introduces forms by reference to what is common to a number of things that exemplify the form. This is also what Kripke does when introducing natural kind terms (predicates) by means of a sample. To be large, for Plato, is to have a certain “character” that is the same in a number of things that are large (Parmenides 132a). To be gold, for Kripke, is to be the same kind of thing as, or to have the same structure as, a number of things that are gold (Kripke 1980, p. 135). I take this idea of applicability conditions, however they may be given, as a general characteristic of predicates. For me, predicates denote properties, or concepts, or functions, in something like Frege’s objective sense of ‘concept’ and ‘function’ — but, unlike Frege, I shall use the term ‘property’ in a broader sense that includes also relations. I think of properties as being identity conditions, in a very abstract conception corresponding to Plato’s idea of a form being what is the same in all the instances. But a characterization of the exact nature of properties will not be necessary for my purposes in this paper, as long as we take them to be objective entities — independent of lan-

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guage and mind — that may be denoted by (or designated by) predicates.7 4. Senses When Frege introduces senses in “On sense and reference” (1892, p. 26) he characterizes them as containing manners of presentation. An object can be presented in different ways, and each way of presenting it is a sense of that object — i.e., a manner of presentation of that object. But even though senses contain manners of presentation, they are “intensional”, and need not present anything. To speak of the series that converges to 0 more rapidly than any other series, is to use a manner of presentation that does not actually present anything. It is quite clear that Frege’s senses are abstract entities that are neither mental nor linguistic, although senses can be grasped mentally, and can be expressed by linguistic signs. If the sense expressed by a linguistic sign — say, a name — contains a manner of presentation of an object, then the sign denotes (or designates, or refers to) that object.8 Thoughts are senses expressed by declarative sentences, which, according to Frege, present one of two objects, the True or the False, if they present anything at all. Senses, including thoughts, may be composed of other senses, and if some of these senses do not present anything, then the complex senses do not present anything either. In particular, a thought that contains non-presenting senses is neither true nor false. Hence, the thoughts expressed by sentences containing non-denoting names — i.e., names that express a sense that does not present an object — are neither true nor false, and so are the sentences expressing them. Although Frege does elaborate his theory of senses to some extent, there are many questions that he does not address. In particular, he says (practically) nothing concerning the nature of senses. What kind of entity is a sense?9 What is the structure of a complex sense? In particular, what is the structure of a thought? How is a complex sense composed of its component senses? These are questions for which we vainly seek answers in Frege’s works, and which have led people to various conceptions of 7. Frege does not allow the use of this terminology for predicates. See, e.g., Frege (1892–95, p. 122). 8. I use ‘denote’, ‘designate’, and ‘refer’ interchangeably. 9. In (1906, p. 192) Frege says that the senses of objects are objects, and the senses of functions are functions (unsaturated).

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senses, the most common of which is the identification of senses with meanings — whatever they may be. Since objects are usually presented either ostensively, or by description, or by a combination of both, there has also been a tendency to infer a connection between senses and definite descriptions, and to think that a sense is “somehow” given by a definite description. I think this idea is correct, though not in the way in which it is usually interpreted. In a famous footnote to “On sense and reference” (1892, p. 27), Frege says: In the case of an actual proper name such as ‘Aristotle’ opinions as to the sense may differ. It might, for instance, be taken to be the following: the pupil of Plato and teacher of Alexander the Great. Anybody who does this will attach another sense to the sentence ‘Aristotle was born in Stagira’ than will a man who takes as the sense of the name: the teacher of Alexander the Great who was born in Stagira. So long as the reference remains the same, such variations of sense may be tolerated, although they are to be avoided in the theoretical structure of a demonstrative science and ought not to occur in a perfect language.

This passage seems to suggest a description theory of proper names, according to which the name ‘Aristotle’ abbreviates a definite description such as ‘the pupil of Plato and teacher of Alexander the Great’ for one person, and the definite description ‘the teacher of Alexander the Great who was born in Stagira’ for another. Of course, even if this were so, it would still not tell us what the sense is. Is it the description itself — the phrase? Is it the sense expressed by the description — i.e., the manner of presentation “contained” in the description? But what is this sense? It is interesting that Frege puts the descriptions after colons, and not within quotes, which emphasizes that the sense is not the phrase. Nor can it be the reference of the phrase, because that is Aristotle, the man himself. And if we take the sense expressed by the name ‘Aristotle’ to be the sense expressed by one or another descriptive phrase, then we are back to square one, because we have no idea of what is the sense expressed by a descriptive phrase. In order to break this impasse, I will introduce my second variation on a Fregean theme. I will say that a sense is an identifying property. More specifically, a singular sense (for an object) is a property denoted by a descriptive predicate of the form

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(14) [Fx šy(Fy o y=x)](x), or a property that is necessarily equivalent to such a property. Thus, corresponding to Frege’s characterizations above we have as candidates for senses expressed by the name ‘Aristotle’ the properties denoted by the descriptive predicates: (15) [x is a pupil of Plato and teacher of Alexander the Great šy(y is a pupil of Plato and teacher of Alexander the Great o y  x)](x) and (16) [x is a teacher of Alexander the Great who was born in Stagira šy(y is a teacher of Alexander the Great who was born in Stagira o y  x)](x). In order to have a brief notation for such predicates, I shall abbreviate (14) as (17) [!xFx](x), and read it as ‘x is the F ’. It is important to notice that the “operator” ‘!x’ is not a variable binding operator, but serves to transform a (first-order) predicate of the form ‘x is F’ into another (first-order) predicate of the form ‘x is the F’. And the same notation can be used for n-ary predicates and for predicates of any order. The reason that the properties denoted by such predicates are said to be identifying is that if they apply to something at all, then they apply uniquely — and this is guaranteed by their logical form. In my view this corresponds well with Frege’s idea that senses contain manners of presentation that may or may not present something. Thus, the property denoted by the predicate (18) [!x(x is a multiple of 7 between 50 and 55)](x), is an identifying property (sense) which does not identify anything; whereas the property denoted by the predicate (19) [!x(x is a multiple of 7 between 50 and 60)](x),

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identifies the number 56, and hence is a sense (manner of presentation) of the number 56. The reason for adding the clause of necessary equivalence in the characterization of senses is that there are identifying properties that are not presented as properties denoted by descriptive predicates. As an example, consider the predicate (20) [x is a prime number greater than 5 and smaller than 10](x). The property denoted by this predicate is necessarily equivalent to the property denoted by the descriptive predicate (21) [!x(x is a prime number greater than 5 and smaller than 10)](x), both being senses (manners of presentation) of the number 7. Evidently, just as we have singular senses for objects we can have n-ary senses for sequences of objects. Thus: (22) [!x !y(x and y are prime numbers greater than 4 and smaller than 10 such that y  x
2)](x, y), is a sense (manner of presentation) of the pair (5, 7).10 And just as we may have manners of presentation for objects, we may have manners of presentation for properties, properties of properties, etc. For example, following Kripke, we may say that a sense (manner of presentation) of the property of being gold can be given by the unary descriptive predicate (23) [Z 1 is the property that is common to the things in this sample in virtue of their physical structure](Z 1), with reference to a specific sample of gold things. Although I do not wish to claim that the notion of sense I introduced above is literally Frege’s notion of sense, I believe that it captures essential features of his notion of sense.11 10. This pair should not be conceived in terms of a set-theoretic characterization. 11. Although it is true that in “Introduction to logic” (1906, p. 192) Frege says that the senses of objects are objects — which goes against my characterization of all senses as properties — he gives no clue as to what objects they might be.

