# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] M. Baaz, A. Leitsch, and R. Zach, Incompleteness of a first-order Gödel logic and some temporal logics of programs. In: Proc. CSL’95 (Paderborn), pp. 1-15 (Springer 1996). [2] M. Baaz, N. Preining, and R. Zach, Characterization of the axiomatizable prenex fragments of Gödel logic. In: Proc. 33 ISMVL Tokyo, pp. 175-180 (2003). [3] Baaz, First-order Gödel logics, Annals Pure Appl. Logic 147 pp 23– (2007) · Zbl 1146.03010 [4] Baaz, Note on witnessed Gödel logics with Delta, Annals Pure Appl. Logic 161 pp 121– (2009) · Zbl 1183.03022 [5] P. Hájek, Metamathematics of Fuzzy Logic (Kluwer, 1998). · Zbl 0937.03030 [6] Hájek, A non-arithmetical Gödel logic, Logic J. IGPL 13 pp 435– (2005) · Zbl 1086.03018 [7] P. Hájek, Arithmetical complexity of fuzzy predicate logic II - a survey. To appear in Annals Pure Appl. Logic. [8] Hájek, On witnessed models in fuzzy logic, Math. Log. Quart. 53 pp 66– (2007) · Zbl 1126.03031 [9] Hájek, On witnessed models in fuzzy logic II, Math. Log. Quart. 53 pp 610– (2007) · Zbl 1126.03031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.