Frege, semantics, and the double definition stroke

Abstract: A survey of the emergence of early analytic philosophy as a subfield of the history of philosophy. The importance of recent literature on Frege, Russell, and Wittgenstein is stressed, as is the widening interest in understanding the nineteenth-century scientific and Kantian backgrounds.

Chapter 8 in The History of Philosophical and Formal Logic. From Aristotle to Tarski, Alex Malpass and Marianna Antonutti Marfori (eds.), Bloomsbury: London New York, 2017

In this paper I advance an account of the necessity of logic in Wittgenstein’s Tractatus. I reject both the “metaphysical reading of Peter Hacker, who takes Tractarian logical necessity to consist in the mode of truth of tautologies, and the resolute” account of Cora Diamond, who argues that all Tractarian talk of necessity is to be thrown away. I urge an alternative conception based on remarks 3.342 and 6.124. Necessity consists in what is not arbitrary (nicht willkürlich), and contingency in what is up to our arbitrary choice (willkürlich), in the symbols we use, in how we picture or model the world. Necessity is not a mode of truth of propositions, but lies in the requirements of their intelligibility. I argue that this conception is implicit in certain “resolute” readings and in some of their critics. Both sides of the dispute are committed to certain logical features of language or thought, patterns of symbolizing constitutive of intelligibility that are not up to us to institute or alter. This conception of non-arbitrary patterns of symbolizing, I argue, is what logical syntax in the Tractatus consists in. I also argue that the well-known Tractarian view of propositions as truth-functions of elementary propositions can be understood in terms of patterns of norms governing our making sense with the affirmation and denial of propositions.

ltury Philosophy The Story of Analytic Philosophy Plot and Heroes Edited by Anat Biletzki and Anat Matar 1998 London and New York 8 Frege, seDlantics, and the double definition stroke! Juliet Floyd I In 1964, in his introduction to his partial translation of Frege's Grundgeset<:l, Montgomery Furth wrote: There is ... rCllson IClr importance today to Frcgc's treatment of [his primitive assertions and his sound rules of inlcrenccJ: that is, Frcgc's explanation of the primitive basis of his system of logic, and particularly of the primitive symbolism, is undertoJrm in terms of a deeply tlwug/U-out semantUal interpretation, which in tum embodies an entirt philosophy rtf language. The influence of the latter upon 1M semantUal structure and even the syntax of the language developed makes iudf fdt steadily throughout the discussion. This phiWsophy of language is very profound, and possesses great interest quite independent of its origin as the handmaiden of "logicism". It tIJo is not gmerally weU rmtlmtood even toti4y. (Frege 1893: vi-vii, my emphases) In a quite different vein,Jean van Heijenoort, in his 1967 paper "Logic as Calculus and Logic as Language," wrote: From Frege's writings a certain picture of IogU emerges, a conception that is perhaps not discussed explidtly but nevertheless constandy guides Frege. In referof logic. ring to this conception I shall speak of the . . . The universality of logic expresses itself in an important feature of Frege's system. In that system the quantifiers binding individual variables range over all objects ... .Frege's unWmt consists rtf aU that tnne is, and it isfixed. . . . Another important consequence of the universality of logic is thaJ nothing can be, or has to be, said outside rtf the !JIstem. And, in fact, Frtge never raises any mdasystematU question (consistency, independence of axioms, completeness). Frege is indeed fully aware that any formal system requires rules that are not expressed in the system; but these rules are void of any intuitive logic; they are "rules for the use of our signs" (note: Beg §13). In such a manipulation of signs,.from which any argumentative logic has been squee;:.ed Oltt, Frege sees precisely the advantage of a formal system. Since logic is a language, that language has to be learned. Like many languages in many circumstances, the language has to be learned by suggestions and clues. Frtge repeatedly states, when introducing his !JIstem, that he is giving 142 }uliet Floyd "hints" to 1M I'UklH; lllat IIu fl. Semantics and the doubll hal to mill and iMUId not b'Itut/fl him " IIIIm rif ''grJoi will", The. problem p to bring the reader to "catch on": he has to get into the language. (note: Here 1M i1ifhunc1 of I+ege on Witfgtnstein is obvious. Also, Frege's rifusal to entertain mtla.lYstematic question.r explains perhaps why he was not too disl urbccl by sl alcllIcnt WIll:Cpt /lorse is not a conccpt". The paradox arises from the fact that, since concepts, being limetions, are not objects, we cannot name them, hence we are unable to talk about them, Some statements that are (apparently) about concepts can easily be translated into the system; thus, "the concept ttx) is realized" becomes "(& )ttx )". 17u statements that resist such a translation are, upon examination, melaf)lslematic; for example, "there arc functions" cannot be translated into the system, but we see, once we have "caught on", that there are function signs among the signs of the system, hence that there are functions.) (Van Heijenoort 1985: 12 13, my emphases) Van Heijenoort insists that for Frege, logic is universal, it cannot pick or choose different universes of discourse, and the formula language (BegriffischtiftJ that is the formalization of this logic is not subject to differing 2 interpretations. Moreover, when correctly formulated this formula language lays bare the framework within which all thought must take place, it lays bare the limits of rational discourse, the limit.r of sense. AllndinR to the Kerry paradox, van Heijclloort treats the issues raised by "the concept horse" problem as paradigmatic for Frege's attitude toward most, if not all "metasystematic" questions. And thus he holds that in the (German) presentation of his formula language Frege takes himself to rely wholly, or at least in large part, upon "suggestions," "clues," and "hints" to help the reader see or "catch on" to "rules ... from which any argumentative logic has been squeezed out." In this way van Heijenoort argues that no "metasystematic questions (consistency, independence of axioms, completeness)" can arise for Frege about his formal system, his formula language. This reading sharply questions Furth's claim that in Frege we fmd "a deeply thought out semantical interpretation" or "semantical structure" embodying "an entire philosophy of language." For although van Heijenoort wrote in a subsequent paper that for Frege the formula language elicits the latent aruJ determinate content of ordinary languagt?-a picture many would caJl "semantical"4----neverthdcss van Heijenoort insisted that Frege conceived himself to be uncovering this latent structure of language, namely logic, from within the one and only universaJ language, namely, logic itself. Hence, unlike Furth, van denied that r:uuld have adopted any sort of "metatheoretical" or model theoretic stance in so proceeding. (For van Heijenoort, "semantics" always meant model theoretic or set theoretic semantics.) The formula language was, van Heijcnoort claimed, intended by Fregc to totally supplant rather than simply to reflect or to aid natural language. This claim explicitly appears in a later stroh 143 pan of "Logic Ai Calculul and Logic au Language," where van HeljenQort buttrelles his reading of Frege by appealing to what he says are parallel lines of thought in Russell: QI.Iestiolls ahout the system are as absent from Principia matlumatica III they are from Frege's work. Smumtic notions are unknown. "I-" is read as " ... is true", and Russell could hardly have come to add to the notion of provability a notion of validity based on naive set theory.... If the question of the semantic completeness of quantification theory did not "at once" arise, it is because of the universality-in the sense that I tried to extricate---of Frege's and Russell's logic. The universalformal language supplants the natural language, and to preserve, outside rif the .rystem, a notion rif validity based on intuilive set theory, does not seem tofit into the scientific reconstruction rif the language. 17u on!>, question rif completeness that ""!y arise ir, to use an expression rif Herbrand's, an experimentlJi question. As many theorems as possible art derived in 1M .rystem. Can we exhaust the intuitive modes of reasoning actually used in science? To answer this question is the purpose of the Frege-Russell enterprise. (van Heijenoort 1985: 14, my emphases) In his 1973 Frege: Philosophy of lAnguage Michael Dummett massively articulated the reading of Frege called for by Furth. Hence, in particular, though without explicitly mentioning van Heijenoort, Dummett strongly with vall Hcijelloort's denial of the pORRihility of metasystematic considerations in Frege: Although Frege did not expressly defme the two notions-semantic and syntactic-of logical consequence, they lit rea4J to hand in his work: for there, on the one hand, is the formal .rystem, with its precisely stated formation rules, axioms and rules of inference; and there, on the other, are the semantic exp/anQ.tions of the sentences rif the formalized language, set out, clearly separated from the formal development, in German in the accompanying text.... The sentential fragment of Frege's formalization of logic is complete, and likewise the flrstorder fragment constitutes the first complete formalization of fIrst-order predicate logic with identity. It was left to Frege's successors to prove this completeness, as also to establish the incompleteness of his, or of any of the effective, formalizations of higher-order logic. Frege had it to hand to raise these questions: but he did not do so ... (Dummett 