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5. Thoughts As mentioned earlier, thoughts are senses that can be expressed by sentences. And, according to Frege, what they present, when they present anything, are the truth-values the True and the False. My last two variations on Fregean themes will consist of giving an account of the structure of thoughts as senses that combine other senses, and giving an account of the truth of thoughts based on their structure and on their relation to reality. Let us reconsider our initial example (24) Plato taught Aristotle. This sentence expresses a sense, which is presumably composed of the senses expressed by the names ‘Plato’ and ‘Aristotle’, and by the sense expressed by the relational predicate ‘taught’. The senses expressed by the names ‘Plato’ and ‘Aristotle’ are identifying properties denoted by firstorder descriptive predicates. These are predicates such as (15), or (16), but for our purposes it does not matter what they are, because we are interested in giving an account of the structure of the thought, and not in its particular nature. So, let us suppose that there are properties Px and Ax such that the senses expressed by ‘Plato’ and ‘Aristotle’ are (25) [!xPx](x) and (26) [!xAx](x), respectively. Let us suppose, moreover, that the first-order relational predicate ‘taught’ expresses a sense that is a second-order identifying property T (that identifies the relation taught). Thus: (27) [!Z 2 T Z 2](Z 2). My proposal now is that the thought expressed by (24) is the (ternary) sense composed of the senses (25)-(27) as follows: (28) [!Z 2 T Z 2 š !xPx š !yAy š Zxy](Z 2, x, y). 210

Hence, the senses expressed by ‘Plato’, ‘Aristotle’, and ‘taught’ are parts of the thought expressed by the sentence ‘Plato taught Aristotle’, but it is also part of the thought that the relation taught relates Plato to Aristotle, in this order. The main objection12 that one can raise against this analysis is that even if one agrees that it works for “atomic” sentences such as (24), it is not clear how to generalize it to the language as a whole. But this is precisely the reason for my initial discussion of predication. If every sentence has a predicative structure, then (28) does give the general analysis, no matter how complex the sentence may be. The thought expressed by a sentence such as (9) Nobody can jump more than 10 feet up in the air, analyzed as (10) [™xZx] (x is a person who can jump more than 10 feet up in the air), has exactly the same structure as (28), except that the sense expressed by the predicate is a third-order identifying property, and the sense expressed by the subject is a second-order identifying property. Hence, this is a completely general account of the structure of thoughts.13 6. The truth of thoughts Now that we have given a general account of the structure of thoughts, we may consider the question of truth. What is it for a thought to be true, or false, or neither true nor false? For reasons that are not altogether clear, 12. Another objection, raised by Dirk Greimann, is that my formulation makes the identity criteria of Fregean thoughts too fine-grained — it rules out, for instance, that (24) and its passive transformation express the same thought. This is an important objection, that has been raised in a different connection by Rosado Haddock (2004, pp. 120–121), and which I consider briefly in Chateaubriand (2004a, pp. 131–133). The point at issue is really the question of the criterion of identity for senses — and, in particular, for thoughts. In a letter to Husserl (Frege 1980, pp. 70–71), Frege suggests a criterion based on logical equivalence, but as I point out in Chateaubriand (2001, p. 144) this is very problematic. 13. Of course, the senses composing the thought may also be composed of other senses, as is normally the case for senses expressed by definite descriptions.

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Frege introduced the objects the True and the False, and maintained that true and false thoughts are manners of presentation of these objects. This is another doctrine of Frege’s that leaves us searching for explanations. It is easy to understand the technical reasons that may have led Frege to hold this view, but it is very difficult to understand the character of truthvalues as objects. What sort of objects are they?14 Do they have any kind of structure? Frege says (1892, pp. 35–36) that we can think of judgments as dividing truth-values into parts, but acknowledges that this is metaphorical talk. And if we do not know what sort of entities are the truthvalues, how can we understand the notion of a manner of presentation of a truth-value? No matter how valuable the introduction of truth-values may be from a technical point of view,15 this metaphorical talk seems to me to undermine both Frege’s account of thoughts and his account of truth. An obvious suggestion for interpreting the True is to think of it as reality as such.16 A thought is true if it contains a manner of presentation of (some aspect of ) reality. This is in the best tradition of Plato and Aristotle, for whom what is true is what is real.17 Given my analysis of the structure of thoughts, we see that a thought is true if it identifies some aspects of reality that combine in agreement with the structure of the thought.18 In fact, a thought is true if, and only if, it is instantiated. The notion of truth is thus subsumed under the notion of instantiation, which, for Frege (1892–95, p. 119) is the most important logical relation. This is my last variation on a Fregean theme, and it raises the question of what to do with the False. The False cannot be “unreality”, and it is unclear how it can be interpreted in a reasonable way. My suggestion is simply to get rid of it. Consider again sentence (9), as analyzed in (10). These are true sentences, whereas (9*) Somebody can jump more than 10 feet up in the air,

14. This is a central theme of Chateaubriand (2001), especially in chapters 2, 8, and 12. A specific difficulty concerning truth-values as objects is also discussed in Chateaubriand (2003). 15. A point that Frege emphasizes in (1893, p. x). 16. Gödel (1944, p. 129) seems to suggest this by comparing Frege’s postulation of the True to Parmenides’ One. 17. See, e.g., Cratylus 385b and Metaphysics * 7 (1011b 26). 18. This is close to Aristotle’s view in Metaphysics 4 10.

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with the predicative analysis (10*) [xZx] (x is a person who can jump more than 10 feet up in the air), are false. In my view, (10*) is false because (10) is true; i.e. because its predicate negation is true. And this is, again, a completely general analysis of falsity, because due to the predicative nature of all sentences, the predicate negation of a sentence is always defined. Hence, to every thought corresponds a negative thought expressed by the predicate negation of a sentence expressing the thought. The negative thought corresponding to (28) is (28™) [!Z 2NTZ 2 š !xPx š !yAy š Zxy](Z 2, x, y), where the predicate !Z 2NTZ 2 identifies the negative relation not-taught. Hence, a thought is true if, and only if, it is instantiated, and a thought is false if, and only if, its corresponding negative thought is instantiated.19 Other thoughts, as well as sentences expressing them, are neither true nor false. Thus, with Frege’s analysis, with which I agree, (12) expresses a thought that is neither true nor false. This is also the case for (11a), and for many other sentences and thoughts which contain parts that do not connect appropriately with aspects of reality. 7. Conclusion I have presented here an abstract theory of senses, thoughts, and truth, which has a purely ontological character, and that may seem very distant from the usual linguistic and/or epistemological interpretations of Frege’s notion of sense and of thought. It seems quite clear to me, however, that there is a good basis in Frege’s works for attributing to him such an abstract theory, even if not necessarily worked out in exactly the way I did. There are other aspects of Frege’s view that I have not developed here. In particular, the connections of the abstract ontology with language and with mind. Senses are expressed by signs and are grasped by us. Frege’s account 19. Some issues about predicate negation and negative properties are discussed in Vallée (2004) and Chateaubriand (2004b).

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of these notions is rather elusive as well, and I have tried to develop them by combining his views with the views of other authors, especially Russell and Kripke.20 21 REFERENCES Aristotle, Metaphysics. Translated by D. Ross. In: The Basic Works of Aristotle, edited by R. McKeon, New York: Random House, 1941. Chateaubriand, Oswaldo, 2001. Logical Forms: Part I —Truth and Description. Campinas: Unicamp (Coleção CLE). — 2002. Descriptions: Frege and Russell combined. Synthese 130, 213–226. — 2003. How is it determined that the True is not the same as the False? Manuscrito 26, 347–357. — 2004a. Syntax, semantics and metaphysics in logic: reply to Guillermo Rosado Haddock. Manuscrito 27, 129–140. — 2004b. Negation and negative properties: reply to Richard Vallée. Manuscrito 27, 235–242. — 2005a. Logical Forms: Part II – Logic, Language, and Knowledge. Campinas: Unicamp (Coleção CLE). — 2005b. Deconstructing “On Denoting”. In: On Denoting: 1905–2005, edited by B. Linsky and G. Imaguire, Munich: Philosophia Verlag, 361–380. — forthcoming. Sense, reference, and connotation. Manuscrito. Denyer, Nicholas, 1991. Language, Thought and Falsehood in Ancient Greek Philosophy. London: Routledge. Frege, Gottlob, 1879. Begriffsschrift, a Formula Language, Modelled upon that of Arithmetic, for Pure Thought. In: J. van Heijenoort, ed. From Frege to Gödel, Cambridge, Mass.: Harvard, 1967. — 1884. The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number. Oxford: Blackwell, 1950. — 1892. On sense and reference. In: Translations from the Philosophical Writings of Gottlob Frege, edited by M. Black and P. Geach, Oxford: Blackwell, 1960, 56-78. (References to the original pagination.) — 1892–95. Comments on sense and meaning. In: Posthumous Writings, edited by H. Hermes, F. Kambartel and F. Kaulbach. Oxford: Blackwell, 1979, 118–125. 20. Some of these ideas are formulated in Chateaubriand (2001, chapter 11) and Chateaubriand (2005, chapter 13), and are developed in Chateaubriand (forthcoming), which complements the present paper. 21. I am grateful to Dirk Greimann for many editorial suggestions on an earlier draft that helped to improve both the content and the style of my contribution to this volume.

