The Composition of Thoughts

Among the many innovations that mark Frege’s Begriffsschrift as a revolutionary work, perhaps the most important is its presentation of the first formal system of logic. Frege believed that the introduction of a new notation, especially for the expression of generality, was necessary if the logical relationships between contents were to be made apparent. This new notation made it possible, at least in an important range of cases, to establish that a given content could be inferred from certain other contents simply by examining the structure of the sentences that expressed those contents.1 In this regard, Frege’s great advance was that, in his system, the logical rules could be stated in purely formal terms, without any reference to the contents expressed by the sentences of his formal language, Begriffsschrift, the conceptual notation. But it is important to recognize that logic, according to Frege, is not ‘formal’ in any sense that would oppose form to content: The sentences of Begriffsschrift are not mere forms. Frege’s goal was “not. . . to present an abstract logic in formulas, but to express a content through written symbols in a more precise and perspicuous way than is possible with words” (AimCN, pp. 90–1). Frege’s formal system is intended to be one we can actually use in reasoning, that is, in inferring truths from other truths, which is to say that we can prove theorems in this system, where theorems are true contents.2 If so, the sentences of Begriffsschrift must express those contents. The point of presenting proofs in a formal system is thus not to empty mathematics of content (FTA, esp. opp. 97ff.),3