1973, 1981: 82, my emphases) Dummett's reading greatly enhanced the wowing interest in Frege by securing Frege's relevance to contemporary philosophy of language and theory of meaning. It has become the dominant reading.. However, Dummett has not gone unchallenged. In his 1979 "Logic in the Twenties: The Nature of the Quantifier," Warren Goldfarb explicitly criticized Dummett5 and agreed with van Heijenoort about the absence of any "metasystematic" standpoint in Frege. Goldfarb based his argument on a detailed historical assessment of the development of quantification theory and concluded: 1« Jr;liIt Floyd &mtmtits and the double definition Jtroke [The] lack of intelligibility [of Independence questions in logic] may be intrinsic to the logicist program. If the system constitutes the universal logical language, then there can be no external standpoint from which one may view and discuss the system. MefaJystematic considtrations are ilkgiJirruJte rather tJuzn simplY undesirahk. This is what Harry Sheffer called "the logocentric predicament" (Sheffer, "Review of Whitehead and RusseU, Principia Mathematica", Isis 8: 226-231), and forms a large part, I think, of the motivations behind Wittgenstein's Tractalus. (Fn: A similar "Iogocentricity" may underlie Frege's curious claim that "Only true thoughts can be premises of inferences" (Frege 1971: 425).) (Goldfarb 1979: 353, my emphasis) Ij.! j)" ; ;J Goldfarb too (1979: 353) buttressed his claim about Frege's "logocentricity" by drawing a parallel between Frege and Russell, pointing out that in §17 of The Principles of Mathematics (1903) Russell had written: 1IIi 1 indemonstrables there must be; and some propositions, such as the syllogism, must be of the number, since no demonstration is possible without them. But concerning others, it may be doubted whether they are indemonstrable or merely undemonstrated; and it should be obseIVCd that the method !!f supposing an if i[ in such cases as the axiom !!f parallels, is here not universal!Y availahk. For all our axioms are principles of deduction; and if they are true, the consequences which appear to follow from the employment of an opposite principle will not really 1,;1 t; .i, t i'; follow, so that arguments from the supposition of the falsity of an axiom are here subject to speeial fallacies. (Russell 1938: IS, my emphasis) ;,1 Thus, though Russell in one sense certainly does raise a question about independence, he seems, as Goldfarb says, "not even to see the intelligibility of stepping outside the system to use an intuitive logic in metasystematic arguments" (1979: 353). Since the 1970s several other philosophers interested in early analytic philosophy have also explicidy differed with Dummett's semantical reading of Frege by offering their own anti-semantical readings. 6 Gordon Baker and Peter Hacker (1984), Cora Diamond (199Ia, 1991b), Burton Dreben Jaakko and Merrill Hintikka (1979, 1986), Thomas Ricketts (1985, 1986a, 1986b, 1996a, 1996b, 1997), Hans Sluga (1980, 1987), and Joan Weiner (1982, 1984, 1990, forthcoming), while differing in various ways with each other, all belong to this anti-semantical tradition. 7 Like van Heijenoort they have argued that far from advocating a theory of meaning or a semantical theory, Frege rejects as nonsense any attempt to account systematically for the nature of logic, the nature of meaning, or the nature of language. Of course, no one denies that Frege uses the notion of the content of a judgment and, later, the notions of Sinn and &tkutung. What is questioned is the idea that in using these notions Frege is theorizing about I, il'r " It, 7 ! , ;1 I * • UP $ ¥ • ;;.•-.. , , - ...,.-...... - 145 language or meaning in general. In order to question this idea, these philosophers emphasize as crucial Frege's use of what he calls "elucidations," "Erlauterungen" (Sluga 1980: 180-2; Weiner 1990: ch. 6; Weiner forthcoming). Frege tells us that elucidations serve to set forth the way we are to understand certain of his undefmable, i.e. logically primitive, notions (1984: 300, 302; 1979: 207, 214, 235).8 In one way or another Frege's anti-semantical readers all seem to agree with van Heijenoort's focus on Frege's explicit uses of elucidation, hints, clues, and metaphors in presenting his formula language. 9 Indeed, Weiner goes so far as to maintain that Frege must deny both that our numerals in everyday arithmetic may be said to have &dtutungen, and that our ordinary, pre-analyzed propositions of arithmetic have determinate truth-values (Weiner 1984: 78). While not all the anti-semantical readers go this far, all emphasize the extent to which Frege explicidy emphasizes that at times he is forced to fall back upon hints, metaphor, and indirect suggestion, relying on his reader for cooperation, good will, and guesswork. 1O Such figurative and colorful language indicates, they argue, that Frege has no antecedent or independent theoretical account of semantical notions. Thus elucidations are to be sharply distinguished from judgments, from theory; they do not and are not intended to play anything like the role of theoretical premises in deductive arguments. Furthermore, on the anti-semantical reading all Frege's judgments, Le. all his genuine theoretical claims, are expressible in the formula language, where proofs are set forth by means of assertions, and assertions are recorded by means of the judgment stroke. As van Heijenoort insisted, Frege does not and cannot so transcribe certain apparendy crucial claims about his basic philosophical notions (e.g. jUnction, concept, object). It is Frege's conception of logic, these philosophers argue, which leads him to treat such apparendy metasystematic and semantical talk as mere pedagogical rhetoric. Thus all such talk must be sharply distinguished from a theory of sense, a theory of meaning, or even a theory of how the truth-values of sentences are determined by semantic features of their constituents. tl For example, in "What Does a Concept-Script Do?" (1983), Cora Diamond emphasized, as had van Heijenoort, Frege's treatment of "the concept HOTJe": What Frege thinks is that through an inadequacy of ordinary language, we can form sentences in it which are acceptable according to its rules but which are not the expression of any thought. It is possible to become clear about what has happened, if we are led to see how thoughts are expressed in a language more nearly adequate by the standards of logic. In grasping the significance of the distinctions embodied in that language, we do not grasp any ineffable troths. A truth is a truth about something; a true thought (that is) is about 146 ]uliLt Floyd whatever logic may construe it as being about. But the distinctions embodied in the concept-script are not what any thought can be about. (Diamond 1991a: 140--1) Ricketts bases hi'l argument for an anti-semantical reading of Frege on what he takes to be Frege's underlying conception of judgment and inference. t2 This conception, he argues, lies behind the way Frege introduces his formula language and is intimately connected with Frege's arguments that truth is indefmable. t3 At best, holds Ricketts, Frege's post-l 890 talk of Sinn and Bedeutung is in "deep tension" with his underlying conception of the universality of logic (Ricketts 1985: 3; 1986a: 172; 1986b: 66-7). Such readings are almost always buttressed by an appeal to the influence of Frege upon Wittgenstein. In 'Tractatus 4.112 Wittgenstein wrote that: The aim of philosophy is the lowcal clarilication of thOllghts. Philosophy is lIot a theory Lut all activity. A philosophical work consists essentially of elucidations The result of philosophy is not a number of "philosophical propositions," but to make propositions clear. (Wittgenstein 1922: 4.112, c( 6.54) The line of descent from Frege appears to be explicit; the anti-semantical tradition sees in Wittgenstein the heir to Frege, and the 'Tractatus show/say distinction is said to have emerged directly from Frege's conception of logic as universal. t4 Dummett remains unmoved by such claims. Indeed, in his 1981 The InJtrpretatUm of Frege's Phiiosop/rl he explicitly based his semantical reading of Frege on what he insisted was Frege's "universal logic": It is, of course, true that Frege did not attempt to explain his logical notation by giving rules of translation from his formulas into sentences of Rather, when you grasp the SelL'IeS of the primitive symbols, you thereby grasp you afe thl'lI ahle 10 the thoughts l'xpres.'Icd by the formula.'I, whkh express in any lauguaRe known to you and (:apahle or cxpn'ssinR tlwlIl. IIl1t it is plain that he did not think that, in the logical notation of he had devised a language in which thoughts could be expressed that could not be expressed in any other way; the logic of was intended as a univl'TJal logic, not one pcculi,lr to a spedallanRwlRc III1(,OI1I1('cll'<1 wilh Ih(' thollRhts WI' are ordinarily concerned to communicate to one another. The meaning-tJuoretical notions used in Part I of GrutuJgeset;;;e are therifore not to be considered as app{ying exclusive{y to Prep's.frmnoJ -!,sttm. They are to be taken as applicable both to it and to natural language, and perhaps as requh'ed for any intelligible language; and the theory of meaning tIuJJ embodies them according{y serves both as a tJuory of sensefor natural and as afoundationfor tlIeJormallngic ... If these conclusions are correct, there is a substantial body of Frege's Smumtit:.1 and tlze doublt definition strokL 147 theory-precise{y tluJt comprising hisphilosOP/rl of Ionguage·-of which no definitive exposition, comparable to Grwu!geset;;;e, or even carrying an authority equal to that of GrwulIogm, exists.... We can do no more than surmise the reason why he never achieved a formulation of his general thMwy of philosophicallogie. (Dummett 1981: 18-19, my emphases) Dummett admits that Part I of Grundgesetze, where Frcge explains to the reader how to use his system, contains "no