Abstract
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.
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Hansoul, G., Teheux, B. Extending Łukasiewicz Logics with a Modality: Algebraic Approach to Relational Semantics. Stud Logica 101, 505–545 (2013). https://doi.org/10.1007/s11225-012-9396-9
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DOI: https://doi.org/10.1007/s11225-012-9396-9