On some semi-intuitionistic logics

Abstract

Semi-intuitionistic logic is the logic counterpart to semi-Heyting algebras, which weredefined by H. P. Sankappanavar in [3] as a variety generalizing the one of Heyting algebraswhile retaining some important features, like the fact that they are all pseudocomplementeddistributive lattices and their congruences are determined by filters. Semi-Heyting algebrasare algebras A = hA,∨,∧,→,>,⊥i that satisfy the conditions:(SH1) hA,∨,∧,>,⊥i is a bounded lattice(SH2) x∧(x → y) ≈ x∧y(SH3) x∧(y → z) ≈ x∧[(x∧y) → (x∧z)](SH4) x → x ≈ >.We present a new, more streamlined set of axioms for semi-intuitionistic logic, which weprove translationally equivalent to the one introduced in [1]. We then study some formulasthat define a semi-Heyting implication, and specialize this study to the case in which theformulas use only the lattice operators and the intuitionistic implication. We prove thenthat all the logics thus obtained are equivalent to intuitionistic logic, and give their Kripkesemantics.This work has been published in Studia Logica [2].