Sentential logics based on k-cyclic modal pseudocomplemented De Morgan algebras

TL;DR

This paper introduces a new class of modal pseudocomplemented De Morgan algebras with an automorphism, studies their algebraic properties, and develops two related sentential logics with distinct features like paraconsistency and algebraizability.

Contribution

It defines and analyzes Ck-algebras, proves their semisimplicity, calculates free algebra cardinalities, and constructs two new logics with unique logical properties.

Findings

Ck-algebras form a semisimple variety and are generated by specific algebras.

The free Ck-algebra cardinality is explicitly calculated.

Two sentential logics, L<k and Lk, are developed with distinct logical properties.

Abstract

The study of the theory of operators over modal pseudocomplemented De Morgan algebras was begun in the papers [15] and [16]. In this paper, we introduce and study the class of modal pseudocomplemented De Morgan algebras enriched by an automorphism k-periodic (or Ck-algebras) where k is a positive integer; for k = 2 the class coincides with the one studied in [15] where the automorphism works as a new unary operator. In the first place, we prove the class Ck-algebras is a semisimple variety and we determine the generating algebras. Afterwards, we calculate the cardinality of the free Ck-algebra with n generator. After the algebraic study, we built two sentential logics that have as algebraic counterpart the class of Ck-algebras that we denote L<k and Lk for every k. Lk is a 1-assertional logic and L<k is the degree-preserving logic both associated with the class of Ck-algebras. Working over these logics, we prove that L<k is paraconsistent, which is protoalgebraic and finitely equivalential but not algebraizable. In contrast, we prove that Lk is algebraizable, sharing the same theorem with L<k, but not paraconsistent.