Algebraic Study of Lattice-Valued Logic and Lattice-Valued Modal Logic

Abstract

In this paper, we study lattice-valued logic and lattice-valued modal logic from an algebraic viewpoint. First, we give an algebraic axiomatization of L-valued logic for a finite distributive lattice L. Then we define the notion of prime L-filters and prove an L-valued version of prime filter theorem for Boolean algebras, by which we show a Stone-style representation theorem for algebras of L-valued logic and the completeness with respect to L-valued semantics. By the representation theorem, we can show that a strong duality holds for algebras of L-valued logic and that the variety generated by L coincides with the quasi-variety generated by L. Second, we give an algebraic axiomatization of L-valued modal logic and establish the completeness with respect to L-valued Kripke semantics. Moreover, it is shown that L-valued modal logic enjoys finite model property and that L-valued intuitionistic logic is embedded into L-valued modal logic of S4-type via Gödel-style translation.