Lattice-valued modal propositional logic and its completeness
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- Shi, H. & Wang, G. Sci. China Inf. Sci. (2010) 53: 2230. doi:10.1007/s11432-010-4098-2
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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 QMR0-algebra is proposed, and both the Boole-typed latticevalued modal propositional logic system B and the QMR0-typed lattice-valued modal propositional logic system QML* are constructed by use of Boole-algebras and QMR0-algebras, respectively. The main results of the paper are the completeness theorems of both the system B and QML*.