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— 1893. The Basic Laws of Arithmetic: Exposition of the System. Translated and edited by M. Furth, Berkeley and Los Angeles: University of California, 1964. — 1906. Introduction to logic. In: Posthumous Writings, edited by H. Hermes, F. Kambartel and F. Kaulbach. Oxford: Blackwell, 1979, 185-196. — 1980. Philosophical and Mathematical Correspondence. Edited by G. Gabriel et al., Oxford: Blackwell. Gödel, Kurt, 1944. Russell’s mathematical logic. In: The Philosophy of Bertrand Russell, edited by P. A. Schilp, New York, N.Y.: Tudor, 125–153. Reprinted in Kurt Gödel: Collected Works, vol. 2, edited by S. Feferman et al, Oxford: Oxford University Press, 1990, 119–141. (References to the original pagination.) Kripke, Saul, 1972. Naming and Necessity. Cambridge, Mass.: Harvard, 1980. Plato, Cratylus. Translated by H. N. Fowler. In: Plato VI, The Loeb Classical Library, Cambridge, Mass.: Harvard, 1953. — Parmenides. Translated by F. M. Cornford. In: F. M. Cornford, Plato and Parmenides, London: Routledge, 1939. Quine, Willard Van Orman, 1939. A logistical approach to the ontological problem. Journal of Unified Science 9, 84–89. Reprinted in: The Ways of Paradox and Other Essays, revised edition, Cambridge, Mass.: Harvard University Press, 1976, 197–202. Rosado Haddock, Guillermo, 2004. Chateaubriand on logical form and semantics. Manuscrito 27, 115–128. Russell, Bertrand, 1905. On denoting. Mind 14, 479-493. Reprinted in: The Collected Papers of Bertrand Russell, vol. 4, edited by A. Urquhart, London: Routledge, 1994, 415–427. — 1911. Knowledge by acquaintance and knowledge by description. Proceedings of the Aristotelian Society 11, 108–128. Reprinted in: The Collected Papers of Bertrand Russell, vol. 6, edited by J. G. Slater, London: Routledge, 1992, 148–161. — 1912. The Problems of Philosophy. London: Williams and Norgate. Vallée, Richard, 2003. On not being a dentist. Manuscrito 27, 227–233.

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Grazer Philosophische Studien 75 (2007), 217–236.

FREGEAN PROPOSITIONS, BELIEF PRESERVATION AND COGNITIVE VALUE 1 Marco RUFFINO Universidade Federal do Rio de Janeiro, Brazil Summary In this paper I argue indirectly for Frege’s semantics, in particular for his conception of propositions, by reviewing some difficulties faced by one of the main contemporary alternative approaches, i.e., the direct reference theory. While Frege’s semantics can yield an explanation of cognitive value and belief-preservation, the alternative approach seems to run into trouble here. I shall also briefly consider the question of whether epistemic issues should be of any concern for semantics, i.e., whether the feature mentioned above should really be regarded as an advantage of Frege’s theory.

Certainly one of the most important aspects of Frege’s conception of truth is his view about the appropriate truth-bearers, i.e., about which kind of entities can properly be called true or false: thoughts or propositions (which are, for him, the senses of complete sentences) are the truth-bearers, and not sentences, facts or anything of the sort. This view is particularly important because it ties in with his views on the nature of belief and of cognitive significance. As I see it, Frege holds three fundamental theses in this connection. First, a complete sentence expresses a proposi1. This paper was first presented in Curitiba at the Universidade Federal do Paraná (UFPR, Brazil) in a colloquium celebrating one hundred years of Russell’s “On Denoting” in August 2005. It was also presented in the Instituto de Investigaciones Filosóficas at the Universidad Nacional Autónoma de México (UNAM) and at the Universidad Autónoma Metropolitana (UAM) in November 2005, and in the symposium “Frege’s Conception of Truth” at the Universidade Federal de Santa Maria (UFSM, Brazil) in December 2005. I want to thank all audiences for comments that helped to clarify many points, especially Michael Beaney, Oswaldo Chateaubriand, Maite Ezcurdia, Dirk Greimann, André Leclérc, Danielle Macbeth, Silvio Pinto and Hans Sluga. Research for this paper was supported by a grant from CNPq (Brazil), to whom I express my gratitude as well.

tion (which he also calls “thought”) which just is a sense. According to his view, a proposition is a composition of (more elementary) senses, so that the proposition expressed by a sentence is wholly determined by the senses expressed by its parts, and no other entities besides senses can take part in a proposition. It is this propositions that we apprehend when we grasp the meaning of a sentence. For example, the proposition expressed by ‘The greatest soccer player of all times was born in Minas Gerais’ can be conceived as a composition of the senses of ‘greatest soccer player of all times’ and ‘being born in Minas Gerais’. This first thesis is primarily semantic, but it has important consequences for epistemology since Frege identifies the cognitive significance of a sentence with the proposition it expresses. The second Fregean thesis is that a belief such as reported by ‘Mary believes that Pelé was the greatest soccer player of all times’ can be seen as a relation holding between a subject (Mary) and a proposition. The third thesis is that a sentence occurring within the scope of a cognitive verb (e.g., ‘believes that’, ‘expects that’, ‘doubts that’, etc.) does not have its ordinary reference (which, in Fregean semantics, would be its truthvalue) but refers to its ordinary sense, i.e., to a proposition. The first thesis explains the difference in cognitive value between ‘The greatest soccer player of all times was born in Minas Gerais’ and ‘Pelé was born in Minas Gerais’. Somebody may believe the second sentence and not believe the first because he knows a little about Pelé’s biography, but doesn’t know much about soccer history (or perhaps got the wrong idea from some Argentinean newspapers that Maradona, and not Pelé, was the greatest soccer player of all times). The second thesis explains the nature of belief, and also why there can be preservation of belief in different contexts: it suffices that the relation continues to hold between the same subject and the same propositional content. That is to say, I can retain today a belief that I had yesterday if I am related today to the same proposition that I was related to yesterday. The third thesis explains the failure of substitution salva veritate of co-extensional terms within belief contexts (e.g., ‘Mary believes that the greatest soccer player of all times was born in Minas Gerais’ might be true, while ‘Mary believes that the greatest soccer player of all times was born in the state where the Vale do Rio Doce Mining Company was first created’ might well be false, in case she is well informed about soccer history, but not as well informed about economic history). Frege’s theory of proposition (and propositional elements) thus emerges from a combination of semantic and epistemic worries, and its explanatory power in epistemic issues is regarded by him as a kind of test of its

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adequacy. Indeed, one of the most attractive features of his theory is that it provides a uniform explanation of language and knowledge. There is a contrast between this perspective and that of the so-called direct reference theory. In this article I shall first recapitulate this alternative to Fregean semantics as developed by Kaplan and Perry, among others.2 Although Kaplan and Perry are not the only philosophers responsible for the direct reference theory, Kaplan and Perry’s works can be seen as paradigmatic in many respects, and I will concentrate on them. As we will see, they have a different view from Frege’s regarding the nature of the truth-bearers (propositions), with important consequences for the nature of belief and cognitive significance. Next I will introduce two problems that arise for this alternative approach, and which have a much less natural solution than the one offered by the Fregean theory sketched above. I shall not offer a solution myself (nor will I claim that no solution at all can be found within this framework); I will simply point out the difficulties for the Kaplan/Perry approach and the relative merits of Frege. Finally, I will briefly consider (taking up a discussion raised by Wettstein (1986)) whether this conclusion can be of any relevance for semantics, i.e., whether it should really be considered as an advantage of Frege’s theory against Kaplan and Perry’s. Although most of this paper will be concerned with the details of Kaplan and Perry’s alternative approach, it is meant as a whole as an indirect defense of Frege’s traditional semantics (in particular of his view on the nature of propositions) by reviewing some difficulties faced by the main alternative. I. Indexicals: Kaplan and Perry The theory based on the three Fregean theses mentioned above would work perfectly well, both on the semantic and on the epistemic level, were it not for the “anomaly” represented by indexical terms, i.e., terms whose referents depend in an essential way on the context of utterance. Following a list presented by Kaplan (1977), we have as typical of this class of terms demonstrative pronouns like ‘this’, ‘that’, personal pronouns like ‘I’, ‘you’, etc., adverbs like ‘here’, ‘now’, ‘today’, etc., and adjectives like ‘actual’, among many other expressions. Some of these terms require 2. There are actually many relevant differences between Kaplan’s and Perry’s theories which I shall skip here; in the aspects that are relevant for our discussion here they are quite similar.

