FUTURE PASTS

The Analytic Tradition in Twentieth-Century

Edited by SANFORD SHIEH

OXFORD VNWERSITY I'RSSS 2001 1879 Publication ofFr:ege's Begriffsschrift

1

Frege's Conception of

WARREN GOLDFARB

Frege is of course an important progenitor of modern logic. The technical ad­ vances he made were compreheusive. He clearly depicted polyadic predication, negation. the conditional, and the quantifier as the bases of logic; and he gave an analysis of and a notation for t.he quantifier that enabled him to deal fully and perspicuously with multiple generality. Moreover. be argued that mathematical demonstrations, to be ti.illy rigorous, must be carried out using only explicitly for­ mulated rules, that is, syntactically specified axioms and rules of inference. Less clear, however. is the philosophical and interpretive question ofhow Frege understands his formalism and its purposes. Upon examination, it appears that Frege had a rather different view of the subject he was creating than we do nowa­ days. ln lectures and seminars as far back as the early 1960s, called attention to differences between how Frege viewed i.he subject mauer of logic and how we do. The point has been taken up by several commentators, beginning with . 1 The technical development historically required to get from a Fregeau conception to our own was discussed in my "Logic in the Twenties: The Nature of the Quantifier."2 Yet there is currently litlle ap­ preciation of the philosophical import of these differences, that is, the role in Prege' s philosophy that his conception oflogic, as opposed to ours. plays. Indeed, some downplay the differences and assign them no influence on or role in the philosophy. Thus Dummett says only that Frege was "impeded" from having the modem view by a particular way oflookiug at the formulas of his Begri.ffsschrift. 3 1 want to urge on the contrary tbaL Frege's conception oflogic is integral to his philosophical system: it cannot be replaced with a more modern conception with­ out serious disruptions in that system. The reasons for this will. I hope, be instruc­ tive about the roots of Frege's philowphi.zing.

I

The first task is that of delineating the differences between Frege's conception of logic and the contemporary one. I shall start with the latter. Explicit elaborations

25 26 BefOre the Wars Frege's Co nccprion ot' J.ogic 27

ofit are surprisingly uncommon. (In most writing on issues in philosophical logic, cation of the set theory in which the definition is to be understood. (However, it it is implicitly assumed: yet many textbooks gloss over it. lor one pedagogical rea­ turns out that for implications between first-order schemata. the definition is son or another.) There are various versions: I will lay out the one formulated by rather insensitive to the choice of set theory. The same implications are obtained Quine in his tcxtbooks4 as it seems to me the clearest. as long as the set tbeory is at least as strong as a weak second-order arithmetic On tb.is conception, the subject matter of logic consists of logical properties of that admits the arithmetically definable sets of natural numbers.)6 sentences aud logical relations among sentences. Sentences have such proper­ (As an aside,letme note that this explication oflogicalconsequence has recently ties and bear such relations to each other by dint oftheir having the logical forms come tmder attack in John Etchemendy's The Concept of Logical ConsetJuence.1 they do. Hence, logical properties and relations are defined by way of the logical Etchemendy argues that. if S is a logical consequence of R, then there is a neces­ forms: logic deals with what is common to and can be abstracted from different sary connection between the truth of R and the truth of S, and the Tarski-Quine sentences. Logical forms are not mysterious quasi-entities, a Ia Russell. Rather. definition does not adequately capture this necessity. Of course. neither Tarski nor they are simply schemata: representations of the composition of the sentences, Quine would feel the force of such an attack. since they both reject tbe cogency constructed from the logical signs (quantifiers and truth-functional connectives. of the philosophical modalities. Moreover. it is only the Tarski-Quine character­ in the standard case) using schematic letters of various sorts (predicate, se!llcnce, ization oflogical consequence in terms ofvarious interpretations ofa schematism and fw1ction letters). Schemata do not state anything and so arc neither true nor that makes lhe notion oflogical consequence amenable to definitive malhemati­ false, but they can be interpreted: a universe of discourse is assi~,rned to lhe qu