definitive exposition" of a Frege's apparatus of philosemalltical theory. Yet, Dummett argues, sophical notions such as jUnction, object, sense, riference, and so on are intended to apply to natural language as well as to language regimented in the formula language, there is in Frege's writings a semantica1 theory which goes far beyond "hints" and "clues" to help the reader "catch on" to the formula language. Dummett takes Frege's conception of the universality of lowe to commit Frege to a theory of meaning for his formula a tltt'my whirh silllllhatWOlIsly applies to sentenc.es of natural language. Thus we have Dummett's implicit challenge to the anli-semantical tradition: What is the point of forcing a reading on Frege which precludes him from adopting anything like a contemporary metasystematic perspective on his formula language? That is, if we were to imagine Frege living a bit longer than he had, and learning of model-theoretic results, on what grounds and for what purposes can we hold that Frege's conception of logic would have had to shift if he had come to accept those results as illuminative of logic? Dummett reiterated this challenge in his 1984 review of Baker and Hacker's liJgi.cal Excavations. Baker and Hacker too had claimed that: Begriffischrifl. contains the first complete axiomatization of the propositional calculus. Of course, Frege did not prove its completeness or its consistency. Indeed, the scorn that he poured on Hilbert's metalogical investigations of axiomatizalion of geomelry he would have turned too against metalogical proofs concerning his own logical system. The only proof of the consistency of a of axioms is a demonstration of their joint truth, the only demonstration the compatibility of the denial of one with the joint of their asscl't ion of Ilw n'sl (On the /-1/1l11d(Itio/lJ iiI and rormal 77,eOlieJ III Arithmetic, 1:1[,104). Lacking any nwans of prccis(·ly dcmarcathlg logical laws apart from their derivation as theorems within his axiomatic system (Cr (Aundgeset;;;e der Arithmetik p. xvii), he was not even in a position to frame an exact l1"cstioll aholll Ihr rOlnplclrncss of his axiomati7.ation. (Uak('f and Hacker 191H: I H) Hence Baker and Hacker concluded that there are "no grounds for asserting that LFrege] advanced ... to any conception that the true busiof logicians is a science of language (semantics) ... The hypothesis that [Fregc] intended to lay the foundation of logical semantics is implausible" (1984: 248---49).t5 Dummett responded: 148 Juliet HOJd The word "semantics" is used in several different ways, but the references to logic and logicians suggest that what [Baker and Hacker] have in mind is a semantic theory for a formal language as conceived in contemporary model theory. If so, their [above-quoted] assertion is very surprising, since Part I of Gnmdgesetzt appears to contain a semantic theory for the formal language, clearly separated from the account of its formation rules, axioms and rules of inference: this theory is stated by stipulating what references the primitive symhols are to have, and laying down how the reference of a complex expressioll is determined Irom cII<: of iL' C()llstitU('lIls. III addition, Frege gives a general framework for such a theory, namely an account of the various possible logical types of expression, of their nature and how they are formed, and of what it is to assign a reference to an expression of anyone such type; this is likewise clearly separated from the specific stipulatiolls governing the primitive symbols of the system. One reason why Baker and Hacker do not see the maUer in this light is that they conllate a semantic theory with a definition of logical conscquellC!'. Thry arc quite right in saying that I,leked the lalter notion. He larknl it hrmuse he <lid IIUt ullI'mte wi. h I ht· mUff'pl iun of a rallKe 01" possihle interpretations of a ... lj: hOlllel,"- lit. "nd.lormed iIIil "onception, lit would Mve Iuld very liule more lIJork 10 do 10 arrive at the semantic notion qf for the blUkgrourui tirlory stated in Pari I l!!' Gruntlgesetzt would immediate!Y Milt yitkkd aformulmion qf wIult, in general, a'!>' one such interpretation should consist in. It is pruisely because r!f 1M presence qf this badrgrowui theory, and its cl1m, though not complete, resemblallu 10 1hz notion used !?y modern logieion.t r!f an interpretation r!f a formal longuage within classUal two-valued Stma1ltics, iIIat Frege's work can beftuiyully compared wiJh tIuU 0/ laJer logieitms. (Dummett 1981: 20 I, my emphasis) , it :1 .,.,. r!, :' ·1 [1 i iii. ,,. Thus Dummetl emphasizes the ease with which Frege's work may be appropriated from a contemporary model-theoretic perspective. And he remains deeply dissatisfied with the notion that Frege's mature philosophy of logic is in tension with model theoretical clarification of logical notions. Dummett again gave voice to this dissatisfaction in his review of a 1987 essay of Dreben and van Heijenoort (Dummett 1987). In this essay, a commentary on Godel's dissertation and subsequent publications on the model-theoretic (semantical) completeness of first-order logic, Dreben and van Heijenoort raise the question: Why was it that fifty years elapsed between the publication of Frege's Begriifsschrffl (1879) and Godel's proof of the completeness theorem (1929)? 16 Their answer is that in order to frame the question of completeness coherently, the algebra of logic tradition-the tradition which denied the universality of logic and emphasized the notion of varying universes of discourse-needed to be combined with thc quitc diffcrclltlogieist tradition stemming from Frcgc and Russell: fbr trege, and then for RusseU and Whitehead, logic was universal: within each explicit formulation of logic all deductive reasoning, including all of classical analysis and much of Cantorlan set theory, was to be formalized. Hence not only was pure quantification theory never at the center of their attention, but &mnnl.ics and ti,e doubli definition stmki I '1 metasystematic questions as such, for example the qUl'!stion of completeness, could not be meaningfully raised. We can give different formulations of logic, formulations that differ with respect to what logical constants are taken as primitive or what formulas are taken as formal axioms, but we have no vantage point from which we can survey a given formalism as a whole, let alone look at logic whole. (Drehen and van Heijenoort 1986: 44) Once again, this reading of Frege is huttressed by quoting from RUllseJl: In the words of Whitehead and Russell 1910 (page 95, or 1925 page 91), "It is to some extent optional what ideas we take as undefined in mathematics.. , . We know no way of proving that such and such a system of undefmed ideas contains as few as will give such and such results. Hence we can only say that such and such ideas are undefined in such and such a system, not that they are indefinable." We arc within IURic and ranllot!ook at it from outside. We arc sllbject to wlmt Slwlfer railed "thr logorcntric prcdic:amC'lIt" ... Thr only way to approaeh the problem of what a limna! system can do is to derive theorems, Again to quote Russell and Whitehead, "ille reasOR inftvor of a'!>' theory on /he prillciple 0/ mathmuJtics must always be inductive, i.e., it must lit in 1hzflUt that tirllhzory in quetion eno.hlts us 10 tkdw:e ordinary matJumoJid' (191 0, page V, or 1925, page v)o (On and this point see van Heijenoort, "Logic as Calculus and Logic as Goldfarb, "Logic in the Twenties: the nature of the quantifier" 0) To raise the question of semantic completeness the Frege--Russell-Whitehead view of logic as all embracing had to be abandoned, and Frege's notion of a formal system had to become itself an object of mathematical inquiry and be subjected to the model-theoretic analyses of the algebraists of logic. (Dreben and van Heijenoort 1986: 44, my emphasis) 17 • • 0 Dreben and van Heijenoort's use of the history of logic to develop an anti-semantical reading of Frege did not convince Dummett: Drehen and van Heijenoort ... discuss [a] question, one of the most interesting in conceptual history, why it took so long for the concept of completeness to be framed. The explanation given by Dreben and van Heijenoort is that, in the tradition descending from Boole through Peirce and SchrOder, the very notion of a formal system was lacking: in his paper of 1915, for instance, Lowenheim worked with exclusively semantic notions. frege, on the other hand, had bequeathed the notion of a formal system to the tradition that stemmed from him; but, according to that tradition, logic is all-embracing: since there is only one logic in accordance with which all reasoning must be conducted, we cannot step outside logic in order to formulate theorems about rather than within it. However accurate a statement of the views of Russell and Whitehead this may he, it appears to me an oversimplification as applied to Frege. True, he never considered the sentential and first-order fragments of his logic as significantly separable from it; and he had repudiated the traditional 0 •• . ., ,: I, ,n - -, --.. _-- • • ,a •• 4 Semantics and th£ double definition stroke 150 juliet Floyd I II t; l I il conception of variable universes of discourse. But, in his Grundgeset;;e, he did attempt a precise formulation of the semantics of his formal system, clearly distinguished from the specification of its axioms and rules of inference. Indeed, he attempted a proof that every term of the formal system had a unique denotation (and every sentence a unique truth-value), which, if correct, would have constituted a consistency-proof. Moreover, he did not content himself with laying down specific interpretations for the primitive symbols, but stipulated what form the interpretation of an expression of each logical type must, in general, take. All tluJt he locked, therifore,for aformulation of the concepts of valUJi!)