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an extra-linguistic element in order to perform their semantic task; this is the case for demonstratives like ‘this’ or ‘that’, which require an accompanying demonstration (typically a pointing or something analogous), and otherwise refer to nothing. Other terms like ‘I’, ‘today’, ‘now’ and ‘here’ require no such demonstration; constitutive facts about the utterance (i.e., the agent, the place and the time) are sufficient to guarantee a reference to these terms.3 Frege was well aware that the mere occurrence of terms of this kind in a sentence is not enough to express a determinate sense (and, hence, to guarantee a reference). In a famous passage of “Der Gedanke” (1918, p. 64) he points out that, in the cases of sentences in which indexicals occur, knowledge of some contextual (non-linguistic) elements such as an act of pointing, or the time of utterance, might be necessary in order to fully grasp the thought expressed. He does not say that the time or the act of pointing themselves are elements of the thought, since instants of time and gestures are not senses, but rather that some indication of the time of the utterance, and some indication of whatever is expressed by the pointing is part of the content of the thought expressed by the sentence. Presumably his view is that indexical expressions acquire a sense in each particular context of use, and this sense is the semantical contribution the indexical makes to the proposition expressed in that particular context. Hence, by contrast with proper names, indexicals express an indefinite number of different senses in different contexts.4 In a sequence of seminal essays, Kaplan (1977 and 1989) and Perry (1977 and 1979) have developed a semantical approach to indexicals (which they generalize to the rest of language as well) that, while preserving 3. Presumably utterances can be made without an accompanying demonstration; but there is no utterance without a speaker, a place and a time. These are the constitutive facts of it. 4. As Frege points out in another essay (1892), there is actually some fluctuation of senses associated with proper names, in that a speaker may associate with the name ‘Aristotle’ a sense that is different from that associated with the same name by another speaker. Hence, we might have a different understanding of ‘Aristotle’ in a history class (e.g., we might think of him as the teacher of Alexander the Great) from that in a philosophy class (here we think of him as the author of the Metaphysics, for instance). However, this dependence on the context is merely accidental, differently from that kind of dependence that we have in the case of terms like ‘here’ and ‘now’, for in these cases the dependence is essential. Whereas it is not absurd to suppose that we might all agree in associating one single sense to ‘Aristotle’ in all contexts, it is absurd to suppose that ‘now’ or ‘here’ could have one single sense in all contexts. In an ideal language, all names should have one single and immutable sense. Although Frege says nothing about the role of indexicals in an ideal language, it is plausible to suppose that, for him, they should have none, and be eliminated altogether and replaced by non-indexicals.

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the Fregean spirit, differs substantially from the classical Fregean doctrine of sense and reference. The main difference between their approach and Frege’s semantics concerns the nature of the contribution made by an indexical to the proposition expressed by a sentence in which it occurs. As we saw, Frege’s view presumably is that this contribution is a Fregean sense that changes from context to context (although he is never explicit about this). One of the problems with Frege’s view, according to Kaplan and Perry, is that it cannot explain the common understanding that all speakers have associated with indexicals like ‘I’ or ‘now’. According to the Fregean view, a different sense (and a different reference) is associated with ‘now’ each time it is uttered, and different senses (and different references) are associated with ‘I’ for different speakers. Strictly speaking, there seems to be nothing in common between the sense of ‘now’ said today and the sense of ‘now’ said a year ago (unless we are using ‘now’ in a special way, referring to a large period of time, as in ‘a long time ago it was impossible to process information quickly, but now we have computers’), and the same is true of ‘I’ as used by two different people: what is expressed by a sentence using ‘I’ in the two cases is completely different. But this seems counter-intuitive. Another problem with the Fregean view, according to Kaplan and Perry, is that a sense supposedly associated with a term like ‘today’ might not be sufficient to pick its reference correctly. Suppose I think that today is August 28, although it is actually August 29. So when I say ‘today’ I associate a certain sense with the word that has August 28 as reference, and nevertheless the reference of my utterance is August 29, i.e., the reference of ‘today’ does not depend on the sense that I associate with the term. Another interesting example presented by Perry is that of Heimson, a somewhat crazy fellow who believes himself to be David Hume. He sees himself so perfectly as Hume that he has all the mental experiences that Hume would have. Hence he has exactly the same contents associated with ‘I’ that Hume would have. Nevertheless, that same content does not guarantee that he refers to Hume when he thinks ‘I’, for only Hume can refer to Hume by thinking ‘I’. Heimson can only refer to Heimson by thinking ‘I’, despite the fact that he associates the same sense to ‘I’ as Hume does. On the other hand, senses do not seem to be necessary to guarantee a reference for an indexical. We can refer to a day by means of ‘today’ without thinking of any sense, as in the situation in which I wake up after a long coma and say ‘today is sunny’ without having any idea about which day I am referring to. Or if I wake up in a dark place without

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having any idea of where I am, and say ‘it is dark in here’. Finally, Kaplan and Perry both reject Frege’s idea that every speaker is presented to himself in a particular and primitive way in which he is presented to no one else, and to which no one else but the speaker has access. Kaplan and Perry propose a theory in which the semantic functioning of indexicals is not explained in terms of the binomial sense/reference as on Frege’s theory, but rather in terms of the binomial character/extension (Kaplan) and role/extension (Perry). Inspired by the Carnapian explanation of intensions as functions that associate extensions to possible worlds, Kaplan describes the character of an indexical as a function that assigns an extension to each context. The character of the indexical ‘I’, for example, is a function that assigns to each context of utterance the agent of it; the character of ‘here’ is a function that assigns to each context the place of utterance; the character of ‘now’ is a function that assigns with each context the time of utterance, and so on. This model can be extended to non-indexical terms like proper names, since these can be seen as having a constant function (i.e., a function that associates the same extension to every context) as character. Because the character of a sentence is a composition of the character of its parts, the character of (say) ‘Marco Ruffino is in Mexico City now’ is a function that assigns to each context, in each possible world, an extension (truth-value). It assigns the False to the context represented by the actual world on April 17, 2006, and the True to the actual world on July 26, 2006. In other possible worlds the opposite could happen, since I could have been in Mexico City on April 17 but not on July 26 of that year. A sentence containing an indexical, when uttered in a particular context, determines what Kaplan calls a proposition, but what we have here is not a Fregean proposition. For indexicals do not, according to Kaplan, have a Fregean sense, and hence cannot contribute with a Fregean sense to the proposition. Neither can their contribution be the corresponding character, for if it were so, I would be expressing the same proposition on April 17 and on July 26 by saying ‘I am in Mexico City now’, and the same proposition would be both true and false, which is impossible. Hence, neither the character nor a Fregean sense is the semantical contribution of an indexical; it is rather the extension that that indexical selects in a particular context. Hence the proposition expressed by a sentence like ‘I am in Mexico City’ said by me has myself as component, together with the property of being in Mexico City. The conception of proposition that Kaplan has in mind is the Russellian one, according to which a proposition is typically formed by an individual (or

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an n-tuple of individuals) and a property (or an n-ary property). What a competent speaker understands as the meaning of an indexical is not the infinity of senses that it might express, nor is it the infinity of extensions (or Russellian propositions, when it is part of a sentence) that it might pick out, but rather its character, which is the same for all speakers in all contexts. Hence, David Kaplan and I understand the same thing when we both say ‘I live in Los Angeles’, although the Russellian proposition isolated is different in each case, since it is true for him, and false for me (and the same proposition could not be both true and false). Perry’s inspiration is not the Carnapian model but rather an intuition about the nature of indexical terms explored by Reichenbach. According to the latter, what is essential to an occurrence of an indexical is that it cannot be understood or have its meaning explained without reference to the occurrence itself. This is why Reichenbach called indexicals tokenreflexive expressions. For example, the meaning of a particular occurrence of the sentence ‘I am a tango-singer’ (call this sentence s) is given by ‘the producer of s is a tango singer’, the meaning of a particular occurrence of ‘here is Mexico City’ (call this sentence t) is given by ‘the place where t is produced is Mexico City’, the meaning of a particular occurrence of ‘now is seven o’clock’ (call this sentence u) is give by ‘the moment when u is produced is seven o’clock’, and so on. In cases like these, the meaning (and, hence, the truth-condition) of each particular occurrence of a sentence that includes an indexical is necessarily spelled out by reference to the occurrence itself. Perry starts just from this intuition, although he focuses less on the production of the token than on the utterance of it. This is important, according to him, because a speaker might utter something using a token that was not produced at the time of the utterance. For example, I might place several times the same note ‘I’ll be back in 15 minutes’ at the door in my office. What is relevant here is not the time in which ‘in 15 minutes’ was written, but the time in which the note is placed at the door. We might also imagine that I use a note written by someone else (the former occupant of my office): what is relevant here is not the agent who actually produces the token but the agent who uses it in the utterance. Reichenbach’s original intuition is not fully endorsed by Perry. One of the reasons for Perry’s partial disagreement is the following: according to Reichenbach, the meaning of ‘I am in Mexico City’ is given by ‘the producer of this token is in Mexico City’, and this seems to require that, for the sentence to be true, someone actually produces (or utters)

