Algebraizability of the Logic of Quasi-N4-Lattices

TL;DR

This paper introduces a new logic, L_QN4, based on quasi-N4-lattices, and proves its algebraizability and equivalence to a class of algebraic structures, expanding understanding of algebraic semantics for non-classical logics.

Contribution

The paper develops a Hilbert-style calculus for L_QN4 and establishes its algebraizability and term-equivalence with quasi-N4-lattices, connecting logic and algebra.

Findings

L_QN4 is algebraizable in the sense of Blok and Pigozzi.

The algebraic semantics of L_QN4 is term-equivalent to QN4-lattices.

The calculus provides a new framework for studying algebraic structures related to non-classical logics.

Abstract

The class of quasi-N4-lattices (QN4-lattices) was introduced as a common generalization of quasi-Nelson algebras and N4-lattices, in such a way that N4-lattices are precisely the QN4-lattices satisfying the double negation law (~~x = x) and quasi-Nelson algebras are the QN4-lattices satisfying the explosive law (x ^ ~x) -> y = ((x ^ ~x) -> y) -> ((x ^ ~x) -> y). In this paper we introduce, via a Hilbert-style presentation, a logic (L_QN4) whose algebraic semantics is a class of algebras that we show to be term-equivalent to QN4-lattices. The result is obtained by showing that the calculus introduced by us is algebraizable in the sense of Blok and Pigozzi, and its equivalent algebraic semantics is term-equivalent to the class of QN4-lattices. As a prospect for future investigation, we consider the question of how one could place L_QN4 within the family of relevance logics.