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On witnessed models in fuzzy logic. III: Witnessed Gödel logics. (English) Zbl 1191.03019
Summary: Gödel (fuzzy) logics with truth sets being countable closed subsets of the unit real interval containing 0 and 1 are studied under their usual semantics and under the witnessed semantics, the latter admitting only models in which the truth value of each universally quantified formula is the minimum of truth values of its instances and dually for existential quantification and maximum. An infinite system of such truth sets is constructed such that under the usual semantics the corresponding logics have pairwise different sets of (standard) tautologies, all these sets being non-arithmetical, whereas under the witnessed semantics all the logics have the same set of tautologies and it is \(\Pi_{2}\)-complete. Further, similar results are obtained
For Parts I and II see ibid. 53, No. 1, 66–77 (2007; Zbl 1110.03013) and ibid. 53, No. 6, 610–615 (2007; Zbl 1126.03031), respectively.

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