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the token in Mexico City. That is to say, the sentence could only be true in those possible worlds in which the agent actually utters the sentence in Mexico City. But this seems counter-intuitive, since it seems possible for the sentence to be true in those possible worlds in which the agent says nothing in Mexico City. In other words, the truth of an utterance containing the token does not seem to depend on its concrete existence. In the same way, ‘I exist’ is true even in those possible worlds in which I do not utter this sentence. According to Perry, an essential feature of a meaningful occurrence of an indexical is that it carries a specific semantic role that elements of the relevant context might assume. The semantic role associated with ‘I’ is assumed by whoever utters the token in a specific context; the semantic role of ‘now’ is assumed by the time of utterance in that context; the semantic role of ‘you’ is assumed by the addressee of the utterance in that context, and so on. Therefore, the understanding that a competent speaker has of an indexical is related neither to the sense nor to the reference of it in a particular context but instead to the role associated with it, in the same way that the understanding of a role in a theater play is not related to the particular person that assumes that role in a particular performance, since that role might be assumed by different actors. Frege’s worry about indexical expressions seems to have been only marginal. His semantical model was thought as primarily applying to expressions with an eternal sense and reference like ‘the German president in July 2005’ or ‘the successor of zero’. The sentences that interest Frege are primarily those that express what he calls eternal thoughts (which are also eternally true or false). He most likely sees the occurrence of indexicals as a pathological feature of natural language, along with other pathological features such as ambiguity, vagueness, etc., although it can be accommodated within his theory of sense and reference. However, for contemporary thinkers like Perry, Kaplan and Castañeda, among others, the opposite is true, i.e., indexical reference is actually the normal and paradigmatic case of reference; non-indexical expressions are exceptional and hard to find outside the realm of mathematics. And the reason for this is that the use of indexical expressions is essential to our location in space and time and hence to our connection to the world. Let us imagine someone for whom there is no difference in the cognitive states associated with ‘here’ and ‘there’: in this case we can hardly say that this person has a sense of spatial orientation. Or someone who does not grasp the difference between ‘today’ and ‘yesterday’: we then can hardly say that this person has a proper

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sense of chronological ordering. Or someone who does not distinguish the states corresponding to ‘I’ and ‘you’ (perhaps a newborn baby looking at his mother): in this case we can hardly say that this person has a sense of self-identity. According to Kaplan and Perry, the cognitive states regarding those sentences that contain indexicals are, furthermore, related primarily to the character, and not to the proposition. For instance, if I am amnesiac, I might have no idea whatsoever about my identity (and, hence, do not associate any obvious sense to the term ‘I’), but nevertheless I might be in the cognitive state associated with ‘I am here’. Another essential aspect of Kaplan’s and Perry’s view concerning indexicals is that human action is primarily guided by characters and not by propositions. To use a famous example of Perry, suppose you are about to be attacked by a bear, so you think ‘I am about to be attacked by a bear’ and I think ‘you are about to be attacked by a bear’. We both entertain the same proposition (which has you as component), but through different characters, and consequently we act differently (you try to be as still as possible, and I run to find help). Now suppose we are both about to be attacked by a bear, so we both think ‘I am about to be attacked by a bear’. Here we entertain different propositions (one has me as component and the other has you) through the same character, and consequently we both act similarly (we both try to be as still as possible). These examples are evidence, according to Perry (and Kaplan, following him), that action is directed not by the proposition apprehended, but rather by the character under which it is apprehended. II. Belief Preservation What does it mean to preserve an indexical belief through different contexts? This question can be better understood by contrasting this case with a belief that is not sensitive to context. If yesterday I believed that the Pythagorean Theorem is true, I believed an eternal and unchangeable proposition, and today I might well continue to believe this same proposition. The explanation of the preservation of my belief seems to be quite simple in this case: the proposition I believed yesterday is identical to the proposition I believe today; it does not change with time. A sentence that is not sensitive to context also has a character, according to Kaplan, but we have here a character that is a constant function, i.e., one that associates the same extension (truth-value) with every possible world. Hence, in the case of the Pythagorean Theorem (which is supposedly insensitive

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to contextual elements), I believed yesterday the same proposition under the same character as I do today. But what now of a belief about a context-sensitive content, e.g., my belief yesterday reported in ‘today is my birthday’? What does it mean to say that today I retain the same belief? I certainly do not retain the same belief if I think the same character today, i.e., ‘today is my birthday’, for in that case I would be thinking that the present day is my birthday, rather than that yesterday was my birthday, as the original belief seemed to be. It also cannot be by thinking the eternal proposition expressed by ‘July 26th is my birthday’ that I retain the belief because merely thinking this proposition seems to leave out something essential to my original belief, which was not simply that a certain day is my birthday, but also that this day is today, the day I am currently living. There are variations of this example. If I meet Breno Hax at a party, talk to him and form the belief expressed by ‘you are a tango singer’, and later change my interlocutor, what does it then mean to say that I retained my original belief? It cannot be simply by thinking again ‘you are a tango singer’ (since ‘you’ has a different extension now), nor it is by believing the eternal proposition expressed by ‘Breno Hax is a tango singer’, for this proposition lacks something that is essential to my original belief, namely, that the person to whom I attribute such a nice talent is (at a certain time) my addressee, someone who has a vivid presence for me at that moment. Kaplan (1989) raised this question as a problem for what he called “cognitive dynamics”. There seems to be a solution suggested by Frege in “Der Gedanke”. He says that “[i]f someone wishes to say today something that he expressed yesterday using the term ‘today’, he should replace it with ‘yesterday’. Although the thought might be the same, its verbal expression must be different, so that differences in sense that could be produced by the different times can be cancelled” (Frege 1918, p. 64). Adapting this suggestion to Kaplan’s theory we might say that to retain a context-sensitive belief we must believe the same (Russellian) propositional content, but by way of different characters. More precisely, we have to believe the same propositional content through an appropriate sequence of different characters. Some sequences of characters are clearly inadequate and indicate loss of the original belief. For example, if I wish to indicate the retention of my belief that today I have an important meeting, the adequate sequence seems to be: yesterday, I thought of the day of the meeting under the character of ‘tomorrow’; today I think of it under the character of ‘today’; tomorrow I think of it under the character of ‘yesterday’; the day after

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tomorrow I think of it under the character of ‘the day before yesterday’, and so on. This model seems to yield a simple and elegant explanation for our capacity of retaining context-sensitive beliefs throughout different contexts: it is simply the capacity of grasping the same content through appropriate sequences of characters. Despite the elegance and intuitive appeal of this solution, Kaplan rejects it using the fictional example of Rip van Winkle, a character of a tale by Dietrich Knickerbocker, who fell asleep for twenty years, and wakes up without knowing that he slept for such a long time. Suppose that the day he fell asleep he believed the proposition expressed by ‘today was a fine day’, and suppose that the day he wakes up twenty years later he thinks ‘yesterday was a fine day’. The proposition expressed by the latter is not that the day when he fell asleep was a fine day, but instead the proposition that the day before he woke up was a fine day. According to the explanation suggested in Frege’s remark, we should say that Rip van Winkle did not retain his original belief, since the character under which he now thinks points to a different proposition from the original one. However, this seems counter-intuitive, for in a way it seems clear that he did retain some form of belief. Again, we can vary this example. Suppose I am talking to Breno Hax at the party and acquire the belief expressed by ‘you are a tango singer’. Later, someone else assumes Breno’s position in front of me without my noticing, perhaps because this person looks pretty much like Breno, or perhaps because I have had a few more drinks than I should. When I formulate the same belief again with ‘you are a tango singer’ the explanation above would lead us to think that I did not retain my original belief, which is counter-intuitive, for I (like Rip van Winkle) certainly did retain something, without noticing the change in context. Kaplan’s response is purely negative: he simply says that the Fregean answer is unsatisfactory on the basis of examples like these, since a mistaken apprehension of contexts is possible without a rupture in belief. But he does not offer an alternative account, i.e., he merely points out the problem without offering himself a criterion of belief preservation. One should keep in mind that the suggestion above was actually not proposed by Frege as a solution for the problem of continuity of belief, since the problem does not arise for him. The point of Frege’s remark is just that some adjustments are necessary if we want to express eternal (Fregean) propositions using indexical terms, where such eternal propositions are the objects of belief for him. It is only when we abandon this principle and postulate the two dimensions, character and proposition (and recognize

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that they do not always go together) that we need an additional explanation for the continuity of belief. Perry (2000a) tries to solve the problem of Rip van Winkle by means of an analysis of what he calls “information games”. These are, on Perry’s conception, ways in which beliefs about an object acquired by means of direct epistemic contact with it can later be used to re-identify this same object again in a new situation, after the object has been absent from our perceptual field for some time. According to Perry, the re-identification of this object usually requires that we recall some stored information about it, and this can be done in different ways, each of them constituting a form of information game. (I shall not here review the different forms of information games described by Perry). Once we acquire an indexical belief about an object by direct epistemic contact with it (e.g., the belief about this man in front of me, to whom I have just been introduced, that he is a tango singer), this becomes a kind of eternal (i.e., non-indexical) belief (that the man I met at that party was a tango singer), and this belief might be used again in a situation in which I want to identify an object (a person) that I meet two years later. The old belief is used in the re-identification, but this doesn’t mean that the object (or the person) is being correctly identified: the object now met might be incorrectly identified, although the old belief was not lost along the way. In the Rip van Winkle case, he uses his old belief (that the day he fell asleep was a fine one) in order to re-identify the day before today, but he is misidentifying that day, although his old belief is still there. Hence, we have an explanation of why the belief was retained, while the proposition expressed by Rip van Winkle is not the one he originally believed. There seems to be something odd about Perry’s solution. For the original problem raised by Kaplan was that of the preservation of a context-sensitive belief, and the Rip van Winkle case was one in which we seem to have preservation associated with misapprehension of the context. Perry’s solution seems to require that contextual beliefs are turned into eternal ones; but if they are made eternal they are not indexical anymore, at least not in the interesting original sense of Kaplan’s formulation. In any case, it seems that the solution proposed is far less natural than the one we would have in Frege’s system, for there we do not actually have indexical beliefs, but only indexical expressions which, together with a context, yield eternal propositions, which are after all the objects of belief. The problem raised (and left unsolved) by Kaplan and solved at a high cost by Perry simply does not arise in Frege’s framework.

