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 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. © 2009 The Author.The collaborative work between the authors of this article was made possible by several research grants: CyT-UBA X484, research CONICET program PIP 5541, AT Consolider CSD2007-0022 LOMOREVI Eurocores Project FP006/FFI2008-03126-E/FILO, MULOG2 TIN2007-68005-C04-01 of the Spanish Ministry of Education and Science, including feder funds of the European Union, and 2009SGR-1433/1434 of the Catalan GovernmentPeer Reviewe