that is, as the univcrsaUy quantified statement "(Vp)('v'q)(p&q - p). ·· Similarly. noli on that logic is wilhouLconLCnl; Lhc luws oflogic. although very general. have logical formulas containing one-place function signs are lo be underslOod not to be seen as substantive. lJldeed, in the Tmctdlus, Wittgensleio breaks with the schematically, but as generalizing over all functions. universalist conception in order to arrive at a view in which Lhe propositions On Frege 's conception U1e business oflogic is lo articuJate and demonstrate certain of logic are empty. Even if Wittgeustein's characterization of logic is rejected, true general statements, ilie logicalla ws. "(Vp )(Vq)(p&q -. p)" is one: it states a Ia w. llie metalinguistic conception will inevitably make the natun: of discourse. or we might say, about all objects. Similarly, "('v'F)(\fG)(VH)(Vx)((Fx--. Gx)&(\fx) of our representations, the locus of any account of logic. A sharp sense of this (Gx-+ Hx)-+ (Vx)(Fx--+ Hx))" is a law aboutallfunctious.11 The business of pure can be obLained by contrasting the remark of Russell's jusl ciLed with this one logic is to arrive at such laws. jusl as the business of physics is lo arrive at physi­ ofDwnmett's. made uuself-consciously and \villi no argument at all, at Lhe start cal laws. Logical laws are as descriptive as physicallaws,n but iliey are more oflaying out his own : "Reality cannot be said to obey a law of logic; general. Indeed, they are supremely general: for, aside from variables. all U1at it is our thinking about reality ilial obeys such a law or flou ts it." 1" On Frege's figure in them are the all-sign, llie conditional. and oilier signs which are not view. as on Russell's. it is precisely reality that obeys llie laws or logic. specific to any discipline. but which figure in discourse on any topic whatsoever. Indeed. the universalistic conception is au essemial background to many of Notions of Lhe special sciences tlrst appear when we apply logic. In applied logic. Frege's ontological views. Frege Look not just proper names but also sentences we infer clainls that conlain more specialized vocabulary on ilie basis ofthe laws and predicates to be referring expressions, that is, to have Bedeutu11g; in ilie lat­ of pure logic. For example, in applied logic we might demonstrate. "lf Cassius is ter case. the referents were of a different logical sort from iliose of proper names lean and Cassius is hungry.then Cassius is lean"; or, "lfall whales are mammals and sentences. From many contemporary viewpoints, it is odd to th ink of sen­ and all mammals are vertebrates, then all whales are vertebrates." These state­ tences as names at all; and if predicates are iliought to refer, it would be to prop­ ments may be inferred from ilie logical laws given at the beginning of this para­ erties or sets or some other entities that need not be sharply distinguished in logical graph. Here we also see a typical situation. that these specialized statements are character from the referents of singular terms. inferred from the logical law by instantiation of universal quantifiers. U should be clear that the universalistic conception demands that sentences and On Frege's universalist co11ception. then. the concern oflogic is the articula­ predicates refer. As we have seen, for Prege the truth-functional laws look like Liou and proof of logical laws, which arc universalLrulhs. Since they are uni­ "(Vp)(\fq)(p&q ~ p)" and will be applied by instantiating the quantifiers with versal. they are applicable to any subject matter, as application is carried out sentences. For "UCassins is lean and Cassius is huugry then Cassius is lean" to by instantiation. For Frege, the laws of logic are general, not in being about couut as a genuine instance of the law, the ex:pressions which instantiate the nothing in particular (about forms). butiu using topic-universal vocabulary to quantllied variables have to refer, to things that are values of the variables, just state truths about everything. as to count as a genuine instance of " ('v'x)(x is a prime number greater than 2 -+ The question arises immediately of how dilJerent iliese conceptions actually are. xis odd)" ilie name replacing "x" has to refer. and what it refers to must be among They can look very close. Both take pure logic to be centrally concerned with gen­ ilie values of "x." (To be is to be llie value of a variable as much for Frege as tor erality. Generality is captured in llie schemaUc conception by definitions iliat in­ Quine.) Similarly, since ilie laws oflogic include many that generalize in predi­ voke all interpretations of the given schemata, and in the universalist conception cate places. and ilieir upplication requires instaruiating those quantified variables by universal quantifiers wiili unrestricted ranges. In the schematic conceplion. wiili predicates. here too we are driven to take predicates as referring expressions. logic is applied by passing from schemata to sentences that are particular inter­ In the case ofsentences, it requires a further argument. based on intersubsUlu tivity pretations of them; in the universalist conception, applications are made by in­ phenomena,L.O conclude that what sentences refer to are their truth-values. and stantiating the quantilied variables of a general law. Given these close parallels, it requires yet oilier considerations to support taking the truili-values to be of the ll is no wonder that many logicians uud philosophers would be inclined to mini­ same logical type as ordinary objects. The former is pretty compelling: the latter mize the distinction bet\veen the two conceptions. has eUcited heated objections. IS Parallels arc not identities. however, and there arc philosophically important For predicates. however. support for the sharp distinction in logical type of the ways that the conceptions differ. First and most obviously, ilie schematic concep­ refereOL can come from the structure of applications of logic. on the universalist Lion is metalinguistic. The claims oflogic are claims about schemata or aboUL sen­ conception. [f the position occupied by n predicate in a statement is taken 1.0 be tences, and thus logic concerns features of discourse. lJl contrast, on the univer­ generalized on direcUy. the distinction in logical type is apparent. since the predi­ salist conceptionlogic sits squarely at the object level, issuing laws thaL are simply cate position has argument places: and if an expression has an argument place statements about the world. What logical laws describe are not phenomena of and so can be used in an instantiation of u quantified predicate variable, then it language or of representation. As Russell put it, ''Logic Is concerned wiili the real cannot be used to instantiate a singular term, without yielding express.ions that world just as truly as zoology. though with its more abstract and general fea­ violaLe the most basic rules of logical (and granlllatical) syntax. Thus we see that tures." 13 This di!Terence will have consequences for Lhe philosophical character­ the univer:mlist conception demands second-order logic.l6 Indeed, it was one of Ization of logic. For example. the universalist conception leaves no room for the Quine's avowedmolivations. in developing the schematic conception, to show that 30 Before the Wars f'reg~:'s Coocepbon of Logic 31