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III. Cognitive value As I said before, one of the attractive aspects of Frege’s semantics is that it yields a simple and elegant explanation of the cognitive difference between expressions that have the same extension as, e.g., the expressions ‘the Morning Star’ and ‘the Evening Star’ do: the difference lies in the sense associated with each. It is also thereby explained why the sentence ‘the Morning Star is the same as the Evening Star’ has cognitive value, while ‘the Morning Star is the same as the Morning Star’ is trivial. An important detail here is that Frege does not restrict this explanation only to definite descriptions, but extends it to proper names as well. The cognitive value of, say, ‘Tully is Cicero’ (as opposed to ‘Tully is Tully’) is explained, according to him, by the fact that ‘Tully’ and ‘Cicero’ express different senses. The theory of Kaplan and Perry for indexicals is part of a broader perspective in contemporary philosophy of language known as referentialism (or direct reference theory) which was developed in the late part of the twentieth century inspired by the works of Donnellan, Putnam and Kripke, among others. The view that Kaplan and Perry advocate for demonstratives and indexicals in general, i.e., that their semantic contribution to a proposition is not a Fregean sense but instead an extensions, is advocated by other direct reference theorists like Kripke and Putnam for proper names and natural-kind terms, and by Donnellan for (a particular use of ) definite descriptions. The idea that proper names are rigid designators typically dispenses with Fregean senses. But although some of these thinkers have focused on the similarities between proper names and demonstratives, there are also important differences between them. For example, strictly speaking, there is nothing in common between the semantic contents of two rigid designators naming two different objects (e.g., two rigid designators ‘Breno’ naming two different persons) except the same combination of letters or sounds. The word ‘Breno’ is, hence, ambiguous, in that it might serve on two different occasions as two different proper names designating two distinct objects. An explanation concerning a particular use of ‘Breno’ could, in principle, eliminate the ambiguity. In the case of indexicals, on the other hand, we do not have ambiguity properly speaking. Two different occurrences of ‘you’ (pointing at different people) have two different references, but still there is a common part shared by their meanings, which is the character (in Kaplan’s terminology) or the role (in Perry’s terminology), and it is this common part that is grasped by competent speakers, independently of the context

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of utterance. It might be that we need to know some aspect of the context of utterance in order to eliminate the ambiguity of a name like ‘Breno’, but in this case the dependence on the context is merely accidental. Once we know the relevant information about the particular context of use we should be able to grasp the meaning of the name and know its extension in every other possible context. In the case of indexicals, on the other hand, the dependence on the context is not accidental, but essential. That is to say, even if we clearly grasp the meaning (character) of an indexical like ‘I’, for instance, we know that the context will be essential in fixing its extension. Despite these differences between indexicals and names, the utterances (1) (2)

‘I am a tango singer’ (uttered by Breno Hax) and ‘Breno is a tango singer’ (uttered by someone referring to Breno Hax)

express the same (Russellian) proposition according to referentialism, since ‘I’ said by Breno Hax, and ‘Breno’ in this particular use refer to the same object. How can we then account for the difference in cognitive value between both sentences? That there is indeed a difference in cognitive value can be shown by the following two imaginary situations (adapted from Perry’s original examples (1997)). In situation 1, Breno Hax is watching a concert in a tango-place in Buenos Aires and talking to an interlocutor who doesn’t know him. They do not agree concerning the quality of the performance. So, in order to reinforce his authority on the matter, Breno says ‘I am a tango singer’. This utterance transmits a kind of information to the interlocutor, which is attested to by his reaction: he feels intimidated and yields to Breno’s opinion about the performance. In situation 2, Breno is at the same tango-place, sitting next to the same interlocutor who doesn’t know anything about his identity. The interlocutor, having heard the crowd calling for someone called ‘Breno’ to perform, turns and asks Breno who the person is that the crowd is calling for. Since Breno is polite enough to give an answer, but also modest and does not want to identify himself as the star of the moment, he simply says ‘Breno is a tango singer’. Although the same proposition is expressed in this situation as in the previous one, the information transmitted by Breno’s utterance is different, since it raises a different reaction in the interlocutor (he simply turns back and waits for the person called ‘Breno’ to perform, or perhaps

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instead rushes to a music store looking for a CD by a singer with that name). The question is: how can referentialism account for the difference in informational content between sentences 1 and 2 uttered in the corresponding contexts if the Russellian proposition expressed in both contexts is exactly the same? At first sight, we could think of an explanation in terms of Kaplan’s notion of character, since the character associated with ‘I’ (which is a function that associates different persons with different contexts) is different from the character of ‘Breno’ (which is a constant function). But Kaplan goes little beyond that, and does not explain how the apprehension of different functions by a speaker corresponds to the apprehension of different cognitive values. Actually Kaplan is more concerned with expressions that involve demonstratives (e.g., an expression like ‘he [pointing at Breno in the tango-house] is him [pointing at Breno in the lecture hall]’ or ‘this [pointing at Venus in the morning] is this [pointing at Venus in the evening]’. In Kaplan (1977) the explanation of the cognitive value of these identities is given by the non-linguistic element (the demonstration), which for him is analogous in several relevant aspects to a definite description, thus having something resembling a Fregean sense. In Kaplan (1989) he changes his mind, and considers the demonstration itself not to be relevant anymore. What is essential to a demonstrative, according to his new view, is that which he calls the “directing intention” accompanying it. But now we have the same difficulty in another form: how can we explain the cognitive value of an identity involving demonstratives with different directing intentions? Kaplan does not deepen his explanation, and sometimes even gives the impression that he recognizes that there is no solution for Frege’s problem within his framework, i.e., no real explanation of the difference in cognitive value in cases of the sort we have discussed. Perry (1977) explains the relation between propositional content and thought in the following way: when we think of something, we not only think of a propositional content, but we think of it in a particular way. Hence, when Breno thinks ‘I am a tango singer’ and when I think ‘Breno is a tango singer’, we both think the same propositional content, but in different ways, since Breno thinks of it under the character of ‘I’ and I think of it under the character of ‘Breno’ (which, leaving aside possible ambiguities, is insensitive to the context).5 Apparently the difference in 5. An interesting detail here is that only I myself can think of propositions that have me as element through the character of ‘I’, i.e., everyone can think of the same proposition, but no one else can think of it through this particular perspective. On the other hand, everyone else, but not

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cognitive value could be explained by the different characters under which we can contemplate the same propositional content. But Perry’s explanation stops at this level, i.e., no further explanation is given about the character according to which a proposition is thought. And for this reason it seems unsatisfactory to some critics, because if his theory is to be more informative and fruitful than the Fregean theory of sense, it should tell us more about this “perspective”, in particular about the privileged perspective that I have to myself.6 In a later essay, Perry (1997) changes his approach, this time exploring in more depth an aspect of indexicals that we have already considered, namely, the fact that they are token-reflexives. He distinguishes two kinds of contents that expressions containing indexicals might have (and, correspondingly, two kinds of truth-conditions): one is the propositional content properly speaking (or, as he calls it, the official content), and the other one he calls reflexive content. Suppose someone hears Breno’s utterance ‘I am a tango singer’, but doesn’t know who said it. Let us call this utterance ‘c’. According to Perry, we can talk of two different truth-conditions for c. One is called by him the reflexive truthcondition, and makes reference to c itself: c is true iff (R ) there is an x such that (i) x is the producer of the utterance c and (ii) x is a tango singer. R is certainly not the proposition expressed by the utterance above, but it corresponds to the understanding of a competent speaker that has incomplete knowledge of the context (i.e., doesn’t know who uttered c). The second kind of truth-conditions Perry calls incremental, and it is given by: c is true iff (P) Breno is a tango singer Notice that the truth-condition here is given by the proposition expressed (i.e., the official content), and does not make reference to the utterance itself (i.e., is not reflexive). P does not simply correspond to the understanding of a speaker with incomplete knowledge of the context; it says more. It is a complete answer to this question: given that we know all the relevant features of the context (in this case, that it was Breno who uttered c), what else (hence the name “incremental”) has to be the case for the me, can think of a proposition that has myself as element under the character of ‘you’. 6. Evans (1982) voices this complaint about the vagueness and insufficiency of Perry’s explanation.

