Subprevarieties versus extensions. Application to the logic of paradox

Journal of Symbolic Logic 65 (2):756-766 (2000)
Abstract
In the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of paradox (viz., the implication-free fragment of any non-classical normal extension of the relevance-mingle logic). In order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule
Keywords Propositional Logic   Extension   Relative Axiomatization   Matrix   Prevariety   Galois Retraction   Kleene Lattice   Paraconsisent Logic   Logic of Paradox   Paraconsistent Kleene Lattice   Classical Logic   Classical Kleene Lattice   Boolean Lattice
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