On the Logics of Perfect Paradefinite Algebras

J. Gomes, V. Greati, S. Marcelino, J. Marcos, U. Rivieccio
LSFA 2021's proceedings, Electronic Proceedings in Theoretical Computer Science

The present study shows how any De Morgan algebra may be enriched by a ‘perfection operator’ that allows one to express the Boolean properties of negation-consistency and negation-determinedness. The corresponding variety of ‘perfect paradefinite algebras’ (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 and L. Cantú. Such equivalence then plays an important role in the investigation of the 1-assertional logic and also the order-preserving logic asssociated to the PP-algebras. The latter logic, which we call PP≤, happens to be characterised by a single 6-valued matrix and consists very naturally in a Logic of Formal Inconsistency and Formal Undeterminedness. The logic PP≤ is here axiomatised, by means of an analytic finite Hilbert-style calculus, and a related axiomatization procedure is presented that covers the logics of other classes of De Morgan algebras as well as super-Belnap logics enriched by a perfection connective.