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utterance to be true? Perry’s reflexive and incremental contents represent different levels of understanding of utterances containing indexicals corresponding to the degree of knowledge that speakers might have of the relevant contexts. If the speaker has no knowledge at all, then he only grasps a purely reflexive content (as, e.g., when I see the inscription ‘I was there with him that day’ in a note forgotten on a table, without any idea about who wrote it or when, where is ‘there’ or who ‘him’ refers to). If, on the other hand, a speaker has complete knowledge of the context, then he grasps the incremental content. But, as Perry notices, between these two levels of understanding there might be degrees of partial understanding corresponding to the degrees of partial knowledge of the relevant contextual factors. E.g., someone might hear the utterance made by Breno at a party ‘I will sing tango in this room today’, recognize the voice and so know that Breno said it, know the day of the party, but not know exactly where (i.e., in which room) the utterance was made. In this case the level of understanding is intermediary in that it does not completely reach the incremental content, and is necessarily limited to some reflexive level (something like ‘c is true iff Breno will sing on April 24th in the room where c was uttered’). Although the distinction between reflexive and incremental truthconditions is more intuitive for utterances containing indexicals, Perry thinks it can be extended to non-indexical terms. The reflexive content of a sentence with no indexicals corresponds to an incomplete knowledge of the semantic rules that fix the meaning of its parts. Consider again the utterance made by Breno at the tango-place in Buenos Aires ‘Breno is a tango singer’ (which I will call b) for the interlocutor who doesn’t know who Breno is (and hence doesn’t know that it is Breno himself uttering b). According to Perry, the truth-condition of b is given by b is true iff (F) there is a person x and a linguistic convention C such that (i) C is employed in b, (ii) C allows one to designate x with the name ‘Breno’, and (iii) x is a tango singer. We have here again a reflexive content. It is not the proposition expressed by b, but it corresponds to the understanding that a competent speaker has if he ignores the rules governing that particular use of the name ‘Breno’. Now let us go back to the question that brought us here: how can we explain the difference in cognitive significance between ‘I am a tango singer’ (uttered by Breno) and ‘Breno is a tango singer’ (uttered by him,

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or by any other person referring to him). Perry’s explanation is in terms of the two different kinds of contents: the understanding of each utterance involves the grasping of different reflexive truth-conditions. If the interlocutor has complete understanding of the meaning of the terms occurring in both utterances, the reflexivity disappears in the second, but not in the first utterance: although all relevant linguistic conventions are known, the interlocutor still has only the reflexive content of the first utterance, although it vanishes (or, better yet, loses importance) in the second. We have here doubtless a sophisticated and elegant explanation, but it seems to be better suited as a theory of the concrete uses of language than as an abstract semantics. The deeper question originally raised by Frege seems to remain, despite the conceptual machinery developed by Perry, and it is, or so I think, the following: assuming that the speaker knows all relevant contextual elements, and all relevant semantic rules, and is therefore able to completely grasp all relevant meanings, how can we explain the different cognitive value of the two utterances? In this (almost ideal) situation it might still make sense to talk about reflexive truth-conditions, but they seem to be completely irrelevant to the speaker, since he is now in possession of the proposition. (Certainly when Frege talks about contextual elements that contribute to the determination of sense he is not interested in real concrete speakers, with incomplete knowledge of contexts, but in ideal speakers, who know all relevant features of the context so that the meaning is completely clear.) In other words, Perry wants to answer a Fregean question, but his answer betrays the spirit (or the presuppositions) of the question. Reflecting on incomplete understanding of the meanings involved is not a good way of answering Frege’s problem. IV. Cognitive value: Does it really matter for semantics? The upshot of the above discussion is that there seems to be an important difference between Frege’s semantics, on the one hand, and Kaplan and Perry’s semantics (and the direct reference theory as a whole) on the other, in that the former has a more natural application to the question of cognitive value, while the latter runs into trouble in this field. That is to say, Fregean semantics has a simple and elegant epistemology as a by-product, but it is not clear that the alternative approach can have it. I take this to be an advantage of Frege’s semantics and of his conception of a proposition in particular as opposed to the Russellian conception of a proposition,

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but I have to register here that this is not free of controversy. Wettstein (1986), for example, does not take this difference to be an advantage of Frege’s semantics, and advocates the view that the explanation of cognitive significance is not (or, better said, should not be) a concern for direct reference theorists. According to him, Fregean semantics and direct reference semantics have different goals with different standards of adequacy: Where Frege’s primary focus was on the connection between language and the mind, or, more accurately, between language and objective thought contents, the new theorist is largely unconcerned with matters cognitive. His interest is in the connection between language and the world, the realm of referents. He is doing anthropology of our institutions of natural language, and he wants to understand the institutionalized conventions in accordance with which our terms refer […] There is no reason to suppose that, in general, if we successfully uncover the institutionalized conventions governing the references of our terms, we will have captured the ways in which speakers think about their referents. (Wettstein 1986, p. 124)

That is to say, for Wettstein Frege’s criterion of adequacy in terms of explanatory power in epistemic issues is part of a paradigm in semantics that was abandoned by direct reference theorists, and hence there is no point in looking for such an explanation if one is a direct reference theorist. I have no space here for an extended discussion of Wettstein’s perspective, and shall only briefly outline some possible worries about it. Wettstein might well be right to think that epistemic explanations is not a concern for the direct reference theorist, but his suggestion seems to depend on a too strong restriction on the field of semantics, and it is certainly at odds with Kaplan’s and Perry’s approaches since, as we saw, both tried hard to solve the problem of epistemic significance. (Wettstein sees this effort as misguided, following his section title, “How Not to Develop the New Theory”.) Alternatively, one could adopt a view diametrically opposed to that of Wettstein regarding the nature of semantics, claiming instead that there is no reason to think that pure semantics as an abstract and a priori philosophical discipline should worry about the institutionalized conventions that govern ordinary language in its concrete use. At any rate, there seems to be a kind of imperative coming from the pragmatics of science that is relevant here: in general, a theory that can unify two distinct and apparently far apart fields is prima facie to be preferred to a theory that explains only one single class of phenomena. Hence a theory that can both account for the connection of language and world, and explain epistemic

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significance, is in principle superior to a more restricted one. It remains to be explained by authors such as Wettstein why semantics would be an exception to this pragmatic imperative.

REFERENCES Almog, Joseph, Perry, John, and Wettstein, Howard, eds., 1989. Themes from Kaplan. Oxford: Oxford University Press. Evans, Gareth, 1985. Understanding Demonstratives. In: Yourgrau (1990), 71– 96. Kaplan, David, 1977. Demonstratives. In: Almog, Perry, Wettstein (1989), 481– 564. — 1989. Afterthoughts. In: Almog, Perry, Wettstein (1989), 565–614. Frege, Gottlob, 1892. Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik, 100, 25–50. — 1918. Der Gedanke. Eine logische Untersuchung. Beiträge zur Philosophie des deutschen Idealismus, III, 36–51. Perry, John, 1977. Frege on Demonstratives. In: Perry (2000), 1–26. — 1979. The Problem of the Essential Indexical. In: Perry (2000), 27–44. — 1997. Reflexivity, Indexicality and Names. In: Perry (2000), 341–54. — 2000. The Problem of the Essential Indexical and Other Essays. Stanford: CSLI Publications. — 2000a. Rip Van Winkle and Other Characters. In: Perry (2000), 355–76. Wettstein, Howard, 1986. Has Semantics Rested on a Mistake? The Journal of Philosophy 83, n. 4, 185–209. Reprinted in: Wettstein (1991), 109–131. — 1991. Has Semantics Rested on a Mistake? And Other Essays. Stanford: Stanford University Press. Yourgrau, Palle, ed., 1990, Demonstratives. Oxford: Oxford University Press.