! and satiffio.bilitY was tJu conceptum of variable interpretations (ironically so close at hand in Hilbert's Foundations of Geometry). The mystery of the half-century that elapsed between the invention of mathematica110gic and the formulation of its fundamental problem is thus not fully dispelled. (Dummett 1987: 573-4, my emphases) In unpublished lectures Dreben has pointed out that in at least two places In Begriffsschrift Frege did raise the question of completeness Frege wrote, Since in view of the boundless multitude of laws that can be enunciated we cannot list them all, we cannot achieve completeness except by searching out those that, by their jxJwer, contain all of them. Now it must be admitted, certainly, that the way followed here is not the only one in which the reduction can be done. That is why not all relations between the laws of thought are set out clearly through the present mode of presentation. There is perhaps another set of judwnents from which, when those contained in the rules are added, all laws of thought could Jik('wisr he deduced. Still, with the method of reduction presented hcre such a multitude of relations is exhibited that any other derivation will be much facilitated thereby. (Frege 1879: §13) And in "On Mr. Peano's Conceptual Notation and My Own" (1897): In order to test whether a list of axioms is complete (vollstiindig), we have to try and derive from them all the proofs of the branch of learning to which they relate. And in so doing it is imperative that we draw conclusions only in accordance with purely logical laws, for otherwise something might intrude unobserved which should have been laid down as an axiom. (FTege 1984: 235; 1967: 221) However, Dreben argues that these passages conftrm the Dreben and van Heijenoort reading of Frege. For in both passages Frege eschews any jump to a metalevel; he operates with what Russell called "inductive" :reasoning and what van Heijenoort, followinR Herhrand, called an "experimental" approach: derive as many theorems as possible within the system in order to answer the question of whether the system is complete. Thus, those questions Dummett holds that Frege merelY didn't raise, but "had it to hand to raise," others say he didn't raise because he couldn't raise, 151 "couldn't" in the sense thal he did not have room, within his conception of logic, for the posing of these questions in their metamathematical sense. In response to a 1992 lecture of Dreben's, Quine expressed scepticism about the anti-semantical reading. IS Does Dreben mean, asked Quine, that if Frege had seen Gode1's proof of the completeness theorem, Frege wouldn't have been able to understand it? Dreben replied: the Frege of the Grundgesetze would have understood it as a piece of mathematics, as showing that a certain set-theoretically defmable class is recursively enwnerable; but Frege would have questioned whether this set was a proper specification of his notion of logical truth, of logical validity.19 Naturally, Quine, devoted as he is to Tarski, was not satisfted with this response. Dreben's reply to Quine is also his reply to the suggestions of Heck, Stanley, and Tappenden that since logic is universal, by the techniques of Godel and Tarski, many metasystematic questions, in particular the completeness theorem, can be carried out within the system. 20 Presumably Heck, Stanley, and Tappenden would share Quine's dissatisfaction, as of course would Dummett. II My aim in what follows is modest: I wish to slightly shift the scope of the interpretive debate about Frege and semantics. Rather than directly addressing those elements of FreRe's conception which have been treated at length in the recent literature, I foeus on one especially puzzling feature of I''rege's formal procedure, namely, his usc of the double stroke (" 11-"), his symbol for explicit definition. Frege's use of this symbol-and his attitude toward his use of it--seem to me to exemplify in an especially perspicuous way how it is that Frege's writings can so easily lend themselves to such radically different interpretations and appropriations of his philosophy. For the double stroke for definitions shows us, I claim, the importance of the fact that at least until 1903 Frege never clearly articulated a distinction between "metalanguage" and "object language," or between a "metasystematic level" and an "object level."21 It helps us to see that even if we can quite naturally regiment Frege's work by means of the distinction between meta- and object level without appearing to go beyond what is written in his logical system, the deeper philosophical questions about Frege's standpoint still remain. My discussion is not intended to or exhaustively characterize the interpretive questions I have canvassed about l''rcgc and scmantics. I intcm], rathcr, to cmphasize that these questions will not be resolved by any argument resting its case on purely formal or mathematical features of Frege's system. If I am right that there is no way to establish how to read uniquely Frege's use of the 152 Juliet Floyd I III! r I ..," !: :1 '\ I, r, r i r .' I: ,I .'" II IIri , II fI; II I double stroke, then the interpretative questions make themselves felt even at the most basic level of Frege's system, in the part of his work one might think is not opell to philosophical interpretation, viz., in his formal system. This confrrms that the debate about how best to read Frege is at root a philosophical one, not amenable to resolution by means of any particular formal or mathematical distinction we may find in Frege. In contemporary logic and philosophy of logic we may distinguish three different issues surrounding identity statements and definitions. First, we may give a syntactical specification of identity statements and definitions in a particular formal language by making statements in a metalanguage (whether metasyntactic or metasemantic). Sccond, we may discuss the nature or purpose of definitions and the nature or purpose of identity statements about the identity relation. Third, we may give an account of the nature of the identity relation itself, the naturc of the truth asserted in a true identity statement. 22 It is striking that in his 1894 review of Husserl's Philosophy of Arithmetic, Frege seems to have conflated these issues. This apparent conflation seems to indicate that Frege did not always sharply distinguish between statements and what statements are about. For in this review Frege forwards the following argument, which purports to show that identity is undefmable: "Since any defmition is an identification [Gleichung] , identity itself [Gleidlheit selbst] cannot be defmed" (Frege 1984: 200; 1967: 184; 1980a: 80).23 The argument appears to be this: in order to present a defmition, one must use the identity sign, and thus there is no way to present an explicit, i.e. eliminative definition of the idcntity sign itsclf. 24 Thereforc, Frcge reasons, identity must be taken as a logical primitive, an undefinedbecause undefmable, and hence logically simple-notion. 25 This 1894 argument for the indefinability of identity is simultaneously an argument that the notions of difinability and of d¢nition cannot be defined. 26 For in the Begriffsschrift and in the Orundgesetze, in order to present a definition Frege must always use, not only the identity sign, but also his special sign for definitions, the double stroke (U 11-"). Hence, by Frege's 1894 line of reasoning there is no way to c1iminate the sign for definition by means of a defmition itself, for in order to frame a defmition one must always use the double stroke. The plausibility of this reasoning and the parallel argument about identity, thus appear to tllrn in part upon Frege's not sharply distinguishing between an object level and a metaIevel, and in . part upon the way in which he handles definitions formally within his logical systems. However, fourteen years earlier Frege had claimed to be able to defme identity, to sho\v, that is, that it is not a 10gicaUy simple notion. Presumably his idea was 10 use Leibniz's law of the icknlily of indiscernihles, Semantics and the double d¢nition stroke 153 according to which a is identical to b if a and b share all their properties. In "Boole's Logical Calculus and the Begrilfsschrift' (1880-81) Frege argued that his own formula language is superior to Boole's partly because Boole uses a greater number of signs. Indeed I too have an identity sign, but I use it between contents of possible judgment almost exclusively to apply the stipulation of sense of a new designation. Furthermore I now no longer regard it as a primitive sign, but would defme [erkliirm] it by means of others. In that case there would be one sign of mine to three of Boole's. (Frege 1979: 35-6; 1969: 40) This appears to indicate both that Frege previously identity as a primitive notion, and that he came, at least for a time, to change his mind about this. 27 This may be connected with his doubts about the treatment of identity he propounded in the Begriffsschrifl (1879). There Frege held that identity is a relation between names and not objects, writing: Identity of content differs from conditionality and negation in that it applies to names and not to contents. Whereas in other contexts signs are merely representatives of their content, so that every combination into which they enter expresses only a relation between their respective contents, they suddenly display their own selves when they are combined by means of the sign for identity of content; for it expresses the circumstance that two names have the same content. Hence the introduction of a sign for identity of content necessarily produces a bifurcation in the meaning (Bedmt:uniJ of aU signs: they stand at times for their content, at time for themselves. (Frege 1879: §8) Of course, Frege is aware that it will strike his readers as odd that names come to change their BtJeutungen in contexts where the identity sign appears between them. He even raises the spectre of an attitude toward identity later pursued by Wittgenstein in the 'TractlltUS: At frrst we have the impression that what we are dealing with pertains merely to the expressiDn and not to the tJwught, that we do not need different signs at aU for the same content and hence no sign whatsoever for identity of content. (Frege 1879: §8) But Frege immediately calls this "an empty illusion" and argues that the sign for identity of content is needed whenever we have two different ways of "determining" a conceptual content, for, "that in a particular case two WI!JIs of detnmining [the content} really yield the same result is the content of a judgment' (1879: §8). He also points out that "a more extrinsic reason for the introduction of a sign for identity of content" is our need for definitions, it being "at times expedient to introduce an abbreviation for a lengthy expression" (1879: §8). But it does not seem that Frege conceives of dther definitions or of the shift in the meaninK of the names which j' I 154 Juliet Floyd I' !; takes place in identity contexts in terms of a clear distinction between meta and object level. Rather, he treats identity contexts as shifting the Bedeutungen of certain names within the very same language. Frcgc came to surrender his Begriifsschrifl theory of identity. 