The Semi-Heyting Brouwer Logic

Abstract

In this paper we introduce a logic that we name semi Heyting–Brouwer logic, SHB, in such a way that the variety of double semi-Heyting algebras is its algebraic counterpart. We prove that, up to equivalences by translations, the Heyting–Brouwer logic HB is an axiomatic extension of SHB and that the propositional calculi of intuitionistic logic I and semi-intuitionistic logic SI turn out to be fragments of SHB.