Truth in Frege

Tarski is widely regarded as the father of formal semantics, and rightly so. But if Tarski is the father, then Frege is the grandfather. Frege’s Grundgesetze der Arithmetik contains a semantic theory for his formal language, his begriffsschrift (or ‘conceptual notation’), that is no less rigorous than the one for the calculus of classes that Tarski develops in the early parts of “The Concept of Truth in Formalized Languages” (Tarski, 1958). Like Frege’s semantics for begriffsschrift, Tarski’s is stated in an informal meta-theory and is in no sense ‘formal’. Moreover, Frege argues in section 31 of Grundgesetze that his semantics is adequate to assign a unique denotation to every expression of begriffsschrift, and this argument has essentially the same purpose, and much the same structure, as Tarski’s proof that his semantics is materially adequate.1 Of course, there are differences between Frege and Tarski. Most of these derive from the fact that they have very different reasons for being interested in semantic theory. Frege wouldn’t have seen much point in formalizing the semantics for his formal langauge,2 since its purpose was not to report the pre-established meanings of the expressions of that language but rather to establish those meanings. Formalizing the semantic theory for begriffsschrift in begriffsschrift would therefore have been unhelpful. Tarski, on the other hand, is quite plainly interested in formalized semantic theories, and, if he does not present the semantics for the calculus of classes as a formal theory, then that is because it would have been obvious enough to his readers how it could be formalized.