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arXiv · 2106.09883

On Logics of Perfect Paradefinite Algebras

Joel Gomes·Vitor Greati·Sérgio Marcelino·João Marcos·Umberto Rivieccio
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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. Cantú 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.

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Joel Gomes, Vitor Greati, Sérgio Marcelino, João Marcos, Umberto Rivieccio. 2022-04-08. On Logics of Perfect Paradefinite Algebras. https://doi.org/10.4204/eptcs.357.5

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