28 By 1890-1 he had divided up his original notion of conceptual content into two aspects, via the notions of Sinn and Bedeutung, and he used these notions to explain the relation between names and an object in both informative and uninformative truths of identity. Whether or not this is connected with his argument that identity is a logically primitive (undefmable) notion, after 1884 he never again claims that identity can be defmed. In the Gruruigeset<,e identity is taken to be a logical primitive. 29 Despite his vacillation about the nature of the identity relation and the nature of our statements about that relation, Frege seems never to have ./iJrmal!v. However, we vacillated ahout how to handk that just as there are competing ways of reading Frege's discussions of truth, meaning, and sense, so there are competing ways of reading his use of the double stroke for defmitions. I shall focus most closely on the use of the double stroke in the Grundgesetze, which, I believe, should be the primary testing ground for any interpretation of Frege. By closely following Frege's textual divisions in volume I of the Grundgeset;:;e, we may and should, at least prima facie, distinguish several different ways in which Frege discusses-and uses-his formula language. First, in his Introduction, he presents, entirely in German, a series of what he calls philosophical arguments about the nature and scope of logic. Second, in Part I, Chapter I he introduces his basic logical notions (such as fimction, concept, obJect, vahle-range, and generality) and presents the primitive signs, the Basic Laws and the formal rules of inference of the formula language. Third, in Chapter 2, Frege discusses the nature of definitions, introduces the double stroke, and then uses it to frame what he calls besondere, that is, "special" definitions. These definitions are applied later in and Basic proo[q through substitutions. Fregc also lists the Baqic Rules and "adds a few supplementary points" (1893: §§47-8). Part II of Volume I, where Frege's proofs occur, falls into two kinds of sections (cf. 1893: §53). On the one hand there are the Z"legungen, or "analyses," where Fl'l'gc sketdu's lhe e!l'Il11'lIts of his pmols ill a mixture of German, signs of the formula language, and names of judgments, that is, (expressions of) judgments surrounded by quotation marks. 30 Each such Z"legung is set off in a separately numbered section, and is immediately followed by another section presenting a precise construction, or AtdfJau. An At.ifbau is used to construct proofs. It contains primitive signs of the formula language, indices and markers to help us read the A'!fbau, and signs that have been explicitly introduced by means of prior defmitions. It Semantics and the double definition stroke 155 contains no words of German or of any other natural language. An A"!fbau always consists of an annotated series of judgments, each prefaced by the judgment stroke, and each one of which either (i) is an identity indexed with a capital Greek letter referring back to an explicit defmition given in a prior Zerlegung, or else (ii) follows by means of Frege's explicit inference rules from Basic Law(s) or from truths previously asserted in earlier Atdhauten. By contrast, the Zerlegungen contain no formal proofs. Indeed, the Zerlegungm contain no judgments in Frege's special sense, no uses of the judgment stroke to recordjudgments. The Zerlegungen contain on{y words f!f German, as weU as names f!f jutigmmts and occurrences f!f identities preceded l!J the double strolce which rm not surrounded l!J quotation marks. Writes Frege, In connection with the following proofs I emphasize that the implementations which I always mention under the heading "Zerlegung' serve only for the ease of the reader; they could fail to appear without lessening the power of the proofs in any way, for that is to be found only under the heading "Acifbau". (Frege: 1893: §53) Now what, precise{y, does Frege say about the use he makes of his double stroke for defmitions? We are interested in the bearing this question has on the appropriateness of ascribing to Frege a sharp distinction between his use and his mention of his formula language, between his procedures at what we might naturally calI his object level and what we might naturally calI his meta level. It is striking that Frege so sharply distinguishes between contexts of formal proof, his AtdfJauten, and contexts in which he gives definitions and discusses his formula language, his Z"legungen. The anti-semantical tradition would, presumably, deny that the Zerlegungm are part of any substantive metatheory for the formula language. 3l I thus ask the following interpretive questions: I Is Frcge's double stroke a (primitive) sign of his formula language? 2 If it is not a sign used within Frege's formula language, but rather a sign ahout the formula language, does Frege's use of the double stroke in his defmitions function as elucidation of his formula language, or rather as part of a metatheory of his formula language? 3 If the double stroke is part of a metatheory, is this a genuine semantical tht'ory or just a mtla.fVntactic tht'ory? The answers to these questions, as I shall now try to explain, are not as readily available as one might think. In both the Begriffischrifl and in the Grundgeset;:;e the double stroke indicates the operation of derming one sign in terms of others, and this operation always has a double role. Frege's double stroke reflects what he used to calls the "double-sidedness of formulas" (Doppelseitigkeit tier d 156 Juliet Floyd j I I' r il ! I ) (i , i express his defmitions (cC 1879: §24). From 1879 on, whenever Frege uses the double stroke to frame a definition, he takes himself to be putting fOlWard a stipulation, something that is not a judgment at all (1879: §24). Yet any such stipulation can, he maintains, be "immediately transformed" into a judgment of identity and used in logical constructions, in proofs within his system. (In the 1879 Begriifsschrifl Frege calls these judgments "analytic"d 1879: §24.)32 And this is how Frege actually proceeds. He frames an explicit definition with the double stroke and then peels away the first of the two vertical strokes in order to assert a corresponding identity judgment in the context of an A,ybau. Thus in its first role, a defmition stipulates that definiens and defmiendum are to share the same content (according to Frege's 1879 view), and the same Sinn and Bedeutung (according to Frege's post-1890 view). On Frege's early account of identity, I II I " !Ii dt! ijl :'I. II L ,I r t n 'I in LA definition is a proposition which] differs from the judgments considered up to now in that it contains signs that have not becn delined before; it itself gives the definition. It does not say "The right side of the equation has the same C'flntent as lhc left," but "It is to have the same content." Hence this proposition is 1I0t II jUtIWlIcllt, 1Il1l1 c'nllsrclw"lItly 'lIll a .!l',,'''dif ';rl/!A""ml rithrr. tn lise thl' Kalltiall expression. (Frege 1879: §24) And on Frege's later account of identity, LThe douhle stroke of definition J is uscd in place of the judgment-stroke where something is to be, not judged, but abbreviated by definition. We introduce a new name by means of a d!finitUm by stipulating that it is to have the same sense LSinn] and the same meaning [Bedeutung] as some name composed of signs that are familiar. (Frege 1893: §27) In its second role, a defmition is immediately "transformed" into ajudgment in an Atifbau. This judgment says that the two signs do share the same content--or, after 1890, the same Sinn and the same Bedeutung. Frege conceives of the double stroke as literally picturing this dual aspect of defilil'st, vertical hal' the stipulation; its verlical har, nitions: which looks just like the judgment stroke, indicates the transition from the stipulation into a judgment of identity. The judgment is only expressed, i.e. asserted, in an A,ybau, in proofs where it is prefaced by the judgment stroke. But the transition from a stipulation to a judgment is immediate and unexplained. 33 How are we to interpret Frege's use of the double stroke? One suggestion is to build on our contemporary distinction between meta- and object language in characterizing Frege's procedures. Then we might hold that the double stroke is not a primilive sign of Frege's formula language. After SemantUs and the double difinition stroke 157 all, it occurs only within the Zerlegungen 34 , and never within Atifbauten. Its role, we might claim, is relegated to a place "outside" the workings of the formula language itself. Furthermore, because an A,ybau conlains defined terms which are not primitive signs of the formula language, we might also hold that an ArdlJau rifers to the "pure" formula language itself, in which formal derivations are written sole{y with the use of primitive signs of the formula language. This I shall call the reading of the double stroke. 