On subreducts of subresiduated lattices and logic

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.

On the variety of strong subresiduated lattices

“…In [5] it was also shown that 𝖱𝟦 ⋆ is an implicative logic in the sense of [11] and that in consequence this logic is complete with respect to the class Alg * 𝖱𝟦 ⋆ , where we included the greatest element 1 in the signature of the algebras. Furthermore, for every algebra in Alg…”

“…Let us begin by recalling the systems 𝖱𝟦 and 𝖱𝟦 ⋆ presented by a Hilbert style system in the language  = {→, ∨, ∧, ¬} of type (2, 2, 2, 1), as it was done in [10] and [5] respectively. In [10], a Hilbert style system for Lewy calculus is proposed by Epstein and Horn.…”

“…The 𝖱𝟦 ⋆ calculus [5] is the calculus, in the language of 𝖱𝟦, which is defined by the same axiom schemes of (𝖱𝟦) by replacing the axiom scheme (𝖠𝟤) by the axiom scheme…”

“…𝖱𝟦 ⋆ = SRL. Thus, we can express the completeness theorem for 𝖱𝟦 ⋆ in the following way [5,Corollary 5.15]:…”

“…In [5] it was shown that 𝖱𝟦 ⋆ has the finite model property with respect to SRL. We finish this paper by proving that □𝖱𝟦 ⋆ has the finite model property with respect to S □ .…”