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