Sense and Proof

1 Frege, Carnap and the contrast between cognitive semantic sense In a paper given at the 1997 SILFS Conference, [Pen98] claims that a strong tension between a semantic and a cognitive notion of sense was already present in Frege's writings. Many authors have discussed recently this tension in Frege, but few of them remark that Carnap was probably the first to attempt to compose the tension. Carnap ([Car47]) soon realized that his conception of intension, although suitable to treat a semantic notion of sense as truth condition on the lines developed by Wittgenstein, was not enough to solve the problem posed by Frege on belief contexts. He devised therefore the notion of intensional structure and intensional isomorphism, as a more fine-grained notion that the semantic notion o intension and intensional equivalence. For many reasons however his notion did not appear successful. In this short paper we try to develop a new perspective which tries to compose the tension between a semantic and a cognitive notion of sense, suggesting the use of proof theory as a mean to interpret the cognitive notion of sense. In the Fregean view, sense is something which is there to be grasped from a speaker. The concept of grasping a sense or understanding is co-original with the notion of sense. However it is not clear in Frege what is the object of understanding. Frege himself made interesting remarks on the limitations of understanding complex mathematical formula with all their connections with other parts of mathematics. In his famous argument on the " intuitive difference of thought " (as it has been named by Gareth Evans) Frege claims that if it is possible to understand two sentences, and coherently believe one and disbelieve the other, then those sentences express different senses or thoughts. The Fregean example is the belief that Hesperus is a planet and Phosphorus is not a planet, given by the ignorance of the identity Hespe-rus=Phosporus. A lack of knowledge permits a person who hold erroneous beliefs to be considered rational, because the erroneous beliefs express two