Paraconsistent and Paracomplete Logics Based on k-Cyclic Modal Pseudocomplemented De Morgan Algebras

Abstract

The study of the theory of operators over modal pseudocomplemented De Morgan algebras was begun in papers [20] and [21]. In this paper, we introduce and study the class of modal pseudocomplemented De Morgan algebras enriched by a k-periodic automorphism (or ${\mathcal {C}}_k$-algebras). We denote by $\lnot _k$ the automorphism where k is a positive integer. For $k=2$, the class coincides with the one studied in [20] where the automorphism works as a new unary operator which can be considered as a negation. In the first place, we develop an algebraic study of the class of ${\mathcal {C}}_k$-algebras; as consequence, we prove the class ${\mathcal {C}}_k$-algebras is a semisimple variety and determine the generating algebras. After doing the algebraic study and using these properties, we built two families of sentential logics that we denote with $\mathbb {L}_{k}^{\le }$ and $\mathbb {L}_{k}$ for every k. $\mathbb {L}_{k}$ is a 1-assertional logic and $\mathbb {L}_{k}^{\le }$ is the degree-preserving logic both associated with the class of ${\mathcal {C}}_k$-algebras. Working over these logics, we prove that $\mathbb {L}_{k}^{\le }$ is paraconsistent with respect to the de Morgan negation $\sim$, which is protoalgebraic and finitely equivalential but not algebraizable. In contrast, we prove that $\mathbb {L}_{k}$ is algebraizable, sharing the same theorems with $\mathbb {L}_{k}^{\le }$, but not paraconsistent with respect to $\sim$. Furthermore, we show that $\mathbb {L}_{k}^{\le }$ and $\mathbb {L}_{k}$ are paracomplete logics with respect to $\sim$ and $\lnot _k$ and paraconsistent logics with respecto to $\lnot _k$, for every k.