Extending ?ukasiewicz Logics with a Modality: Algebraic Approach to Relational Semantics

摘要

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 ?_n-valued frames. The latter structures are frames in which we specify in each world u the set (a subalgebra of ?_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.