logic did not require us to take there to be anything designated by the predicates in developing an area of mathematics axiomatically could lead us to recognize a new our statements. logical inferential principle.20 The clo:;est Frege comes to providing a notion oflogi­ Logic. as construed on the universalist conception, is also in back of a doctrine caJ consequence occurs in "On the Foundations ofGeometry," where he defines one of Frege's that many have found pll.Z71ing, namely. that all functions be detined truth's being logically dependent on another. The definition is: when the one can everywhere; for the special case of concepts, this is the requirement !.bat concepts be obtained by logical laws and Inferences from the other(Pregc 1906, p. 423). No "have sharp boundaries." 17 For Frege, ail quantified variables bave unrestricted further characterization oflof:,ricallaws and inferences is made. Thus, in direct con­ domain. Given this, and given that ''(V'F)(\ix){Fx v - Fx)" is a logical Jaw. Frege's trast to the situation in the schematic conception, Prege's notion bert! rests on the requirement follows at once. ff something is a concept. then an expression for it provision of the logical laws and inference rules. can instantiate the quantifier in this law: thus we can logically derive that. for Now Frege does say, "Logic is the science of the most general Jaws of truth. "21 every object. either the concept holds of it or the concept does not. This is just what But he does nor intend this as a demarcation oflogic. only as a "rough indication Frcgc means by ··sharp boundaries." of the goal oflogic.'' As we have seen, generality and absence of vocabulary from any specialized science are, on the universalist conception, features of the logi­ cal. Frege does not attempt to give any specification of the vocabulary allowable II in logic: moreover, there is no reason to think that he would take truth and ab­ sence of specialized vocabulary as sufficient for logical status. 22 Yet there is a A second in1portant difference between the two conceptions concerns the role of dt!eper reason that his phrase gives only a "rough indication." and that has Lo do a truth predicate. Clearly, the .schematic conception employs a truth predicate: with the anomalous status of "true" when used as a predicate. the definitions of validity and logical consequence talk of the truth under ail in­ Prege repeatedly calls attention to that anomalous status. in "Oer Gedanke," he terpretations of schemata.1 8 Since the predicate is applied to an inlinite range of presents a regress argument to show that any attempt to define truth must fail. and sentences, it cannot be elinlinated by disquotalioo. On the universalist concep­ concludes that ''the content of the word 'true' is sui generis and indefinable. "23 Both tion. in contrast, no truth predicate is needed either to frame the laws of logic or the argument and his subsequent considerations show that he does notmeansinl­ to apply them. Moreover. although Frege sometimes calls logical laws the "laws ply thai the notion of truth is a primitive notion, not to be defined in terms of any­ of truth." he does not envisage using a truth predicate to characterize the nature thing more basic. After reflecting that "I SJllell the scent of violets" and "lt is true of those laws. that l smell the scent of violets" have the same content, so that the ascription of On the schematic conception, logic starts with the delinitions of validity and truth adds nothing, he concludes: "The meaning of the word 'true' seems to be al­ consequence and goes on to pronounce that a given schema is val.id or is a coo­ together sui generis. May we not be dealing here with something whil:h cannot be sequence of other schemata. formal systems may be introduced as u means to called a property in the ordinary sense at all?" (Frcge 1918. p. 61). In" [ntroduc­ eslabl.ish such facts, but this then requires a demonstration ofsoundness to ~how lion to Logic,'' 24 Frege goes farthest in suggesting that truth is not a property at ail: that what the system produces are, in fact, validities and consequcn.ces. The in­ "If we say 'the thought is true' we seem to be ascribing truth to the thought as a troduction of a formal system also raises the (less urgent) question of complete­ property. If that were so, we should have a case of subswnptlon. The thought as ness, of whether ail validities and aJJ inlplications can be obtained by means of an object would be subsumed under the concept of the true. But here we are mis­ the system. Thus it is the overarchiog notions of validity and consequence that led by language. We don't have the relation ofan objectto a property" (PW, p. 194:). set the logical agenda and provide sense to the question of how well a system for In "My Basic Logical Insights. ~ 25 be connects the use of "true" in characterizing inference captures logic. On this conception, the notion of logical inference rule logic with the idea that the ascription of truth to a thought adds nothing: is posterior to !.bat of consequence: a logical inference rule is one whose premises imply its conclusion or, in the context of a system for establishing validities only, So the sense of the word "true" is such that it does not make iluy essential is one that always leads from valid premises to a valid conclusion. contribution to the thc>ught. If [ assert "it is true that sea-water ls salty." I lnPrege's universalistconception. there is no analogous cbaracterizalion of what assen the same [hiog as if I assert "sea-water is salty." This enables us to is a logicnl law or what follows logically from what. Frege's conception of logic is recognize that the assertion is not to be found in the word "true" but in the retail, not wholesale. He simply presents various laws oflogic and logical inference assertoric force with which the sentence is uttered.... "[T]rue'' makes only rules, and then demonstrates other logical laws on the basis ofU1ese. He frames no an abortive attempt to indicate the essence oflogic. since what logic is reaJJy ovcrarcbing characteristic that demarcates the logical laws from otbers.19 Conse­ concerned \vith is not contained in the word "true" at all but in the assertoric queoUy. the only sense that the question has of whether the Jaws and rules Frege force with which a sentence is uttered. (Prege •1915, pp. 251-252) presents are complete is an "experimental" one--whether they suffice to derive all the particular results that wehavesetourselves to derive. For example, at one point, Thus, rubrics like ··general laws of truth" cannot serve to give a real character­ Frege entertains the possibility that a failure to obtain established results while ization of logic or a demarcation of the realm of the logical. The notion of truth is 32 Botore the Wars f'rcgo.:'s Conception of Log1c 33

