On the failure of standard completeness in \(\Pi\)MTL for infinite theories. (English) Zbl 1117.03032

Summary: It is well-known that Hájek’s basic fuzzy logic (BL), Łukasiewicz logic, and product logic are not strongly standard complete. On the other hand, Esteva and Godo’s monoidal t-norm-based logic (MTL) and its involutive extension IMTL are strongly standard complete. In this paper we show that \(\Pi\)MTL (an extension of MTL by the axioms characteristic of product logic) does not enjoy the strong standard completeness theorem like BL, Łukasiewicz, and product logic.


03B52 Fuzzy logic; logic of vagueness
03B50 Many-valued logic
Full Text: DOI


[1] Cignoli, R.; Esteva, F.; Godo, L.; Torrens, A., Basic fuzzy logic is the logic of continuous t-norms and their residua, Soft comput., 4, 106-112, (2000)
[2] Esteva, F.; Godo, G., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy sets and systems, 124, 3, 271-288, (2001) · Zbl 0994.03017
[3] Esteva, F.; Gispert, J.; Godo, G.; Montagna, F., On the standard completeness of some axiomatic extensions of the monoidal t-norm logic, Studia logica, 71, 2, 199-226, (2002) · Zbl 1011.03015
[4] Fuchs, L., Partially ordered algebraic systems, (1963), Pergamon Press Oxford · Zbl 0137.02001
[5] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Academic Publisher Dordrecht · Zbl 0937.03030
[6] Hájek, P., Observations on the monoidal t-norm logic, Fuzzy sets and systems, 132, 107-112, (2002) · Zbl 1012.03035
[7] Horčík, R., Standard completeness theorem for \(\operatorname{\Pi} \operatorname{MTL}\), Arch. math. logic, 44, 4, 413-424, (2005) · Zbl 1071.03013
[8] R. Horčík, Algebraic properties of fuzzy logics, Ph.D. Thesis, Czech Technical University in Prague, 2005.
[9] R. Horčík, P. Cintula, F. Montagna, Archimedean completeness and subvarieties of \(\operatorname{\Pi} \operatorname{MTL}\)-algebras. In: Proc. Internat. Conf. Algebraic and Topological Methods in Non-Classical Logics II, Barcelona, 2005, pp. 40-41.
[10] Jenei, S.; Montagna, F., A proof of standard completeness for esteva and Godo’s logic MTL, Studia logica, 70, 2, 183-192, (2002) · Zbl 0997.03027
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