A GODEL MODAL LOGIC 1 ( revised Dec . 2008 )

Sometimes it is needed in approximate reasoning to deal simultaneously with both fuzziness of propositions and modalities, for instance one may try to assign a degree of truth to propositions like “John is possibly tall”or “John is necessarily tall”, where “John is tall”is presented as a fuzzy proposition. Fuzzy logic should be a suitable tool to model not only vagueness but also other kinds of information features like certainty, belief or similarity, which have a natural interpretation in terms of modalities. We address in this paper the case of pure modal operators for Gödel logic, one of the main systems of fuzzy logic arising from Hájek [11] classi…cation. For this purpose we consider a many-valued version of Kripke semantics for modal logic where both, propositions at each world and the accessibility relation, are in…nitely valued in the standard Gödel algebra [0,1]. We provide strongly complete axiomatizations for the -fragment and the 3-fragment of the resulting minimal logic. These fragments are shown to behave quite asymmetrically. Validity in the …rst one is univocally determined by the class of frames having a crisp (that is, two-valued) accessibility relation, while validity in the second requires truly fuzzy frames. In addition, the -fragment does not enjoy the …nite model property with respect to the number of worlds or the number of truth values while the 3-fragment does. The results of this paper were announced at the meeting on "Logic, Computability and Randomness", Cordoba, Argentina, Sept. 2004. Publication was delayed, aiming to axiomatize the full logic with both modal operators, which resulted elusive. Since the results have been quoted in some publications based in an incomplete preliminary manuscript, we have chosen to circulate this revision. We obtained recently the strong completeness of the full logic, result which will appear elsewhere.