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Fuzzy logic and arithmetical hierarchy. (English) Zbl 0857.03011
For fuzzy logic in the sense of J. Pavelka [Z. Math. Logik Grundlagen Math. 25, 45-52, 119-134, 447-464 (1979; Zbl 0435.03020, Zbl 0446.03015, and Zbl 0446.03016)], i.e. for an extension of Łukasiewicz’s \([0,1]\)-valued propositional logic with graded notions of proof and of entailment and with truth degree constants for all truth degrees, the author gives a considerably simplified axiomatization. He also proves completeness for the restricted case that only truth degree constants for the rational truth degrees are accepted and that the fuzzy sets of premisses have only rational membership degrees. Finally he proves \(\Pi_2\)-completeness for the notion of provability for this logic.

MSC:
03B52 Fuzzy logic; logic of vagueness
03B50 Many-valued logic
03F20 Complexity of proofs
03B25 Decidability of theories and sets of sentences
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