ABSTRACT
P: Should semantic theories be deflated, as Robert Brandom believes? Q: What is a semantic theory, and what is it for? P: The notion of a semantic theory was invented by logicians. The first system-
atic semantic theory was given by Gottlob Frege, the founder of modern mathematical logic. A semantic theory holds for a particular language, and Frege’s semantic theory applied to the formal language of his magnum opus, Grundgesetze der Arithmetik. Frege had some quite special ideas about the nature of a semantic theory (in his terminology, a specification of the Bedeutungen of the expressions of the language); but we need not follow these, or even explain them, at this point. To formalise logic, you have in the first place to have a syntax, which consists of a specification of the different categories of primitive expression and an account of how these may be put together to form sentences. Of these, a few may be designated logical constants relative to the formalisation. Typically, these consist of the sign of identity, together with expressions that serve to form complex sentences out of simpler ones. The latter typically consist of the sentential operators and first-and perhaps higher-order quantifiers, but may include other operators such as modal and temporal ones. Since generality is always expressed by logicians by means of the quantifiers introduced by Frege, the notation will need bound variables to be governed by them; when only first-order quantifiers are considered, the notation for the formalisation will also need schematic letters, representing indeterminate expressions of the different categories. The formalisation will lay down basic rules of deduction and provide effective means of deriving further rules. It will apply to formal languages constructed in accordance with the given syntax, but of course having specific non-logical primitives in place of schematic letters, and to such fragments of natural language as can be represented by means of that syntax.
