On possibilistic modal logics over Gödel logic

Possibilistic logic (see e.g. [4]) is a well known formal system to reasoning graded beliefs by means of necessity and possiblity measures. From a logical point of view, possibilistic logic can be seen as a graded extension of the nonnested fragment of the well-known modal logic of belief KD45. When we go beyond the classical framework of Boolean algebras of events to many-valued frameworks, one has to come up with appropriate extensions of the notion of necessity and possibility measures for many-valued events [3]. In this abstract, we consider the problem of definining a proper Gödel modal logic capturing a suitable possibilistic semantics for the possibility and necessity operators and its relation to the generalized KD45 Gödel modal logic recently defined by Caicedo and Rodriguez [1]. After this short introduction we first summarize the main results by CaicedoRodriguez about a complete many-valued Gödel modal logic KD45(G) with respect to a given Kripke style semantics and then we consider our many-valued possibilistic Kripke semantics, and pose an open problem: are the two semantics equivalent? This problem is addressed and solved in the final section for the particular case over finite-valued Gödel logic with ∆ and truth constants.