Page 537 - Encyclopedia of Philosophy of Language

he was awarded the degree of Doctor of Philosophy for a dissertation in geometry. He returned to Jena almost immediately to take up the post of Privatdozent in the Faculty of Mathematics. As the post was unsal- aried, Frege had to rely on his mother for financial support; during this period he also supplemented his means by keeping pigs. He remained in the math- ematics faculty at Jena—receiving promotion to aufierordentlicher Professor in 1879, and to Hon- orary Professor in 1896—until his retirement in 1918. He died in 1925 after a life in which the main adventures were, it seems, adventures of the mind.
2. TheFregeanProgram
Most of Frege's intellectual life was devoted to a single, narrowly circumscribed project: the reduction of arithmetic in its entirety to pure logic. He attempted to prove, that is, that there belong in arithmetic no concepts, objects, procedures, or truths which cannot be accounted for given only the resources of pure, deductive logic.
Clearly, however, the logic that Frege inherited was too weak for this purpose. So in his first publication, the Begriffsschrift of 1879, he set out to extend and strengthen it. To this end he invented a notation cap- able of expressing not only multiple generality, using quantifiers and bound variables, but also relations of any degree of complexity; he introduced a truth- functional account of the logical connectives, an anticipation of the theory of types, the beginnings of a categorial grammar, and a complete formalization of the first-order predicate calculus with identity. For logic, consequently, 1879 is now widely regarded as 'the most important date for the subject' (Kneale and Knealel962:511).
Frege's next major publication, the Grundlagen (1884), contains an informal defense of the logicist program. In it Frege argues that an empirical ascrip- tion of number (e.g., There are three people in the next room) is an assertion about a concept, and that a proposition of arithmetic (e.g., 3+ 5= 8) is a logical truth, knowable a priori, concerning 'logical objects' called 'numbers.' Numbers, it turns out, are abstract objects, namely classes of equivalent classes.
In his two-volume magnum opus, Grundgesetze der Arithmetik (1893,1903), Frege attempted nothing less than a formal proof of the logicist thesis, by providing a deductively valid derivation of the truths of number theory from just seven axioms or basic laws, each of which, he claimed, was itself a truth of logic. This formal proof is in fact invalid; for, as Bertrand Russell communicated to Frege in 1902, axiom V permits derivation of the logical contradiction that has since come to be known as Russell's Paradox.
3. TheLinguisticTurn
If Frege's unprecedented additions to traditional logic were not to seem arbitrary or ad hoc, it was necessary
to provide an informal, intuitively accessible defense of them. They had to be shown to be genuine 'laws of thought,' governing the significance, the truth or falsity, and the validity of our thoughts, judgments, and inferences. The attempt to provide such a defense led Frege to confront the philosophical problems that arise in connection with such notions as identity, exis- tence, generality, logical form, truth, sense, reference, object, concept, function, assertion, thought, and judgment.
Logic studies and codifies certain formal, structural properties of thoughts, according to Frege. But 'this would be impossible,' he believed, 'were we unable to distinguish parts in the thought corresponding to parts of a sentence, so that the sentence serves as a model of the structure of the thought' (Frege 1923: 36). Within this perspective, the investigation of language becomes the most fundamental part of any philo- sophical enquiry into the nature of such diverse notions as those of, say, existence, object, truth, and thought.
4. LogicalSyntax
Frege introduced a procedure of analysis, designed to isolate those components of an arbitrary sentence which are responsible for whatever logical powers the sentence possesses. The syntax that results is 'logical,' for 'only that which affects possible inferences is taken into account. Whatever is needed for valid inference is fully expressed.'
The procedure of analysis is functorial: it takes sen- tences and singular terms as complete expressions, and assigns all others to the category of incomplete or functional expressions. All syntactically complex expressions are construed as the values of component functional expressions for other component expressions as their arguments. Predicates, relational expressions, and prepositional connectives are thus construed as first-order, and quantifiers as second- order, functional expressions.
5. Semantics
Frege's logical syntax is matched by a 'logical seman- tics,' which assigns extralinguistic entities of an appro- priate kind to expressions of different syntactic categories: truth-values are assigned to sentences; objects to singular terms; truth functions to prop- ositional connectives; second-order functions to quan- tifiers, and so on. The extralinguistic entity assigned to an expression Frege calls the reference (Bedeutung) of that expression; and expressions with the same ref- erence can be intersubstituted anywhere salvaveritate. Reference, then, is simply what an expression must possess if it is to participate in classically valid deduct- ive inference.
In addition to the notion of reference, Frege's theory of meaning also assigns to expressions of every category a sense (Sinn), and to free-standing, unem-
Frege, Gottlob
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