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.
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Maruyama, Y. (2008). Algebraic Study of Lattice-Valued Logic and Lattice-Valued Modal Logic. In: Ramanujam, R., Sarukkai, S. (eds) Logic and Its Applications. ICLA 2009. Lecture Notes in Computer Science(), vol 5378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92701-3_12
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DOI: https://doi.org/10.1007/978-3-540-92701-3_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92700-6
Online ISBN: 978-3-540-92701-3
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