unavailable for the role ofsetting the agenda for logic. Moreover. if we take Frege's of tie to the content of individual judgmentS. Thus we can see here a prow-sche­ scruples seriously, it follows that the schematic conceplion oflogic Is simply un­ matic conception. (TWs is particularly visible in Balzano.) Frege rarely speaks of available to him. To formulate it, as we have seen, use has to be made of a truth forms of judgment. It is not hard to surmise some reasons. predicate. That predicate figures nol as a suggestive way of talking, nor as a term First, talk of forms of judgment and of abstracting from individual judgments whose usefulness arises only from the "imperfection of language," as Prege puts has a dangerously psychological ring to it. The very locution "lo['mS of judg­ it in (Frege *1915), but as a scienll{ic term in the definitions of the most basic ment" suggests that the forms arc of mental acts and so are prime material for concepts of the discipline. Clearly, Frege would not think that legitimate. psychologistlc treatment. Moreover, Frege argued vigorously against any notion The question then arises ofwhether Prege's scruple:> are well-placed. or wbelher of abstraction as needed to get from particulars to general notions. 27 Indeed, elimi­ they can be dismissed as merely peripheral phenomena. with no deep systematic nation of any role for abslraction is central not just to Prege's anLipsychologism, conoectiocs. Addressing this question requires a careful examination of the ar­ but also to his anti-Kantianlsm. To eliminate abstraction is to eliminate the ques­ guments Frege adduces. I shall not attempt this here: for a detailed treatment, tion. How do we attain the general? Frege replaces it with the question of the 2 see Thomas Ricketts' "Logic and Truth in Frege." & I limit myself to mentioning relation between the (already given) general and the particular, a question to be the philosophical outlook wWch I take to be expressed in Frege's scruples about answered by logic. a lrulh predicate. It is that objective truth is not to be explained or secured by an 'I'Ws leads us to lhe second reason Frege has for discarding talk of the forms of ontological account. Such an account would take us to have a concepllou of judgment. He has no need ofsuch talk. precisely because his devising oflhe quan­ lh i ugs "out there" and of their bebaviors or configurations that exist independent tilier gives him a rigorous tool to capture the generality that "forms of judgment" of our knowledge. and it would depict those behaviors or conl1gurations as being gestures toward. The generality is directly expressed by the quantifier. The rela­ that wWch renders our thoughts true or false. Such an account is often ascribed tion of general to particular Is given by the logical rule ofinstantiallon from lormer LO Prege, for it Is just what is Involved in ascribing a truth-conditional semantics to latter. not by some imprecise. psychological notion of abslruclion from lhe latter to Wm. But this ascription is incompatible with Frege's remarks on truth. To take to the former. E'rege's scruples seriously is to appreciate that there is no general notion of These two reasons are not relevant to the modern schematic conception. which something's making a truth true-that is, that there is no theory of bow the has found precise nonpsychologistic notions to replace "forms ofju dgment" and thoughts expressed by sentences are determined to be true or false by the items "abstraction" and wWch uses quantification (in the metalanguage) to capture referred to in them. It is thus to put us in a position to appreciate the extraordi­ the desired generality. There is, however, another consideration at work in Frege narily subtle view Frege can be read as unfolding. On tWs reading. Frege is not a that is not simply obsolete. realist. on the usual philosopWcal characterizations of that position. He is com­ Frege's conception oflogic is intertwined with his notion of justification. A cor­ nutted to the Qbjectivity of truth and its independence of anyone's recognition of nerstone ol'Frege's thinking Is the sharp distinction between the rational basis of that truth. but the conception oftrulh here is immanent within our making of a claim- the truths thut il presupposes or depends upon- and what we might judgments and inferences, our recognitions of truth. call concomitants of thinking or making the claim: the psychological phenom­ ena that occur when a person thinks of the claim, or believes lt. or comes to ac­ cept it. the empirical conditions someone must satisfy in order to know the claim. III the history of the discovery of the claim. and so on. The distinction is emphasized throughout Frege's writings, and particularly vividly in the Introduction to the Earlier I noted that the most obvious difference between the universalist and the Powult1tious ofArilhmel ic. Remarks like these abound: "Never Jet us lake a descrip­ schematic con ceptions is that iu the former logic operates at the object level. tion of the origin of an idea for a definition, or an account oft he mental and physi­ whereas in the latter it operates at the meta-level. Even tWs by itself has conse­ cal conditions on wWch we become conscious of a proposition for a proofo fit. "18 quences. and it can be used to get at an important role the universalist concep­ The point is more general than antipsychologism, or a distinction between ob­ tion has in Frege's system. jective and subjective. as lhe following shows: Of course. it is important to avoid anachronism here. At the lime Frege was writing. a distinction between object level and meta-level could hardly have been A delightful example of the wuy in which even mathematicians can confuse drawn: in fact. it was not lo become clear until the 1920s. Nonetheless, we can the grounds of proof with the mental or physical conditions lobe satisfied if see a precursor oftbe distinction as being at issue. Many traditional logicians the proofis to be given is to be lound in E. Schroder. Under tbc beading "Spe­ spoke of logic as being about the forms of judgment, wWch were to be obtained cial Axiom" he produces the following: "The principle f have in mind might by abstraction from judgments. Although this conception was far from precise well be called the Axiom ofSymbolic Stability. It guarantees us that through­ and traditional logic lacked the machinery to work it out. it seems clear that forms out all our arguments and deductions lbe symbols remain constant in our of judgment were invoked as a way of capturing the gencralicy of logic and lack memory-or preferably on paper." and so on. (Frege 1884, p. viii) 3-1 Before the Wars 1· rcgt:'s Conl.cpuon of Luglc 35

