Deducibility of Identicals, Reflection Principle and Synthetic Connectives

Abstract

Sambin et al. (2000) introduced Basic Logic as a uniform framework for various logics. At the same time, they also introduced the principle of reflection as a criterion for being a connective in Basic Logic. In this paper, we make explicit the relationship between Hacking's deducibility of identicals condition (Hacking, 1979) and the principle of reflection by proving their equivalence. Moreover, despite Sambin et al.'s conjecture that only six connectives satisfy the principle of reflection, we show that a logical connective satisfies the principle of reflection if and only if it is Girard's synthetic connective.