On some semi-intuitionistic logics

JUAN MANUEL CORNEJO; IGNACIO VIGLIZZO

Bahía Blanca

Congreso; XIII Congreso Dr. Antonio Monteiro; 2015

Departamento de Matemática (Universidad Nacional del Sur) INMABB (Conicet / Universidad Nacional del Sur)

Semi-intuitionistic logic is the logic counterpart to semi-Heyting algebras, which were defined by H. P. Sankappanavar in [Hanamantagouda P. Sankappanavar. Semi-Heyting algebras. Amer. Math. Soc. Abstracts, page 13, January 1985.] as a variety generalizing the one of Heyting algebras while retaining some important features, like the fact that they are all pseudocomplemented distributive lattices and their congruences are determined by filters. Semi-Heyting algebras are algebras A = (A,∨,∧,→,1,0) that satisfy the conditions: (SH1) (A,∨,∧,1,0) is a bounded lattice. (SH2) x ∧ (x → y) ≈ x ∧ y (SH3) x ∧ (y → z) ≈ x ∧ [(x ∧ y) → (x ∧ z)] (SH4) x → x ≈ 1. We present a new, more streamlined set of axioms for semi-intuitionistic logic, which we prove translationally equivalent to the one introduced in [Juan Manuel Cornejo. Semi-intuitionistic logic. Studia Logica, 98(1-2):9?25, 2011.]. We then study some formulas that define a semi-Heyting implication, and specialize this study to the case in which the formulas use only the lattice operators and the intuitionistic implication. We prove then that all the logics thus obtained are equivalent to intuitionistic logic, and give their Kripke semantics. This work has been published in Studia Logica [Juan M. Cornejo and Ignacio D. Viglizzo. On some semi-intuitionistic logics. Studia Logica, Online first:1?42, 2014.].