Clearly, we would nol be able to arrive at correct mathematical arguments if our reel applicability of logic: there are no pre:.uppositions. no implicit. steps. in the inkblots were constantly to change. Yet that does not imply that mathematics application of logical laws. To illustrate Lhls, let us examine how. on Frege's pic­ presupposes the physical laws of inkblots, that those laws would figure in the ture.logic would be used to jusli(y lhe conclusion that all whales are vertebrates justifications of mathematical laws. on the basis of the claims that all whales arc mammals and that aU mauuoaJs are It is important to note that something must give content to the distinction be­ vertebrates. We start with Lhe assertions: tween rational basis and mere concomitant; something must provide a means for saying what counts as showing that one proposition is the rational basis for an­ ( 1) All whales are mammals. other. and showing when one proposition presupposes another. It is Fregc's logic that plays this role. Logic teUs us when one claim is a ground for another. namely. (2) All mammals are Vt'rlebrates. when the latter can be inferred. using logical laws, from the former. Explanalion and justilication are matters of giving grounds. For Frege, then, the explanation We then provide a logical demonstration from tirst principles that ends with: of a truth is a logical proof of that truth from more basic truths: the justification of a truth is a logical proof of that claim from whatever first principles are its ul­ (3) (\fF)(V'G)('v'H}[(V'x)(Fx ...... Gx) ...... ((V'x)(Gx ...... Hx) - · (V'x)(Px _, Hx)J].~9 timate basis. Thus the laws of logic are el.'Plicatory of explanation and justifica­ tion: on Lbls rests their claJm to the bonorillc litle "logic." Three instantiation steps then license us in the assertion of: Given this role for logic, it should occasion no surprise that Frege's conception of logic aud tbedemands be puts on the nolioo of justification are closely llnked. ('v'x)(x is a whale--. x is a mammal) ...... ((V'x)(x is a mammal -. xis a Now the notlon of justiilcalion plays a philosopblcally very important role for vertebrate}-> (\fx)(x is a whale-. xis a vertebrate)). Prege. as il is key to his argument for the logicist project Although we might start off thinking Lbat arithmetical discourse is completely understood, transparent. Or. in ordinary English: and poses no problem. Frege urges that we lack knowledge of the ultimate jus­ tification of the truths of arithmetic. In order to "afford us insight into the depen­ (4 ) If all whales are mammals. then u· all mammals arc vertebrates then aU dence or truths upon one another," we must analyze the seemingly simple con­ whales are vertebrates. cept of number and find the ·'primitive truths to which we reduce everything" (l'rege 1884, p. 2). Frege also brings up "philosopblcal motives" for the logicist By modus ponens from (4) and (1) we obtain: project, asking what looks to be the traditional philosophical question ofwhether arilhmetic is analytic or synthetic. But actuaUy he redefines these notions (as weU (5) If all mammals are vertebrates then aU whales are vertebrates. as those of a priori and a posteriori) so that they concern .. not the content of the judgment bul the justification for making the judgment" (Fregc 1884. p. 3 ). Here Finally. by modus pottens from (5) and (2). we arrive at: too it is the notion ofjustification that is doing the work. Essential to the role ofthis notion ofjustificationlo supporting the logicist project, (6) All whales are vertebrates. and to the plausibility offrege's redelinltioos of traditional philosophical terminol­ ogy. is the applicability LO all knowledge of the standards of justification. The can­ Taken together. aU these assertions, including those in the logical proof of (3 ). ons ofjustification must be universal in their purview: ''Thought is in essentials the constitute the justillcation of the assertion of "AU whales are vertebrates" on same cveryw here: ll is not true that there are different kinds of laws of thought lo lhe basis of the assertions of"All whales are mammals" and "All mammals are suit the d111er.cnt kinds of objects thought about'' (Frege 1884. p. ill). Another im­ vertebrates." portautfeuture ofjustification is explicitness: a justification must display everything The requirement o(explicitness and tht! need for the logical laws to be direclly on which the truth oft.ht! claim being justified depends. To lnsure that ''some olher applicable can be highlighted by consideration of an argument against logicism type of premis~.: is not involved at some point without our noticing it." a )uslifica­ devised by Heurl Poincare. JO The version r summarize here is formulated by Uoo must provide "a chaln of inferences with no link missing, such that no step in Charlt:s Parsons.Jl In order to show that arithmetic is logic. one must devise a it is taken wWch does not conform to some one of a small number of principles of formal system oflo!,ric and show how the theorems of arithmetic can be obtained inference recognized us purely logical" (Frege 1884. p. 102). in that formal system. Now. to give a formal system is to speci(y.lirst. the class of Obviously, these dem

