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Linear Kripke frames and Gödel logics. (English) Zbl 1118.03016
First-order Gödel logics are determined by closed subsets of the unit interval [0,1], containing 0 and 1, as their truth-value sets, which are called Gödel sets in this paper. Recently, the logics have received increasing attention particularly in relation to fuzzy logic and, in fact, several interesting results have been obtained in the last decades. Since Gödel logics are intermediate predicate logics enjoying both the so-called constant domain axiom and Dummett’s linear axiom, Kripke frames for them should be those linear with constant domains.
In this paper, the authors establish a correspondence between Gödel sets and countable linear Kripke frames so as to determine the same logics. That is, for any such Kripke frame there is a Gödel logic which coincides with the logic defined by the Kripke frame on constant domains and vice versa. The correspondence allows us to transfer results on one side to those of the other, respectively, among which two remarkable applications from observations on Gödel logics are mentioned here: one is a complete characterization of axiomatizability of logics based on countable linear Kripke frames with constant domains, and the other that the total number of logics defined by countable linear Kripke frames on constant domains is countable.

MSC:
03B50 Many-valued logic
03B55 Intermediate logics
03B52 Fuzzy logic; logic of vagueness
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