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A new axiomatization for involutive monoidal t-norm-based logic. (English) Zbl 0998.03024
The paper gives an alternative axiomatization for the monoidal t-norm-based residuated logic originally introduced by F. Esteva and L. Godo [Fuzzy Sets Syst. 124, 271-288 (2001; Zbl 0994.03017)]. The presented approach stresses the geometric understanding of Girard monoids, especially the rotation invariance \(T(x,y)\leq z\) iff \(T(y,N(z))\leq N(x)\), where \(T\) is a left-continuous triangular norm and \(N\) is a strong negation. The rotation invariance is transformed into axiom schemata for logical calculi, and then serves as a substitute for the standard adjointness condition linking conjunction and implication connectives in t-norm-based residuated logics. The proposed new (and equivalent) axiomatization may give new interesting results in the recently introduced residuated fuzzy logic based on left-continuous t-norms.

MSC:
03B52 Fuzzy logic; logic of vagueness
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[1] Esteva, F.; Godo, L., Monoidal t-norm-based logictowards a logic for left-continuous t-norms, Fuzzy sets and systems, 124, 271-288, (2001) · Zbl 0994.03017
[2] Esteva, F.; Godo, L.; Hájek, P.; Navara, M., Residuated fuzzy logic with an involutive negation, Arch. math. logic, 39, 103-124, (2000) · Zbl 0965.03035
[3] Gottwald, S., Axiomatizations of t-norm based logics—a survey, Soft comput., 4, 63-67, (2000)
[4] Gottwald, S., A treatise on many valued logics, studies in logic and computation, (2001), Research Studies Press Baldock, Hertfordshire, UK
[5] Hájek, P., Metamathematics of fuzzy logic, trends in logic, vol. 4, (1998), Kluwer Acad. Publ. Dordrecht · Zbl 0937.03030
[6] Jenei, S., Geometry of left-continuous t-norms with strong induced negations, Belg. J. oper. res. statist. comput. sci., 38, 5-16, (1998) · Zbl 1010.03520
[7] Jenei, S., Structure of left-continuous triangular norms with strong induced negations. (I) rotation construction, J. appl. non-classical logics, 10, 83-92, (2000) · Zbl 1033.03512
[8] S. Jenei, Structure of Girard monoids on [0,1], in: S. Rodabaugh, E.P. Klement (Eds.), Topological and Algebraic Structures in Fuzzy Sets, Kluwer Acad. Publ., Dordrecht, to appear. · Zbl 0993.68123
[9] Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms, (2000), Kluwer Acad. Publ. Dordrecht · Zbl 0972.03002
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