Por Daniele Chiffi (Politecnico di Milano).
In Die Grundgesetze der Arithmetik (I, §48) Frege introduces his rule of the fusion of horizontals, according to which if an occurrence of the horizontal stroke is followed by another occurrence of the same stroke, either in isolation or “contained” in a propositional connective, the two occurrences can be fused with each other. However, the role of this rule, and of the horizontal sign more in general, is controversial; Michael Dummett notoriously claimed, for instance, that the horizontal is “wholly superfluous” in Frege’s logical system. In this talk I challenge Dummett’s view by providing a comprehensive analysis of the significance of the horizontal stroke. After some preliminary remarks, I argue that even if Frege’s connectives “contain” somehow the horizontal, yet they are total functions. Then, I take up the question of the sense expressed by the horizontal, and I claim that unlike other sentential operators, the horizontal is not sense-compositional. Finally, I consider the semantic and pragmatic aspects of Frege’s horizontal in connection to his judgment stroke and the double judgment stroke for definitions. Contra Dummett, I argue that the horizontal is a special and indispensable element of Frege’s logic.