certain formulas are stipulated to be axioms; the derivable fomlUlas are specified We then assert, along with whatever grounds needed tO show It from lirst principles: as those and only those formulas obtained from the axioms by finitely many appli­ cations ofcertain inference rules. Thus these specifications are inductive in nature: (3) There is an interpretation of "(V'x)(Px -. Gx)," "(V'x)(Gx -+ l!A),'' and the notion of a finite number of applications ofgiven operations is essential to them. "(V'xj(F:x-+ Hx}" under which these schemata become (regimented Therefore. number is presuppose.'<.! in the lobricist foundation for ariUunetic. This is versions ol) the sentences "All whales are mammals," "AU mammals a pelWo /11'illcipil. Thus there is a logical circle in the logicist reduction. are vert.ebrates," and "All whales are vertebrates," respectively. I bolievePoincan!'s objection falls, und it iS important to see why. The objection would succccdifFrege construed the justilicalion of arithmetic to involve. for one We now adduce a matbematkal proof culminating in: or another arithmetical claim, the following assertion: ''This claim is provable in such-and-such formal system." That assertion iS a metatheorctic one. IL is about (4) Any interpretation that makes "(V'x)( Fx ~ Gx)" and ''(Vx)(Gx --+ Hx)" the formal system; since Poincare iS quite right that inductive definiliom are used true also makes "(V'x)(Px -. Hx)" true. to :;pecify the formal system, it follows that the assertion relies on number theory. That iS not. however, how Frege conceives ofjustificalion. To give a justification of Using some logical laws and intermediate steps lor makings the transition. we can an arithmetical claim is ro give the claim with its grounds. lt is not lo assert that assert on the basis of(3) and (4): the claim is provable; it is to give Ute proof. Now, ofcourse, one might wautlo verUy thar what has been given is, in fact. a proof by the ligh ts of the formal system. Such (5) If "All whales arc mammals" and ''All mammals are vertebrates" are a vcrllication wou ld proceed by syntactic means, and does presuppose the specifi­ true, then "All whales are vertebrates'' is true. cation of the system. The verification is not constitutive of the argwnenr's being n justUicalion; lt iS just a means for us to ascertain that it is. In order for ns lo be To apply (5). we must adduce the Tarskl paradigms: psychologically sure that what we are giving are justilicalions, we have to use our knowledge of the formal system, that is. our mctatheoretic knowledge which is of (6) "All whales are mammals" is true if and only if all whales are an inductive nature. But that Is different from what the justification of the claim mammals. actually is. Here Frege is relying precisely on the distinction berween what we might have (7) "All mammals are vertebrates" is true if and only if all mammals are to do, in fact, by our natures, in order lo be in a position to do mathematics, vertebrates. and what the justification of mathematics iS. That we need l.o set out a formal system to be sure of our justifications Is no more relevant to the ralionul grounds (1), (2), (S), (6), (7), and truth-functional laws will allow us to obLain: of mathematics than our need to write down proofs because otherwise we will nut remember them. (8) "All whales are vertebrates" is true. The Fregeau rebuttal to Poincare requires that in whatFrege would call a justlll­ cation, say of an arithmetical truth. everything that is presupposed by the truth Finally. adducing does play a role. This Ues in back of hiS demand for "gap-free" deductions. To gain an appreciation of the role of U1e universalist conception oflogic in this. ( 9) "All whales are vertebrates" is true if and only if all whales are it is instructive to contrast bow a ju~tification abiding by the Fregeau requirement vertebrates. of explicitness would have to proct!cd ii Logic were taken on the schematic con­ ception. Let us once again undertake a j uslification of .. All whales are vertebrates" we obtain: on lbc basis of .. All whales axe mammals" and "All mammals are vertebrates." We can't simply pass from the latter to the former, with a note ("oO'to the side," (10) All wh ales are vertebrates. so to speak) that the latter two jointly in1ply the former. since this does uot make explicit what is involved in the inference. Rather, matters have LObe laid out as Needless to say, from Prege's point of view this outline already looks terribly cir­ follow~. As before. we starl by asserlin~: cuitous. and the amount that has to be tilled in to providejustificalions for (3) and (4) will make matters worse. Even ignoring L':rege's scruples about a truth predi­ cate. the status of the disquotalional biconditionals iS also troublesome. for. in what ( 1) All whales arc marnma1s. is outlined. those biconditionals figure among the grounds of"All whales are mam­ mals" as much as do assertions (1 ) and (2). U, for example, they are meant to be (2) All mammals are vertebratt!S. consequences of a substantial semantic theory. then we arc in the position of re- 38 Before r.he Wars Frt!gt.:'s Conception of Lo!,>iC 39

