Characterizing Belnap's Logic via De Morgan's Laws

Mathematical Logic Quarterly 41 (4):442-454 (1995)
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Abstract

The aim of this paper is technically to study Belnap's four-valued sentential logic . First, we obtain a Gentzen-style axiomatization of this logic that contains no structural rules while all they are still admissible in the Gentzen system what is proved with using some algebraic tools. Further, the mentioned logic is proved to be the least closure operator on the set of {Λ, V, ⌝}-formulas satisfying Tarski's conditions for classical conjunction and disjunction together with De Morgan's laws for negation. It is also proved that Belnap's logic is the only sentential logic satisfying the above-mentioned conditions together with Anderson-Belnap's Variable-Sharing Property. Finally, we obtain a finite Hilbert-style axiomatization of this logic. As a consequence, we obtain a finite Hilbert-style axiomatization of Priest's logic of paradox

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References found in this work

A useful four-valued logic.N. D. Belnap - 1977 - In J. M. Dunn & G. Epstein (eds.), Modern Uses of Multiple-Valued Logic. D. Reidel.
Theory of Logical Calculi: Basic Theory of Consequence Operations.Ryszard Wójcicki - 1988 - Dordrecht, Boston and London: Kluwer Academic Publishers.
Distributive Lattices.Raymond Balbes & Philip Dwinger - 1977 - Journal of Symbolic Logic 42 (4):587-588.

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