35 On this reading Frege is seen to treat the double stroke for defmitions as a metasyntactic device: it is not part of his formal system. Every use of Frege's double stroke in a Zerkgung is coded into a corresponding Atifbau through its index, a capital Greek letter. This letter must be written in the margin at every line of an Arifbau where the corresponding judgment of identity appears. So, on the metasyntactic reading, the double stroke is an explicit sign of an algorithm which "decodes" every ArdlJau of the Grundgesetz:.e into a proof couched solely in primitive signs of the formula language. (fhe ArdlJauten on this view are like "programs" which can be alF;orithmic.ally carried into an "assembly Ir.vellanguage.") In the primitive fbnllula lanKllaKc' il:wll' all "ddinitional" ilklllily vanish. This might be held to explain why Frege repeatedly remarks that Irolll the perspective of the formula language, definitions are "mere abbreviations," convenient shorthand devices used to make logical constructions, A,ybautm, perspicuous. Differently put, Frege's Zerlegungen, his analyses, use definitions in order to present his logical constructions in such a way that they can be taken in by us: or The need of defmitions never ceases to be apparent in any attempt of this sort.... The definitions do not really create anything, and in my opinion may not do so; they merely introduce abbreviated notations (names), which could be dispensed with were it not that lengthiness would then make for insuperable external difficulties. (1893: Preface, vi, my emphasis) In the Begriffsschrifl the same point appears to be made twice: IiII' the iutrotluclillll of a lor of l:nnlent is A ilion' exhinJic that it is at times expedienl to introduce an ahhreviation for a lengthy expression ... (Frege 1879: §8, my emphasis) [A defmition] is not a judgment, and consequendy not a either, to use the Kantian expression.... If [it] were a synthetic judgment, so would be the propositions derived from it. But we can do without the notation introduced by proposition and hence without the proposition itself as its definition; nothing follows from the proposition that could not also be inferred without it. Our sole purpose in introducing such definitions is to bring about an extrinsic simplification by stipulating an abbreviation. They serve besides to Semantics and the double definition stroke 158 Juliet Floyd emphasize a particular combination of signs in the multitude of possible ones, so that our faculty of representation can get a firmer grasp of it. (Frege 1879: §24, my emphasis) According to the metasyntactic reading, an ArdlJau of the GnmdgesetJ:J which is not written exclusively in the primitive formula language represents a genuine proof; that is, it presents a precise set of rules telling us how we are to write down a particular proof in primitive signs of the formula language. We can take it in as a representation because it uses abbreviations. But we know it successfully represents a genuine proof only insofar as every definitional judgment of identity occurring in it can be eliminated in a mechanical way, with no appeal to elucidations, hints, or guesswork. Of course, this "metasyntactic" reading of the double stroke cannot wholly sever the signs it speaks of from their Sinne and from their &deutungen. When Frege speaks of a defmition as a "stipulation," he does not mean that a defmition tells us to substitute one uninterpreted sign for another uninterpreted series of signs. Rather, the stipulation appeals to the content, to the Sinn and the Bedeutung, of the definiens. As he writes in "Function and Concept," "In defmition it is always a matter of associating with a sign a sense or a meaning. Where sense and meaning are missing, we cannot properly speak either of a sign or of a definition." Thus the metasyntactic reading inevitably spills over into some sort of metasemantical reading. The double stroke may be said to function as a paradigmatically semantical sign insofar as it tells the reader how to assign Sinne and &deutungen to signs. Nevertheless, even though the assignment of Sinne and &deutungen to newly introduced signs may take place in a metalanguage, the terms of this language have, at least so far, no more, no less and no different semantical content than what is already gotten--directly and immediate1y-out of the object language. Thus, the question as to whether this metalanguage consists of mere elucidation, as opposed to an articulate semantical or philosophical theory, remains unresolved-and perhaps unresolvable-by this reading. Moreover, in an unpublished manuscript which further develops his antisemantical reading of Frege, Ricketts explicitly denies that Frege's double stroke for defmitions is in any sense a metasyntactic device. 36 According to Ricketts, just as the judgment stroke uses the formula language to make judgments-· something we do within IUIIRuagc -so the double stroke uses the formula language to defme one sign in terms of another-something else we do within language. On his view, the double stroke is part of Frege's formula language, and defmitions take place within the universal formula .. 159 language. Thus does Ricketts interpret the immediacy of Frege's transitions from stipulations to judgments of identity. If Ricketts is correct, then Frege's system has no precise, closed, vocabulary, and there is no set of well-formed formulas given by Frege once and for all. Instead, through the use of the double stroke, the formula language is systematically enlarged in a step-by-step way. This enlargement helps us to work with the formula language more easily, and this is the point of Frege's definitions. On this reading there is no room to view the Atifbauten as metasyntactic representations of the primitive formula language, for there is no formula language with a fIXed vocabulary. Frege had no closed formalism at all! Instead, the ArdlJauten function by means of the formula language in the context of a carefully regulated, growing vocabulary. The additions of new vocabulary in defmitions function like special axioms of identity which are always conservative extensions of Frege's original primitive system-though as Ricketts emphasizes, Frege nowhere attempts to prove such a metatheoretic statement. On Ricketts' reading, the uses of the double stroke, though not part of Frege's logical constructions, are formalized in the Arifbauten by means of the judgments of identity which are indexed to the explicit definitions given in the Zerlegungen. Defmitions are set off from ArdlJauten because they are not judgments, but, rather, stipulations. But they are still "part of the system" in the sense of being framed within the universal formula language, they are not stipulations or abbreviations made from a standpoint external to the system or the language. Thus Ricketts holds that Frege did not have a metasyntactical theory, but only rules for writing proofs down in his formula language.37 This shows how deeply the debate between the semantical and the anti-semantical readings of Frege may be said to penetrate our understanding of Frege's philosophy and logic. For we have been told--even by van Heijenoort himself-that Frege was the first to frame precisely the notion of a formal system (1985: 12). If Ricketts is right, then even this understanding of Frege may have to be revised, even if only slighdy. My point has been to emphasize the difficulties we face in attempting to extract defmitive answers about Frege's overarching enterprise, and even about the structure of his formula language. To read Frege we are constantly thrown back on our own philosophical predilections---even at 38 those points where Frege's procedures may seem most clear. NOTES This chapter is slated to appear in French in Frege, Logique MathimaJique et Philosophie, ed. Mathieu Marion and Alain Voizard (paris: I'Harmattan). I 160 2 3 4 5 6 7 8 9 }ulift Hr!vd hereby give my permission for the paper to be printed there and for my rights to it to be transferred to Routledge for the present volume. I am indebted to the English translations of Frege cited, though I have occasionally altered them when I felt it necessary. Unless otherwise specified, page numbers refer to the original page or section number of Frege's works. Van Heijenoort, "Logic as Calculus and Logic as Language," (1985: 12-13): "Boole has his universe class, and De Morgan his universe of discourse, denoted by '1'. But these have hardly any ontological import. They can be changed at will. The universe of discourse comprehends only what we agree to consider at a certain time, in a certain context. For Frege it cannot be a question of changing universes. One could not even say that he restricts himself to one universe. His universe is the universe." Van Heijenoort, "Frege and Vagueness," (1985: 95). Van Heijenoort is quoting from Quine's "Facts of the Matter" (1977), and holds, with Quine, that these words accurately describe Frege's conception of his own enterprise. Of course, the whole point of Quine's thought experiment about the indeterminacy of translation is to undercut the view that there is a "latent and determinate content" to be elicited from ordinary language. Compare Burton Dreben, "Pumam, Quine and the Facts" and Hilary Pumam's reply in Hill (ed.) (1992). Compare Michael Dummett, The Interpretation c!f Frege's Philosopf!y (1981: 13, 17-19). Goldfarb wrote: "Frege has often been read as providing all the central notions that constitute our current understanding of quantification. For example, in his recent book on Frege (Frege: Philosophy c!f Language, ftrst edn), Michael Dummett speaks of 'the semantics which [Frege] introduced for interpretation