quiring that theory in the justification of·· All whales are vertebrates'' on the basis ables ranging over such collections. This process continues with oo end. To of" All whales are manunals" and" All mammals arevertebrates." l\lau.en; look less avoid a vicious regress. we have to be able to lake Lhe logical rules used in the peculiar if the truth predicate is meant lo come merely from a Tarski-style defini­ justification for granted. Yet. on this conception, it has to be admitted that a tion; but eveu here an oddly large body of mathematics must figure in order to jus­ fuiJer justification, one amplilied by a further soundness proof. is alw<~ys pos­ tify what is, after all. a rather simple lo(:.'.ical inference. All this Is LO say tbaltbe sche­ sible. In passing to that fuller justification. we also pass to a larger universe of matic conception oflogic fits poorly with the Fregeau picture of justification. u discourse. The upshot is tbal at no level can one think of the quantifier as rang­ This lack ofllt comes out in another difficulty as well. In Lhe justification us just log over everything; there Is no absolutely unrestricted quantifier. All the while. outlined, various transitions, like that from (3) and (4) to (5). will be made by tilough. in enunciating the claims at any level, one is not (yet) in a position to applying logical rules. On the l>Chem

9. "Ober dicGrundlagcn der Geometric."' ]alm1£bericht d~r Deutsclum Matlwmatiker­ 2.2. Richard Heck suggests that Frege may have formulated the principlt: ofcount­ VtJreinigung 15 (1906): 293-309, 377-403, 423-430, p. 38-! (hereafter cited as able choice to himself and found himself unable to derive it. "The Finite and the Lnfi­ Frcge 1906). nile ln Frege's Grrmdgesetze der llrltftmctik." iLl Plzilosoplty of Matlrcnwtics Today, ed. 10. Throughout this paper I use modern logical notation rather than Prege's. M. Sch.lrn (Oxford: Oxford University Press, 1998). Uso. Frege may have wondered 11. Here. in using modern notation, lam eliding a nicety required by Frege's whether that principle is an example of a truth statable without using nonlogical quantifying over all functions, not just all concepts, namely. his use of lhe horizon­ vocabulary. but not itself a truth of logic. tal. 23. Beitrlige zur Philosophie tlesdeutsclum Idealism us 1 11918): 58-77. p. 60 (here- 12. GnmdgeseLzeder Aritl11netik, Begrijfsschrlftl!cil abgeleitet. vol. 1 Oena: H. Pohle, after cited as Frege 1.918). 1893), p. xv. 2-!. In Prege (1979). pp. 185- l96. L3. fntrocluction to Mathematical Philosophy (London: Allen & Unwin. 1919). 25. In Frege (1979). pp. 251 - 252 (hereafter cited as Frege •1915). p. 169. 2 6. Proceedings ofthe AristoLelitm Society. ~upplemen tary volume 70 ( 1 9 9 6 ): 121- 14. Tile Logictll Basis uf Metaphysics (Cambridge, Mass.: Harvard University Press. 140. 1991), p. 2. 27. See. for example. ''Review of ll. G. Husserl's l'ililosop!Jie dcr Arithmetik." 1 S. For example. sec Dummett, Frege: Pltilo~>ophy ofLnnguage (London: Duckworth Zeitscilrift fiir Philosoplzie und philosophische Kritik 103 ( 1894 ): 31 3- 33 2. 1973). p. 180 and following. Dummettcalls Prege's view on the matter a "gratu­ 28. Die Grwzdlagcn der Arlthmctik (Breslau: W. Koebner, 1884). p. vi (hereafter Itous blunder." cited as Prege 1884). 16. Charles Parsons canvasses an objection to Frege's conclusion In his "Frcge's 29. See Frege, Begrijfssclzrift. ei11e der arltlunetisclreu mzc/zyebildctc f'ormelspmche Theory of Number" (in C. Parsons. Mathematics in Plti/osophy [Ithaca: Cornell Uni­ des reineu Deukens (Halle: L Nebcrt. 18 79). §23. versity Press. 