A relational formalisation of a generic many—valued modal logic

In the paper we define a generic many valued modal logic, in which modalities are defined in the most general way possible following the idea of Thomason. Both the valuation of formulae and the accessibility predicates — replacing the usual accessibility relations — can be many valued. We present two types of semantics of the logic: the standard (Kripke) one and a relational one. In connection with the latter, we define a special calculus of relations corresponding to the connectives and modalities of the logic, and we develop a complete deduction system in Rasiowa-Sikowski style for this calculus. We illustrate the results by applying our formalisation to the two-valued modalities of possibility and necessity, and to a general class of many-valued modal logics defined by Fitting.