概念の外延・文脈原理・フレーゲ構造

Abstract

What are mathematical objects? Frege's answer was that they are extensions of concepts. In this paper I will examine Frege's notion of "extension of concept", and thereby reconsider significance of his project, Logicism. Frege's theory of extension was given as semantics for his formal language (i. e. Begriffsschrift) in Grundgesetze. Although the semantics based on so-called "Context Principle" is very complicated, the idea is simple. What he tried to do is to represent mathematical objects and structures by the logical structure of language. The notion of extension was the medium that makes the representation possible. As we know, Frege's system is inconsistent because of Russell's paradox. But it does not show that the notion of extension itself is inconsistent. Rather what the paradox tells us is that the notion of proposition needs more detailed formulation than Frege's. The remedy is, I think, Aczel's Frege structure. Referring to it, I will try to correct and reformulate Frege's argument while avoiding the paradox.