1982 . original date of publication 19 6 5], pp. I 50-1 72, [hereafter cited 30. [n "La logiquede l'infini." Re\'uecfe MelC!plrysiquat de t\llomle 17 (1909): 461- as Parsons 19651). on pp. 499- 501. Suppose we take it that predicates are general­ 482. Poincare was responding to 's "Mat.hematical Logic as Based ized not directly but only via "nominalization," that is, only once they are trans­ on the Theory of Types," Amerlcau Journal of Mathematics 30 ( 1908): 222- 262. formed into names of qualities. properties. sets. or lhe like: for example. only once .. x 3 l. In (Parsons 1965). See also Mark Steiner. Mathematical KnoiVledge (Ithaca: is malleable" is transformed into "x has {the property of) malleability." Since ··mal­ Cornell University Press. 1975). p. 28 and following. leability" lacks argument places. it may be taken as generalizable using a variable of 32. Not that it was meant to: Tarski's and Quine's views of justification are differ­ the same logical type as those over objects. Parsons notes a problem with this: since ent from Frege's. the assertions with which one starts and ends will contain the unnominalized predi­ 33. This line of objection is suggested by a remark by Charles Parsons in "Objects cate, one needs a principle to underwrite the transformation. that is, a general prin­ and Logic," Mouist 65 (1982): 491-516. on p. 503. ciple a particular instance ofwhich will be "(Vx)(x is malleuble H x has the property 34. Here I draw on Thomas Ricketts. "Prege. the Tmct.atus, and t:he Logoccutric of malleability)." But any such principle, it seems clear, will have to contain a gcm­ Predicament," Noris 19 {1985): 3-1.4, p. 7. See also Ricketts. "Logic and Truth," eralization directly in the predicate place occupied in this instance by "Is malleable." section 2. Hence the Fregeau conclusion stands. As Parsons notes (1965 p. 503). there is one 35. The position that there can be no such lhing as a truly unrestricted quanti­ way around this conclusion. That Is to use the method of semantic ascent and un­ fier is due to Parsons in "The Uar Paradox" lin C. Parsons, Mathematics i11 Plriloscr derstand the principle underwriting lhe nominaHzatioo not as a direct generaliza­ phy [Ithaca: Cornell University Press, 1982, originally published 1974], pp. 221- tion over what predicates refer to. but as an assertion that any assertion of a certain 250. Parsons's argumentis based on phenomena associated wtlb lhe Liar Paradox. form may be transformed into another form salva verilate. To adopt this strategy Is The use of the idea In the current context is due to Ricketts. slmply to give up the universalist conception. as it requires a metalinguistic prin­ 36. I am greatly indebted ro Thomas Ricketts forcoumless conversations and com­ ciple that makes inelimlnable use of a truth predicate. ments. as well as for access to his unpublished works. Needless ro say. he does not 17. Funkliou uud Begrijf Gena: H. PoWe, J 891). p. 20. agree with all the formulations given in this paper. ] 8. The truth predicate needed is a predicate of sentences. For Prege, it was not sentences but rather thoughts (senses of sentc:nces) that were true or false. Conse­ quently, those who take Frege to have a form of the schematic conception treat im­ plication as a relation among thoughts. In this version, the definition would require a truth predicate of thoughts. 19. 1n this Frege dllfers from Russell, who from the first tries to formulate neces­ sary and sufficient conditions for a truth to be a logical truth. Russell tries to use generality as Lhe key element ofthe characterization. Wittgenstein takes up the prob­ lem. but criticizes Russell's invocation of generality. He takes himself Lo have solved the problem with the notion of tautology; that notion gives him a general concep­ tion of the logical 20. "Uber die BegrilTsscbrift des Herm Peano uod meinc eigene," Bericllte iiber clie Verlumdlungen der Kiinlgliclz Sticlzslschen Gese/lsclwften der Wlssensclzaften zu Leipzig (Mathematisch-physische Classe) 48 (1897): 362-378, p. 363. 21. "Logic." in G. Frege. Postlmmous Writings (Oxford and Chicago: Blachvell and Press. 1979). pp. 126- 151. p. 128 (hereafter cited as Frege 19 79).