This paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their “A General Algebraic Semantics for Sentential Logics”. The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo. Landini and Ziliani, at the suggestion of the late Antonio Monteiro.
We study the class of abstract logics projectively generated by the class of logics defined on tetravalent modal algebras by the family of their filters. These logics are fourvalued in the sense that they can be characterized by a generalized matrix on the four-element tetravalent modal algebra which generates this variety together with a family of homomorphisms. They can be called modal as this four-element algebra can be given a nice epistemic interpretation as an extension of Belnap's four-valued logic. We also characterize them by their abstract properties and prove a completeness theorem with respect to a sequent calculus suggested by the abstract version.
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