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Journal of Logic and Computation Advance Access published online on October 7, 2009
Journal of Logic and Computation, doi:10.1093/logcom/exp062
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© The Author, 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Papers

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

Félix Bou, Francesc Esteva and Lluís Godo

Institut d’Investigació en Intel.ligència Artificial, IIIA - CSIC, Campus UAB, Bellaterra 08193, Spain.
E-mail: fbou{at}iiia.csic.es; esteva{at}iiia.csic.es; godo{at}iiia.csic.es

Ricardo Oscar Rodríguez

Dpto. de Computación, Fac. Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina. E-mail: ricardo{at}dc.uba.ar

Received 23 October 2008.

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.

Keywords: Many-valued modal logic; modal logic; many-valued logic; fuzzy logic; substructural logic



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