of formulas of the language of predicate logic.' That is, such a formula ... is obained by assigning entities of suitable kinds to the primitive nonlogical constants occurring in the formula ... [T]his procedure is exactly the same as the modern semantic treatment of predicate logic' (pp. 89-90). Indeed, 'Frege would therefore have had within his grasp the conrepts necessary to frame the notion of the completeness of a formalization of logic as well as its soundness ... but he did not do so' (p. 82). This common appraisal of Frege's work is, I think, quite misleading" (Goldfarb 1979: 351). I use the term "anti-semantical" as shorthand for "anti-theory of meaning" or "anti-theory of semantics." It should be clear to the reader from what has already been quoted that in a loose or intuitive sense of the term, Frege lives in semantics. My overview of this "anti-semantical" tradition is highly schematic, and obviously cannot do justice to the specifics of each individual member's view of Frege. Others who have discussed and characterized this tradition and its relation to Dummett include Richard Heck (forthcoming); Jason Stanley (forthcoming); and Sanford Shieh (forthcoming). James Conant's article "The Search for Logically Alien Thought: Descartes, Kant, Frege and the Tractatus" is replied to by Hilary Putnam (1992). See also Kemp (1995). Note that "Erlliuterung" is not systematically translated (Cf. 1969: 224,232, 254). An especially subtle working out of the sorts of paradoxes Frege faces in introducing his system is contained in de Rouilhan (1988). Sm/antics and the double ddinition stroke 161 10 Passages where Frege calls attention to such ideas occur in "On Sense and Reference" (1980a: 61; 1984: 161; 1967: 145); see also "Concept and Object" (1980a: 54; 1984: 193; 1967: 177; 1980b: 37; 1976: 63; 1893, 1903: Appendix 2, n I, 1969: 288ft). II For example, in "What Nonsense Might Be" (1991a: 97) Cora Diamond explicitly criticized Dummett's reading of Frege, emphasizing that: "On the Frege-Wittgenstein view, if a sentence makes no sense, no part of it can be said to mean what it does in some other sentence which does make sense-any more than a word can be said to mean something in isolation. If 'Caesar is a prime number' is nonsense, then 'Caesar' does not mean what it does when it is in use as a proper name, and the last four words do not mean what they do in sentences which make sense" (Diamond 1991 a: 100). "I have wanted parallels (between Frege and Wittgenstein) to be suggested. I have wanted to suggest ... a distance between Frege's view and what might be called a Tarskian view" (199Ia: 112). 12 All of Ricketts' work is relevant, but see especially 1985 and 1986b. 13 Frege's arguments that truth is indefmable are canvassed by Ricketts in "Objectivity and Objecthood" (1 986b), as well as in "Logic and Truth in Frege" (1996a). 14 See Diamond (199Ia: ch. 4), Sluga (1980: 182), Ricketts (1985), and Weiner (forthcoming). 15 Like Dreben, Goldfarb and van Heijenoort, Baker and Hacker draw a parallel between Frege and Russell: "It is salutary ... to remember how modern the semantic conception of validity is. Although Russell acknowledged a profound debt to Frege 'in all questions of logical analysis', Principia Mathemati£a did not formulate a clear distinction between logical truth and provability within its axiom system, and hence it did not envisage the possibility of proofs of consistency, independence, and completeness" (1984: 373). 16 Compare van Heijenoort, "Logic as Calculus and Logic as Language" (1985: 14), and Goldfarb (1979). 17 Compare van Heijenoort, "Logic as Calculus and Logic as Language" (1985: 14). Notc that in a footnote on p. 95 of ft'incipia Mathtmatica (1910 ednj p. 91 1925 edn) Russell and Whitehead explicitly refer to the 1903 passage quoted by Goldfarb (see p.l44 above). It might be added that even as late as Russell's Introduction to Mathmuzti£al Philosopf!y (1919) we ftnd Russell maintaining that "The theory of deduction ... and the laws for propositions involving 'all' and 'some,' are of the very texture of mathematical reasoning: without them, or something like them, we should not merely not obtain the same results, but we should not obtain any results at all. We cannot use them as hypotheses, and deduce hypothetical consequences, for they are rules of deduction as well as premisses. They must be absolutely true, or else what we deduce according to them does not even follow from the premisses" (Russell 1919,1920: 191). 18 Dreben, "Frege on FOundations," 4/6/92 lecture to the Boston University Colloquium for the Philosophy of Science. Quine responded as a member of the audience. 19 See, for example, Thomas Ricketts (1996a). Ricketts argues that for Frege truth is not a genuine property, and truth not a genuine concept. Hence Frege would not have been able to accept, without significant modification of his standpoint, the Tarskian criterion of material adequacy as relevant to any lUI r'" ........ ".I • . '" ... . , - - - - . - - - . - .. * e.... IS. 20 21 22 23 24 25 26 27 28 29 ...........-.- ... Semantics and the double definition stroke 162 Juliet Floyd truth definiJion. A fortiori, from Frege's point of view the Godel completeness theorem, formalized in the &griffischrijl, could not give any explanation or account of the notion of "logical truth." Compare Ricketts' "Objectivity and Objecthood" (1986b: 76, especially footnote 18). For an investigation of a similar response, namely Wittgenstein's response to Godel's first incompleteness result, see my "On Saying What You Really Want to Say: Wittgenstein, (j()del and the Trisection of the Angle" (Floyd 1995). In the very late "Logical Generality" (1923 or later) Frege began to articulate a distinction between what he calls Hilfssprache and Darlegungssprache, which could be taken to anticipate Tarski's later distinction between metalanguage and object language. See Frege 1979: 260lf., 1969: 287lf. We can also discuss, in set theory, the capacity of a formal theory to define, i.e., to contain, formulas model-theoretically satisfiable by certain relations and Such a discussion would I Ihink be quilc fon'iRJl 10 Frege, but I shall nol argue Ihis point here. I am grateful to J. Weiner for having pointed passage out 10 me in her and the OriRins of Analytic Philosophy" talk. (1!)/llla: /I(); 1%7: 1/11). holds that Scc H'cgc's revicw of Frege's argument in the Husser! review that identity is a logically primitive notion is "not very convincing" since "it is only possible for Frege to say this because he takes the sign of idenlity to do duty also for the biconditional, which is in turn possible only because he assimilates sentences to names, viz., of truth-values; and in any case it seems more natural to take a defmition as a stipulation of the interchangeability of two expressions, rather than of the truth of a sentence connecling them.... [rJhe thesis of the indefinability of identity does not seem to play any important role in Frege's philosophy" (Dummett 1981: 543). Frege in one way does and in another way does not countenance a sharp distinction between stipulating interchangeability among (linguistic) expressions and capturing, analyzing, proving, and asserting by means of a real defmition. Compare Benacerraf, "Frege: the Last Logicist" (1995). What is "logically simple" cannot, according to Frege, be defmed (c( 1903: §146, n. I; 1980a: 42). Rosalind Carey pointed out to me that in "Insolubilia and Solution by Symbolic Logic" Russell writes that "the notion of defmition ... oddly enough, is not defmable, and is indeed not a defmite notion at all" (1906 essay, reprinted in Russell 1973: 209). The remark occurs in the context of Russell's treatment of the Richard Paradox and the Vicious Circle principle. Tappenden discusses Frege's paper on Boole's logical calculus, and suggests that between at least 1880 and 1884 Frege developed an interesting and novel account of the fruitfulness of definitions and concepts-albeit a view which came into tension with the later Sinn/BedeutuTlf. distinction. Tappenden argues that qualllificational cOlllplexity in a corresponds to fruitfulness of a defmition at this stage of Frege's development. This may be related to Frege's claim to be able to define identity, although I cannot pursue this conjecture here. See Tappenden 1995. Compare Picardi 1988. For an informative account of the difficulties facing Frege's early view, see Mendelsohn 1982. Basic Law III in the formalizes Leibniz's law, but does not purport in any way to be a defmition of identity. Section 65 of the GnmJItJgen could be •.