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On theories and models in fuzzy predicate logics. (English) Zbl 1111.03030
This is a nice contribution to first-order mathematical fuzzy logic. The basic system is the monoidal t-norm logic MTL, i.e., the logic of the left-continuous t-norms. This well-known system is treated here in an a bit more general form, which allows the authors to subsume a whole family of further logics to these considerations, mainly extensions of MTL. All these logics have algebraic semantics.
First, the authors give a slight generalization of the completeness theorem, mainly saying that models with degree sets without unnecassary degrees suffice. Then they study conservative extensions of elementary theories via different forms of Henkin extensions. And they give a syntactic characterization of witnessed models. Finally they transfer most of these results to extensions which contain also the \(\triangle\)-operator.

03B52 Fuzzy logic; logic of vagueness
03B50 Many-valued logic
03C07 Basic properties of first-order languages and structures
Full Text: DOI
[1] DOI: 10.1023/A:1011958407631 · Zbl 0985.03014
[2] An algebraic approach to non-classical logics (1974) · Zbl 0299.02069
[3] DOI: 10.1016/S0168-0072(01)00041-0 · Zbl 1004.03020
[4] Journal of Multiple-Valued Logic and Soft Computing 12 pp 9– (2006)
[5] DOI: 10.1007/s001530050173 · Zbl 0966.03022
[6] DOI: 10.1007/s001530050006 · Zbl 0965.03035
[7] DOI: 10.1016/S0165-0114(00)00118-4 · Zbl 1001.68922
[8] A propositional calculus with denumerable matrix 27 pp 97– (1959) · Zbl 0089.24307
[9] DOI: 10.1007/s00153-002-0152-0 · Zbl 1026.03017
[10] Archive for Mathematical Logic (2006)
[11] Gödel’96: Logical foundations of mathematics, computer science, and physics 6 pp 23– (1996)
[12] Rational Pavelka logic is a conservative extension of Łukasiewicz logic 65 pp 669– (2000) · Zbl 0971.03025
[13] The liar paradox and fuzzy logic 65 pp 339– (2000)
[14] Proceedings of Linz Seminar 2005
[15] DOI: 10.1016/j.fss.2005.03.005 · Zbl 1094.03014
[16] DOI: 10.1007/s00500-004-0448-6 · Zbl 1093.03012
[17] First-order logic revised pp 107– (2004)
[18] DOI: 10.1023/A:1011906423560 · Zbl 0988.03042
[19] Metamathematics of fuzzy logic 4 (1998) · Zbl 0937.03030
[20] Logical, algebraic, analytic and probabilistic aspects of triangular norms pp 275– (2005)
[21] DOI: 10.1023/A:1024621922509 · Zbl 1057.03058
[22] Intuitionistic fuzzy logic and intuitionistic fuzzy set theory 49 pp 851– (1984)
[23] DOI: 10.1007/s00153-004-0214-6 · Zbl 1059.03011
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