Mathware & Soft Computing 15 (2008) 175-188 Exploring a Syntactic Notion of Modal Many-Valued Logics ∗ F. Bou, F. Esteva and L. Godo IIIA - CSIC 08193 Bellaterra, Spain {fbou,esteva,godo}@iiia.csic.es Abstract We propose a general semantic notion of modal many-valued logic. Then, we explore the difficulties to characterize this notion in a syntactic way and analyze the existing literature with respect to this framework. Keywords: modal fuzzy logic, modal many-valued logic. 1 Introduction The purpose of this paper is the search for a syntactic notion of modal many- valued logic that generalizes the notion of classical (normal) modal logic [8, 3]. In particular, modal fuzzy logics 1 will be inside this class. The addition of modal operators to fuzzy formalisms has been previously considered in the literature for several semantic purposes; for instance, in [34, 12, 26, 30, 23, 13, 21]. This paper is also motivated by semantic issues since we understand modal many-valued logics as logics defined by Kripke frames (possibly with many-valued accessibility relations) where every world follows the rules of a many-valued logic given by a residuated lattice, this many-valued logic being the same for every world. We point out that the semantics of this general framework is doubly many-valued because both the accessibility relation and the non-modal fragment are many- valued. We could have considered the logics introduced either by many-valued accesibility relations over classical logic (for instance [20, 9, 30, 13, 35]) or by classical accessibility relations over many-valued logics (for instance [24, 22, 25]); but we prefer to keep to the general semantics because these other two logics are extensions of the general one; indeed, the general logic is minimal. Unfortunately, within this general framework we have not been able to find a syntactic characterization of the notion of modal many-valued logic that works in all ∗ The present paper is a revised and slightly extended version of [5]. 1 In the sense of considering [0, 1] as the set of truth values. 175