On subreducts of subresiduated lattices and logic

TL;DR

This paper explores subreducts of subresiduated lattices, their algebraic properties, and introduces a new calculus with these algebras as semantics, advancing the understanding of subintuitionistic logics.

Contribution

It studies implicative and implicative-infimum subreducts of subresiduated lattices and proposes a new calculus with algebraic semantics based on these structures.

Findings

Characterization of implicative subreducts

Development of a calculus with algebraic semantics

Analysis of expansions and properties of the calculus

Abstract

Subresiduated lattices were introduced during the decade of 1970 by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. These logics are examples of subuintuitionistic logics, i.e., logics in the language of intuitionistic logic that are defined semantically by using Kripke models, in the same way as intuitionistic logic is defined, but without requiring of the models some of the properties required in the intuitionistic case. Also in relation with the study of subintuitionistic logics, Celani and Jansana get these algebras as the elements of a subvariety of that of weak Heyting algebras. Here, we study both the implicative and the implicative-infimum subreducts of subresiduated lattices. Besides, we propose a calculus whose algebraic semantics is given by these classes of algebras. Several expansions of this calculi are also studied together to some interesting properties of them.