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

Studia Logica 110 (5):1291-1325 (2022)
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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 -algebras). We denote by \ the automorphism where k is a positive integer. For \, 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 \-algebras; as consequence, we prove the class \-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 \ and \ for every k. \ is a 1-assertional logic and \ is the degree-preserving logic both associated with the class of \-algebras. Working over these logics, we prove that \ is paraconsistent with respect to the de Morgan negation \, which is protoalgebraic and finitely equivalential but not algebraizable. In contrast, we prove that \ is algebraizable, sharing the same theorems with \, but not paraconsistent with respect to \. Furthermore, we show that \ and \ are paracomplete logics with respect to \ and \ and paraconsistent logics with respecto to \, for every k.



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Paraconsistent Logic: Consistency, Contradiction and Negation.Walter Carnielli & Marcelo Esteban Coniglio - 2016 - Basel, Switzerland: Springer International Publishing. Edited by Marcelo Esteban Coniglio.
Defining LFIs and LFUs in extensions of infectious logics.Szmuc Damian Enrique - 2016 - Journal of Applied Non-Classical Logics 26 (4):286-314.

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