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Future Pasts FUTURE PASTS The Analytic Tradition in Twentieth-Century Philosophy Edited by JULIET FLOYD SANFORD SHIEH OXFORD VNWERSITY I'RSSS 2001 1879 Publication ofFr:ege's Begriffsschrift 1 Frege's Conception of Logic 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, Burton Dreben 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 Jean van Heijenoort. 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<m­ cal treatment.) tiliers. predicate letters are replaced by predicates or assigned extensions lofthe On this scl!emat!c t'onception of logic. the fom1allan.guage of central concern is appropriate r-ilies) over the universe, sentence letters can be replaced by sentences that of logical schemata. Pure logic aims at ascertaining logical properties and or assigned truth-values. Under interpretation, a schema will receive a truth­ lo!,rical relations of these formulas. and also at demonstrating general laws about value. We may then del:ine: a schema is valid if and only if it is true under every the properties and relations. AppUed logic. we might say, then looks at sentences­ interpretation: one schema implies another, that is, the second schema is a logi­ of one or another formal language for maLhematics or science or of (regimented cal collsecjuetu:e of the first, if and only if every interpretation that makes the first versions of) everyday language-to see whether they may be schematized by true also makes the second true. A more general notion of logical consequence. schemata having this or that logical property or relation. Thus. there is a sharp between sets ofschemata and a schema, may be defmed similarly. Finally. we may distinction between logical laws, which are at the metalevel and are about sche­ arrive at the logical properties or relations between sentences thus: a sentence is mata. and logical truths. which are particular sentences that can be schematized logically true if <md only if it can be schematizcd by a schema that is valid: one by valid schemata. The pivotal role in this conception of schemata. that is, of sentence implies anotherifthey can beschematized by schemata the first of which uninterprcted formulas that represent logical forms. gives a specific cast to the implies the second. generality of logic. Logic deals with logicallorms, which schemarue away the The notion of schematization is just the converse of interpretation: to say that particular subject matter of sentences. Thus logic is tied to no particular subject a sentence can be schematized by a schema is just to say that there is an inter­ matter because il deals with these "empty" forms rather lhan wilb. particular pretation under which the schema becomes the sentence. Thus. a claim that a contents. sentence R implies a sentenceS, that is. that S is a logical consequence of R. has Such a schematic conception is foreign to Frege (as well as lo Russell). This two parts. each of which uses the notion of interpretation: it is the assertion that comes out early in his work. in the contrast he makes between his begrijfsschrift there are schemata R* and S* such that and the formulas of Boole: "My intention was not to represent an abstract logic in formulas, but LO express a content through written signs in a more precise and (1) R' and S*, under some interpretation, yield RandS: and clear way than it is possible to do through words. "8 And it comes out later io his career irt b.is reac[ion to Hilbert's Foundations of Geometry: "The word 'interpre­ (2) under no imerpt·etation is R• true and s• false. tation' is objectionable, for when properly e.x.'Pressed, a thougbtleavt!S no room for different interpretations. We have seen that ambiguity [Vieldeutigkeit] simply This is often called the Tarski-Quine definition, or (in the Tarskian formulation) has to be rejected. ''9 There are no parts of b.is logical formulas that await inter­ the model-theoretic definition. of logical consequence. 5 Itis precise enough to al­ pretation. There is no question of providing a universe of discourse. Quant.Wers low the malhematical investigalion of the notion. For example, using this notion in Frege's system have fixed meaning: they range over all items of th.e appropri­ oflogical consequence, we can frame the question of whether a proposed formal ate logical type (objects. one place functions of objects. two place functions of system is sonnd and complete, and this question may then be treated with math­ objects, ctc.) The letters that may figure in logical formulas, lor example. in ''(p&q ematical tools. Better put, though. we should say that the definition is capable of -+ p)" are nol schematic: they are not sentence letters.IO Rather. Frege under­ being made precise. For the definition quantifies over all interpretations. This is stands them as variables. Elere they are free variables. and hence irt accordance a set-theoretic quantification; hence. complete precision would require a specifl- with Frege's general rule the lormula is tO be understood as a universal closure, 28 B•~ton: the Wars Fre~;e ' b Com.eption of LOl:,'i t. 29 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.
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