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JOHANN GOTTLIEB FICHTE Gesamtausgabe der Bayerische Akademie der Wissenschaft Edited by Reinhard Lauth, Erich Fuchs and Hans Gliwitzky †. 1962 et sqq. Cloth. Ca. 40 volumes. Price per volume (subscription) € 246,-; single volume € 291,-. ISBN 978 3 7728 0138 9. 37 volumes available ›ADDRESSES TO THE GERMAN NATION‹ VOLUME I,10: Works 1808-1812. Edited by

Reinhard Lauth, Erich Fuchs, Hans Georg von Manz, Ives Radrizzani, Peter K. Schneider, Martin Siegel and Günter Zöller. 2005. XVI, 476 pp., Available 2 ill. ISBN 978 3 7728 2170 7. In 1964, the first volume of series I (works) was published. With the publication of volume 10, the editors have now completed the edition of Fichte’s works. Volume 10 contains publications of the last five years of Fichte’s life. The famous ›Addresses to the German Nation‹ that Fichte delivered publicly in Berlin in 1807/8 succeed the ›Grundzüge des gegenwärtigen Zeitalters‹ and the ›Anweisung zum seeligen Leben‹ and thus form the last part of the trilogy which marks Fichte’s change of direction towards a Philosophy of History. Following Pestalozzi, Fichte appeals for a new education of the German people in order to achieve an intellectual reformation. At the same time, this work is shaped by Fichte’s intention to participate – within the bounds of his possibilities – in the course of political events. Fichte connected Germany’s loss of political autonomy due to Napoleon’s dominance with an imminent intellectual decline in Germany, and was therefore urged to take action. He realised his right and duty to speak out. Additionally, this volume contains Fichte’s adaptations of Petrarca and Camões, his elaborated final lecture of 1810 ›Wissenschaftslehre in ihrem allgemeinen Umrisse‹, the speech ›Ueber die einzig mögliche Störung der akademischen Freiheit‹ (which Fichte delivered when he took up his office as headmaster at Berlin University in 1811), as well as his two summer term lectures of 1812 ›Über die Bestimmung des Gelehrten‹. The volume is completed by 14 reviews from 1788.

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Interpretation and Transformation Explorations in Art and the Self Michael Krausz

In this book, Michael Krausz addresses the concept of interpretation in the visual arts, the emotions, and the self. He examines competing ideals of interpretation, their ontological entanglements, reference frames, and the relation between elucidation and selftransformation. “This book marks a decisive moment in the philosophical scholarship on interpretation. Krausz is a unique figure in the current philosophical climate, equally capable of theoretical sophistication, eloquence, and compelling argumentation. Widely acclaimed for his major contributions to interpretation theory, he has now added a crucial dimension to his work, and to the field itself.” Andreea Deciu Ritivoi, Carnegie-Mellon University (Author: Yesterday’s Self; Editor: Interpretation and Its Objects: Studies in the Philosophy of Michael Krausz)

Amsterdam/New York, NY, 2007 XIII-154 pp. (Value Inquiry Book Series 187) Paper € 34 / US$ 46 ISBN-13: 9789042021808

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Contemporary Pragmatism

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Edited by John Shook, Paulo Ghiraldelli, Jr.

Contemporary Pragmatism is an interdisciplinary, international journal for discussions of applying pragmatism, broadly understood, to today’s issues. CP will consider articles about pragmatism written from the standpoint of any tradition and perspective. CP especially seeks original explorations and critiques of pragmatism, and also of pragmatism’s relations with humanism, naturalism, and analytic philosophy. CP cannot consider submissions that principally interpret or critique historical figures of American philosophy, although applications of past thought to contemporary issues are sought. CP welcomes contributions dealing with current issues in any field of philosophical inquiry, from epistemology, philosophy of language, metaphysics and philosophy of science, and philosophy of mind and action, to the areas of theoretical and applied ethics, aesthetics, social & political philosophy, philosophy of religion, and philosophy of the social sciences. CP encourages work having an interdisciplinary orientation, establishing bridges between pragmatic philosophy and, for example, theology, psychology, pedagogy, sociology, economics, medicine, political science, or international relations. Current issue: Vol. 3, Issue 2 Online access is included in print subscriptions.

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The Self-Correcting Enterprise Essays on Wilfrid Sellars Edited by Michael P. Wolf and Mark Norris Lance

This volume presents ten new papers on the work of Wilfrid Sellars and its implications for contemporary philosophy. Contributors run the gamut from established voices in the Sellarsian literature to the newest voices in the field. It addresses topics ranging from cognitive science and philosophy of mind to epistemology and the philosophy of language. This volume is of interest to those studying cognitive development, perception, justification and semantics. It will also be of great interest to anyone following the recent work of John McDowell or Robert Brandom.

Amsterdam/New York, NY, 2006 274 pp. (Pozna n´ Studies in the Philosophy of the Sciences and the Humanities 92) Bound € 60 / US$ 81 ISBN-10: 9042021446 ISBN-13: 9789042021440

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Selected Writings on Ethics and Politics Translated by Paul Rusnock and Rolf George Bernard Bolzano

Celebrated today for his groundbreaking work in logic and the foundations of mathematics, Bernard Bolzano (17811848) was best known in his own time as a leader of the reform movement in his homeland (Bohemia, then part of the Austrian Empire). As professor of religious science at the Charles University in Prague from 1805 to 1819, Bolzano was a highly visible public intellectual, a courageous and determined critic of abuses in Church and State. Based in large part on a carefully argued utilitarian practical philosophy, he developed a non-violent program for the reform of the authoritarian institutions of the Empire, which he himself set in motion through his teaching and other activities. Rarely has a philosopher had such a great impact on the political culture of his homeland. This volume contains a substantial collection of Bolzano’s writings on ethics and politics, translated into English for the first time. It includes a complete translation of the treatise On the Best State, his principal writings on ethics, an essay on the contemporary situation in Ireland, and a selection of his Exhortations, dealing with such topics as enlightenment, civil disobedience, the status of women, anti-Semitism and Czech-German relations in Bohemia. It will be of particular interest to students of central European philosophy and history, and more generally to philosophers and historians of ideas.

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Education for a Democratic Society Central European Pragmatist Forum, Volume Three Edited by John Ryder and Gert Rüdiger Wegmarshaus

This book is the third volume of selected papers from the Central European Pragmatist Forum (CEPF). It deals with the general question of education, and the papers are organized into sections on Education and Democracy, Education and Values, Education and Social Reconstruction, and Education and the Self. The authors are among the leading specialists in American philosophy from universities across the U.S. and in Central and Eastern Europe. Studies in Pragmatism and Values (SPV) promotes the study of pragmatism’s traditions and figures, and the explorations of pragmatic inquiries in all areas of philosophical thought.

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Science in Culture Piotr Jaroszyn´ski

This book tries to uncover science’s discoverer and explain why the conception of science has been changing during the centuries, and why science can be beneficial and dangerous for humanity. Far from being hermetic, this research can be interesting for all who want to understand deeper what really conditions the place of science in culture.

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Essays in Logic and Ontology Edited by Jacek Malinowski and Andrzej Pietruszczak

The aim of this book is to present essays centered upon the subjects of Formal Ontology and Logical Philosophy. The idea of investigating philosophical problems by means of logical methods was intensively promoted in Torun by the Department of Logic of Nicolaus Copernicus University during last decade. Another aim of this book is to present to the philosophical and logical audience the activities of the Torunian Department of Logic during this decade. The papers in this volume contain the results concerning Logic and Logical Philosophy, obtained within the confines of the projects initiated by the Department of Logic and other research projects in which the Torunian Department of Logic took part.

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Contemporary Pragmatism 3:1 Edited by John R. Shook and Paulo Ghiraldelli, Jr.

Contemporary Pragmatism is an interdisciplinary, international journal for discussions of applying pragmatism, broadly understood, to today’s issues. This journal will consider articles about pragmatism written from the standpoint of any tradition and perspective, but it will concentrate on original explorations of pragmatism and pragmatism’s relations with humanism, naturalism, and analytic philosophy. The journal welcomes both pragmatism-inspired research and criticisms of pragmatism. We cannot consider submissions that principally interpret or critique historical figures of American philosophy, although applications of past thought to contemporary issues are sought. Contributions may deal with current issues in any field of philosophical inquiry. CP encourages interdisciplinary efforts, establishing bridges between pragmatic philosophy and, for example, theology, psychology, pedagogy, sociology, medicine, economics, political science, or international relations.

Amsterdam/New York, NY, 2006 182 pp. (Contemporary Pragmatism Vol. 3, Issue 1) Paper € 37 / US$ 46 ISBN-10: 9042021225 ISBN-13: 9789042021228

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Ectogenesis Artificial Womb Technology and the Future of Human Reproduction Edited by Scott Gelfand and John R. Shook

This book raises many moral, legal, social, and political, questions related to possible development, in the near future, of an artificial womb for human use. Is ectogenesis ever morally permissible? If so, under what circumstances? Will ectogenesis enhance or diminish women’s reproductive rights and/or their economic opportunities? These are some of the difficult and crucial questions this anthology addresses and attempts to answer.

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