Belnap's four-valued logic and De Morgan lattices

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Josep Maria Font
Title: Belnap's four-valued logic and De Morgan lattices
Journal: Logic Journal of the IGPL
Volume 5
Pages: 413-440
Year: 1997


This paper contains some contributions to the study of Belnap's four-valued logic from an algebraic point of view. We introduce a finite Hilbert-style axiomatization of this logic, along with its well-known semantical presentation, and a Gentzen calculus that slightly differs from the usual one in that it is closer to Anderson and Belnap's formalization of their logic of first-degree entailments.

We prove several Completeness Theorems and reduce every formula to an equivalent normal form. The Hilbert-style presentation allows us to characterize the Leibniz congruence of the matrix models of the logic, and to find that the class of algebraic reducts of its reduced matrices is strictly smaller than the variety of De Morgan lattices. This means that the links between the logic and this class of algebras cannot be fully explained in terms of matrices, as in more classical logics. It is through the use of generalized matrices (here called abstract logics) as models that we are able to confirm that De Morgan lattices are indeed the algebraic counterpart of Belnap's logic, in the sense of Font and Jansana's theory of full generalized models for sentential logics.

Among other characterizations, we prove that its full generalized models are the abstract logics that are finitary, do not have theorems, and satisfy the metalogical properties of Conjunction, Disjunction, Double Negation, and Weak Contraposition. As a consequence, we find that the Gentzen calculus presented at the beginning is strongly adequate for Belnap's logic and is algebraizable in the sense of Rebagliato and Verd\'{u}, having the variety of De Morgan lattices as its equivalent algebraic semantics.