zbMATH — the first resource for mathematics

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.

03B52 Fuzzy logic; logic of vagueness
03B50 Many-valued logic
03F20 Complexity of proofs
03B25 Decidability of theories and sets of sentences
Full Text: DOI
[1] Biacino, L.; Gerla, G., Weak decidability and weak recursive enumerability for L-subsets, (1986), Universita degli Studi di Napoli, preprint No. 45
[2] Gottwald, S., Mehrwertige logik, (1988), Akademie-Verlag Berlin
[3] Novak, V.; Novák, V., On the syntactico-semantical completeness of first-order fuzzy logic II, Kybernetika, Kybernetika, 26, 134-152, (1990) · Zbl 0705.03010
[4] Novák, V., Ultraproduct theorem and recursive properties of fuzzy logic, (1993), Preprint · Zbl 0826.03011
[5] Novák, V., Fuzzy logic revisited, (1994), Preprint
[6] Pavelka, J.; Pavelka, J.; Pavelka, J., On fuzzy logic III, Zeitschr. f. math. logik und grundl. der math., Zeitschr. f. math. logik und grundl. der math., Zeitschr. f. math. logik und grundl. der math., 25, 447-464, (1979) · Zbl 0446.03016
[7] Rogers, H., Theory of recursive functions and effective computability, (1967), McGraw-Hill New York · Zbl 0183.01401
[8] Rose, A.; Rosser, J.B., Fragments of many-valued statement calculi, Trans. AMS, 87, 1-53, (1958) · Zbl 0085.24303
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.