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Kripke-style semantics for many-valued logics. (English) Zbl 1035.03010
Summary: This paper deals with Kripke-style semantics for many-valued logics. We introduce various types of Kripke semantics, and we connect them with algebraic semantics. As for modal logics, we relate the axioms of logics extending MTL to properties of the Kripke frames in which they are valid. We show that in the propositional case most logics are complete but not strongly complete with respect to the corresponding class of complete Kripke frames, whereas in the predicate case there are important many-valued logics, like BL, Ł and \(\Pi\), which are not even complete with respect to the class of all predicate Kripke frames in which they are valid. Thus, although very natural, Kripke semantics seems to be slightly less powerful than algebraic semantics.

MSC:
03B50 Many-valued logic
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03G25 Other algebras related to logic
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