• 30 31 32 33 34 35 36 37 38 163 read as explicidy advocating Leibniz's law as a defmition of identity, although even in the Husser! review Frege mentions the law as expressing something important about identity (see Frege 1984: 20). Jan Harald Alnes has persuaded me that it is doubtful that in the Grundlngen Frege held that identity could be defined. More precisely, the Zerlegungen contain names of judgments which will be made in the corresponding Al!fbau, and some of these judgments will contain defmed terms, as well as primitive terms of the formula language. This denial is quite explicit in Weiner 1990: ch. 6. The word "analytic" does not occur in Grundgesetze. However, in 1914 Frege says that the sentence expressing the judgment of identity corresponding to a defmition is "a tautology" ("Logic in Mathematics," in 1979: 208). (Presumably he writes this qfter he and young Wittgenstein had spoken.)Jan Harald Alnes has suggested in convenation that sameness of sense on two sides of a lrue identily statement only whrn Ihe identity springs from an explicit defmition or is an instance of a basic law. This would explain why, even alier introducing the Sinn/lkdeutung Frege holds in and that Ihe Iwo sid('s of an inslance nf Ra.qjc lA'\w V "express the same sense, but in a dillerenl way," while simultaneously holding that derived truths (such as "2+2=4") involve expressions with differing scnses on eilher side of the identity sign (Frege 1984: 143; 1967: 130). This formal point points toward several unclarities in I'rege's attitude toward defmitions in the context of his "reduction" of arithmetic to logic. See, in particular, lknacerraf 1995, and the response to Benacerraf by Weiner 1984. Sec also Picardi 1988, Dummell 1991b (ch. 2), and Tappenden 1995. And, of course, in the section on "special definitions" in Grundgeset.c:.e Chapter I and in the table of definitions in Appendix. I am aware that there is a long established usage in which the metalanguage is dubbed the "syntax language" of the object language. In this sense what I am calling "metasyntactic" would be called "syntactic." In conversation, correspondence and in Chapter I section 3 of his unpublished manuscript on Frege. Further evidence for Ricketts' view, as Ricketts himself has suggested, is §48 # 12, labeled "Citation of Definitions." Here, according to Ricketts, Frege sets out the usc of the double stroke as a rule of inference of the formula language. A critic of Ricketts' view would have to claim that this section, rather than setting forth a rule of inference, is giving what Frege called in §47 a "supplementary point." Note, however, that in Grundgesetze §53, Frege does speak of "the rules to which I appeal in the Zerlegungen" and refers the reader back to 48. I was stimulated to think about Frege's treatment of definitions as a result of responding to papers by Hans Sluga and Joan Weiner at a December 1995 American Philosophical Association symposium on "Frege and the Origins of Analytic Philosophy." Both Sluga and Weiner argued against 5CVtral widespread interpretations of Frege's distinction between Sinn and and their arguments got me to think more carefully than I had about Frege on identity and defmitions. Thanks are due to them for their stimulation, both at the symposium and in other published works. I also thank Anat Biletzki and Anat Matar for their enthusiasm and generosity in organizing the January 1996 Tel Aviv conference at which an earlier version of this in 1 • Semantics and the double tkfinition stroke 164 Juliet Floyd paper was read. The audiences at Tel Aviv and at Oslo University in June 1996, contributrd for which I am Kl'ateful. My understanding of Frege and controversies about how to read him is enormously indebted to Burton Dreben, and I thank him for our many hours of discussing Frege since 1983, and for his sage editorial and philosophical advice at every stage of this paper's composition. Jan Harald Alnes, Rohit Parikh, and Sanford Shieh have also offered helpful comments on dralls of my paper, as has Thomas Ricketts, manuscript 011 Frege's philosophy has been especially instructive, especially its Chapter I, §3 on Frege on defmitions. BmUOGRAPHY Baker, G. P. and Hacker, P. M. S. (1984) 1JJgieal Excavations, Oxford: Oxford University Press. Benacerraf, P. (1995) "Frege: The Last Logicist," in W Demopoulos (ed.) Frege's of MathemoJics, Cambridge, Mass.: Harvard University Press. Conantj. (1992) "The Search for Logically Alien Thought: Descartes, Kant, Frege and the Tractatus," in Hill, C. (ed.) Philosophical Topics 20/1. de Rouilhan, P. (1988) Frtge: us Paradoxes de La Rtprbmlation, Paris: I,rs Editions de Minuit. of Mathematics, Cambridge, Mass.: Demopoulos, W (ed.) (1995) Frege's Harvard University Press. Diamond, Cora (199Ia) The Realistic Spirit, Cambridge, Mass.: MIT Press. ---{I 991 b) "Ethics, Imagination and the Method of Wittgenstein's Tractatus," R. Heinrich and H. Vetter (eds) Bilder tier Philnsophie, Wzener R£ihL 5, 55-90. Dreben, B. (1962-96) Unpublished lecture transcriptions. --"Frege on Foundations," 4/6/92 lecture to the Boston University Colloquium for the Philosophy of Science, unpublished transcription. ---{1992) "Putnam, Quine and the Facts," in Hill, C. (ed.) Philosophical Topics 20/1. Dreben, B. and van Heijenoort,j. (1986) "Introductory note to 1929, 1930 and 1930a," in S. Feferman, J. W Dawson, Jr, S. C. Kleene, G. H. Moore, R. M. Solovay,J. van Heijenoort (eds) Kurt GOrkl, Collected Papers, L New York, Oxford: Clarendon Press. of Language, Cambridge, Mass.: Dummett, M. (1973, 1981) Frege: Harvard University Press. --- (1978) Truth and Other EniRma-r, Cambridge, Mass.: Harvard University Press. --(1981) The Interpretation of Frege's Philosophy, Cambridgc, Ma.,s.: Harvard University Press. --{1984) Unsuccessful Dig: Critical Notice of G. Baker and P. Hacker, lAgieal Excavations" in C. Wright (ed.) Hege: Tradition and Iriflumce, Oxford: Blackwells. 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Austin as The Foundations of Arithmetic (2nd edition, Oxford, 1953; reprinted by Northwestern University Press, 1980). ---{1893, 1903) Grundgeset<.e rkr Arithmetik, Vol. I, Vol. II; reprinted by Hildesheim: Georg Olms Verlagsbuchhandlung (1966); partially translated into English (through s.52) as The Basic Laws of Arithmetic by M. Furth, Berkeley: University of California Press (1964). ---{I 967) Kleint Schriftm, I. Angelelli (ed.) Hildesheim: Georg Olms Verlagsbuchhandlung. ---{I 969) Nachgelassene Schriftm, H. Hermes, et aL (eds) Hamburg: Felix Meiner. - - ( 1971) On the RJundations of Geometry and Formal Theories of Arithmetic, trans. Eo Kluge, New Haven: Yale University Press. --{I 976) WlSsenschtifllicher Bril.fivechsel, G. Gabriel et at. (eds), Hamburg: Felix Meiner. ---{1979) Posthumous Writings, Hans Hermes, Friedrich Kambartel, Friedrich Kaulbach (eds), trans. P. Longand R. White, Chicago: Universityof Chicago Press. ---{1980a) Translationsfrom tire Philosophical of Gottlob Frege, trans. M. Black and P. Geach (eds), third edition, Totowa, 1'{J: Rowman and Littlefield. - - (1980b) PhilosophicalandMathematical Correspondence, G. Gabriel etal. (eds), abridged by B. McGuinness, trans. H. Kaal, Chicago: Universityof Chicago Press. --(1984) Collected Papers on Mathematics, 1JJgie and B. McGuinness (ed.), trans. M. Black et al., Oxford: Blackwell. Gt'idel, K. (1986) Kurt GOrkl, Collected Papers, 1, S. Feferman,J. W Dawson,Jr., S. C. Kleene, G. H. Moore, R. M. Solovay, J. van Heijenoort (eds) New York: Oxford, Clarendon Press. Goldfarb, W. (1979) "Logic in the Twenties: the Nature of the Quantifier," Journal of Symbolic Logie 44,3: 351-68. Heck, R. (forthcoming) "Frege and Semantics," in Ricketts (ed.) The Cambridge Companion to Frege, Cambridge: Cambridge University Press. Hill, C. (ed.) (1992) Phiiosophical10pics 20, I: 114-80. Hintikka,j. (1979) "I<'rcge's Hidden Semantics," Revue Inttrnationale de Philosophie 33: 716-22. -and Hintikka, M. (1986) Investigating Wlttgenstein, Oxford: Blackwell. Kemp, G. (1995) "Truth in Frege's 'Law of Truth'," SyntJrese 105: 31-51. Marion, M. and Voizard, A. (eds) (forthcoming) Frege, 1JJgifjue MaJJrJmatique et Philosophie, Paris: I'Harmattan. Mendelsohn, R. (1982) "Frege's Begrilfsschrift Theory of Identity," Journal of tire History of Philosophy XX, 3: 279-99. Picardi, E. (1988) "Frege on Defmition and Logical Proof," in Atti rkl Congresso Temi e prospettive rklla logiea e rklla filosofia della scien;:,a contemporanu, Cesena 7-10 gennaio 1987, Vol. I CLUEB, Bologna Italy: 227-30. 166 Juliet Floyd Putnam, Hilary (1992) "Reply to Conant," "Reply to Dreben," in PhilJJsophi£al Topics 20, I. Quine, W V. 0. (1977) "Facts of the Matter," in R. W Shahan and K. R. Merrill (eds) American from Edwards to Qyine, Norman, Ok.: University of Oklahoma Press. Ricketts, 1: (1985) "Frege the Tractatus, and the Logocentric Predicament," NoUs 9,1: 3-15. --(1986a) "Generalit); Meaning and Sense in Frege," Pacift PkiUisophical QyarterlY 67,3: 172-95. ----{1986b) "Objectivity and Objecthood: Frege's Metaphysics of Judgment," in L. Haaparanta and J. Hintikka (eds) Frege SyntksiQd, Dordrecht: D. Reidel: 65-95. ---(1996a} "Logic and Truth in Frege," Proceedings of the Aristotelian Society. ----{1997) "Truth-Values and Courses-of-Value in Frege's GrwuigesetQ," in W W Tait (ed.) EarlY Analytic PhiJDsopf!y: Essays in honor if Leonard Linsky, Chicago: Open Court Press. ---(forthcoming) unpublished manuscript on Frege's philosophy. Ricketts, 1: (ed.) (forthcoming) Th CambriJige Companion to Frege, Cambridge: Cambridge University Press. 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(1995) "Extending Knowledge and 'Fruitful Concepts': Fregean Themes in the Foundations of Mathematics," NoUs-. 427-67. van Heijenoort,j. (1985) Sel«ted Essays, Naples: Bibliopolis. van Heijenoort, J. (ed.) (1967) From Frege to Giid8/: A Sourcebook in Math.em.atiud Logic 1879-1931, Cambridge, Mass.: Harvard University Press. Weiner,j. (1982) "Putting Frege in Perspective," Ph.D. thesis, Harvard University. --{1984) "The Philosopher Behind the Last Logicist," in C. Wright (ed.) Fregt Tradition and Irifluence, Oxford: Blackwell. --{1990) Frege in Perspeaice, Ithaca: Cornell University Press. ---{forthcoming)"Theory and Elucidation," in J. Floyd and S. Shieh (eds) Fu/.IIrt Pasts, Cambridge, Mass.: Harvard University Press. Wittgenstein, Ludwig (1922) Tractatus LJgico PhiUisophicu.r, with an introduction by Bertrand Russell, trans. C. K. Ogden, London: Routledge and Kegan Paul (corrected edition 1933). Wright, C. (ed.) (1984) Frtge: Tradition and Irifluence, Oxford: Blackwell. 9 Ar o lli Gil, Since its founders, : its conceIT been done analytic p: , believed tl: logically t • and presu · quences. . Contrlll · concerned , linguistic" , lite . reason . profound!) '. The idea' tool of c( • language<wrong ane tions. And , held to be ,1Ionds of 1; language, < again ; important . everywhen .", But byr· lions it SUi

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JF 1998 Frege, Semantics and the Double-Definition Stroke

JF 1998 Frege, Semantics and the Double-Definition Stroke

JF 1998 Frege, Semantics and the Double-Definition Stroke

JF 1998 Frege, Semantics and the Double-Definition Stroke

JF 1998 Frege, Semantics and the Double-Definition Stroke

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