The Semi Heyting–Brouwer Logic

Abstract

In this paper we introduce a logic that we name semi Heyting---Brouwer logic, $${\mathcal{SHB}}$$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 $${\mathcal{HB}}$$HB is an axiomatic extension of $${\mathcal{SHB}}$$SHB and that the propositional calculi of intuitionistic logic $${\mathcal{I}}$$I and semi-intuitionistic logic $${\mathcal{SI}}$$SI turn out to be fragments of $${\mathcal{SHB}}$$SHB.