On Logics of Perfect Paradefinite Algebras
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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.
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