On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice FELIX´ BOU, FRANCESC ESTEVA and LLU´IS GODO, Institut d’Investigaci´oen Intel.lig`enciaArtificial, IIIA - CSIC, Campus UAB, Bellaterra 08193, Spain. E-mail:
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[email protected] RICARDO OSCAR RODR´IGUEZ, Dpto. de Computaci´on,Fac. Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina. E-mail:
[email protected] Abstract 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 only evaluated in the idempotent elements and the ones 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. Keywords: many-valued modal logic, modal logic, many-valued logic, fuzzy logic, substructural logic. 1 Introduction The basic idea of this article is to systematically investigate modal extensions of many-valued logics. Many-valued modal logics, under different forms and contexts, have appeared in the literature for different reasoning modeling purposes. For in- stance, Fitting introduces in [22, 23] a modal logic over logics valued on finite Heyting algebras, and provides a nice justification for studying such modal systems for dealing with opinions of experts with a dominance relation among them.