Two families of many-valued modal logics are investigated. Semantically, one family is characterized using Kripke models that allow formulas to take values in a finite many-valued logic, at each possible world. The second family generalizes this to allow the accessibility relation between worlds also to be many-valued. Gentzen sequent calculi are given for both versions, and soundness and completeness are established.

We introduce Kripke semantics for modal substructural logics, and provethe completeness theorems with respect to the semantics. Thecompleteness theorems are proved using an extended Ishihara's method ofcanonical model construction (Ishihara, 2000). The framework presentedcan deal with a broad range of modal substructural logics, including afragment of modal intuitionistic linear logic, and modal versions ofCorsi's logics, Visser's logic, Méndez's logics and relevant logics.

We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply , , , . The postulates (and theorems) are all deducibility statements . The only postulates that might not be obvious are.

1.1 In standard modal logics, the worlds are 2-valued in the following sense: there are 2 values that a sentence may take at a world. Technically, however, there is no reason why this has to be the case. The worlds could be many-valued. This paper presents one simple approach to a major family of many-valued modal logics, together with an illustration of why this family is philosophically interesting.

This paper presents an algebraic approach of some many-valued generalizations of modal logic. The starting point is the definition of the [0, 1]-valued Kripke models, where [0, 1] denotes the well known MV-algebra. Two types of structures are used to define validity of formulas: the class of frames and the class of Ł n -valued frames. The latter structures are frames in which we specify in each world u the set (a subalgebra of Ł n ) of the allowed truth (...) values of the formulas in u. We apply and develop algebraic tools (namely, canonical and strong canonical extensions) to generate complete modal n + 1-valued logics and we obtain many-valued counterparts of Shalqvist canonicity result. (shrink)

Suppose there are several experts, with some dominating others (expert A dominates expert B if B says something is true whenever A says it is). Suppose, further, that each of the experts has his or her own view of what is possible — in other words each of the experts has their own Kripke model in mind (subject, of course, to the dominance relation that may hold between experts). How will they assign truth values to sentences in a common modal (...) language, and on what sentences will they agree? This problem can be reformulated as one about many-valued Kripke models, allowing many-valued accessibility relations. This is a natural generalization of conventional Kripke models that has only recently been looked at. The equivalence between the many-valued version and the multiple expert one will be formally established. Finally we will axiomatize many-valued modal logics, and sketch a proof of completeness. (shrink)

We prove strong completeness of the □-version and the ◊-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this (...) logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants. (shrink)