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

Authors:
Yoshihiro Maruyama
Affiliations:
Faculty of Integrated Human Studies, Kyoto University, Japan
Venue:
ICLA '09 Proceedings of the 3rd Indian Conference on Logic and Its Applications
Year:
2009

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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.