On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum mod...

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Bibliographic Details
Published in:Journal of logic and computation 2011-10, Vol.21 (5), p.739-790
Main Authors: Bou, Félix, Esteva, Francesc, Godo, Lluís, Rodríguez, Ricardo Oscar
Format: Article
Language:English
Subjects:
Chains
Computation
Frames
Lattices
Logic
Mathematical analysis
Operators
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Summary:This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones only evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truth-constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.
ISSN:0955-792X
1465-363X
DOI:10.1093/logcom/exp062