On Logics of Perfect Paradefinite Algebras

TL;DR

This paper introduces perfect paradefinite algebras, shows their equivalence to involutive Stone algebras, and develops a 6-valued logic (PP<=) that captures their properties, including a finite axiomatization.

Contribution

It establishes the equivalence between PP-algebras and involutive Stone algebras and develops a finite axiomatization for the associated logic PP<=.

Findings

PP-algebras are term-equivalent to involutive Stone algebras.

The logic PP<= is characterized by a 6-valued matrix.

PP<= is a Logic of Formal Inconsistency and Undeterminedness.

Abstract

The present study shows how to enrich De Morgan algebras with a perfection operator that allows one to express the Boolean properties of negation-consistency and negation-determinedness. The variety of perfect paradefinite algebras thus obtained (PP-algebras) is shown to be term-equivalent to the variety of involutive Stone algebras, introduced by R. Cignoli and M. Sagastume, and more recently studied from a logical perspective by M. Figallo-L. Cant\'u and by S. Marcelino-U. Rivieccio. This equivalence plays an important role in the investigation of the 1-assertional logic and of the order-preserving logic associated to PP-algebras. The latter logic (here called PP<=) is characterized by a single 6-valued matrix and is shown to be a Logic of Formal Inconsistency and Formal Undeterminedness. We axiomatize PP<= by means of an analytic finite Hilbert-style calculus, and we present an axiomatization procedure that covers the logics corresponding to other classes of De Morgan algebras enriched by a perfection operator.