, Volume 53, Issue 11, pp 2230-2239
Date: 22 Oct 2010

Lattice-valued modal propositional logic and its completeness

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Abstract

Based on the concept of the complete lattice satisfying the first and second infinite distributive laws, the present paper introduces the semantics of the lattice-valued modal propositional logic. It is pointed out that this semantics generalizes the semantics of both classical modal propositional logic and [0, 1]-valued modal propositional logic. The definition of the QMR 0-algebra is proposed, and both the Boole-typed latticevalued modal propositional logic system B and the QMR 0-typed lattice-valued modal propositional logic system QML* are constructed by use of Boole-algebras and QMR 0-algebras, respectively. The main results of the paper are the completeness theorems of both the system B and QML*.