# zbMATH — the first resource for mathematics

Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. (English) Zbl 1168.03052
This is an algebraic study of t-norm based logics. Hájek’s Basic Logic is the logic of all continuous t-norms and their residual implications (see [P. Hájek, Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)]). Esteva and Godo observed that left-continuity (rather than continuity) is sufficient, and necessary, for a t-norm to have a residuum, and proposed a weaker logic, called MTL, conjecturing that this is the logic of all left-continuous t-norms and their residual implications. This conjecture was proved by Jenei and Montagna. Interestingly, MTL can be characterized as Full Lambek calculus plus exchange, weakening and prelinearity. The authors consider three completeness properties with respect to the semantics given by linearly ordered MTL-algebras. Five different kinds of semantical generators are considered, namely the real, the rational, the hyperreal unit interval, the strict hyperreals (i.e., those ultrapowers which are a proper extension of the real unit interval) and finite chains. In a final section all these completeness properties and semantics are discussed for first-order logics, and a number of results are proved using a variety of techniques, including formal grammars and languages.

##### MSC:
 03G25 Other algebras related to logic 03B50 Many-valued logic 03B52 Fuzzy logic; logic of vagueness 06F05 Ordered semigroups and monoids
Full Text:
##### References:
 [1] Aglianò, Paolo; Montagna, Franco, Varieties of BL-algebras I: general properties, Journal of pure and applied algebra, 181, 105-129, (2003) · Zbl 1034.06009 [2] Alsina, Claudi; Trillas, Ernest; Valverde, Llorenç, Some logical connectives for fuzzy set theory, Journal of mathematical analysis and applications, 93, 15-26, (1983) · Zbl 0522.03012 [3] Avron, Arnon, A constructive analysis of RM, Journal of symbolic logic, 52, 4, 939-951, (1987) · Zbl 0639.03017 [4] Baaz, Matthias, Infinite-valued Gödel logic with 0-1-projections and relativisations, (), 23-33 · Zbl 0862.03015 [5] Baaz, Matthias; Ciabattoni, Agata; Montagna, Franco, Analytic calculi for monoidal $$t$$-norm based logic, Fundamenta informaticae, 59, 4, 315-332, (2004) · Zbl 1057.03019 [6] Běhounek, Libor, On the difference between traditional and deductive fuzzy logic, Fuzzy sets and systems, 159, 10, 1153-1164, (2008) · Zbl 1175.03012 [7] Blok, Willem J.; Pigozzi, Don, () [8] Blok, Willem J.; van Alten, Clint J., The finite embeddability property for residuated lattices, pocrims and BCK-algebras, Algebra universalis, 48, 253-271, (2000) · Zbl 1058.06016 [9] Burris, Stanley; Sankappanavar, H.P., () [10] Chang, Chen Chung, Algebraic analysis of many-valued logics, Transactions American mathematical society, 88, 456-490, (1958) · Zbl 0084.00704 [11] Chomsky, Noam, Syntactic structures, (2002), Walter de Gruyter Berlin · Zbl 0052.24706 [12] Ciabattoni, Agata; Esteva, Francesc; Godo, Lluís, $$t$$-norm based logics with $$n$$-contraction, Sofsem2002, Neural network world, 12, 5, 453-460, (2002), (special issue) [13] Cignoli, Roberto; D’Ottaviano, Itala M.L.; Mundici, Daniele, () [14] Cignoli, Roberto; Esteva, Francesc; Godo, Lluís; Torrens, Antoni, Basic fuzzy logic is the logic of continuous $$t$$-norms and their residua, Soft computing, 4, 2, 106-112, (2000) [15] Cignoli, Roberto; Torrens, Antoni, Standard completeness of Hájek basic logic and decompositions of BL-chains, Soft computing, 9, 12, 862-868, (2005) · Zbl 1094.03013 [16] Cintula, Petr, Weakly implicative (fuzzy) logics I: basic properties, Archive for mathematical logic, 45, 6, 673-704, (2006) · Zbl 1101.03015 [17] Petr Cintula, Erich Peter Klement, Radko Mesiar, Mirko Navara, Fuzzy logics with an additional involutive negation, Fuzzy Sets and Systems (2009) (in press) · Zbl 1189.03028 [18] Czelakowski, Janusz, () [19] Czelakowski, Janusz; Dziobiak, Wieslaw, Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class, Algebra universalis, 27, 128-149, (1990) · Zbl 0695.08016 [20] Di Nola, Antonio; Esteva, Francesc; Godo, Lluís; Montagna, Franco, Varieties of BL-algebras, Soft computing, 9, 12, 875-888, (2005) · Zbl 1092.03036 [21] Dummett, Michael, A propositional calculus with denumerable matrix, Journal of symbolic logic, 27, 97-106, (1959) · Zbl 0089.24307 [22] Esteva, Francesc; Gispert, Joan; Godo, Lluís; Montagna, Franco, On the standard and rational completeness of some axiomatic extensions of the monoidal $$t$$-norm logic, Studia logica, 71, 2, 199-226, (2002) · Zbl 1011.03015 [23] Esteva, Francesc; Gispert, Joan; Godo, Lluís; Noguera, Carles, Adding truth-constants to continuous $$t$$-norm based logics: axiomatization and completeness results, Fuzzy sets and systems, 158, 597-618, (2007) · Zbl 1117.03030 [24] Esteva, Francesc; Godo, Lluís, 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 [25] Esteva, Francesc; Godo, Lluís; Hájek, Petr; Navara, Mirko, Residuated fuzzy logics with an involutive negation, Archive for mathematical logic, 39, 2, 103-124, (2000) · Zbl 0965.03035 [26] Esteva, Francesc; Godo, Lluís; Montagna, Franco, The $$\operatorname{Ł\Pi}$$ and $$\operatorname{Ł\Pi} \frac{1}{2}$$ logics: two complete fuzzy systems joining łukasiewicz and product logics, Archive for mathematical logic, 40, 1, 39-67, (2001) · Zbl 0966.03022 [27] Esteva, Francesc; Godo, Lluís; Noguera, Carles, On rational weak nilpotent minimum logics, Journal of multiple-valued logic and soft computing, 12, 1-2, 9-32, (2006) · Zbl 1144.03020 [28] Francesc Esteva, Lluís Godo, Carles Noguera, First-order $$t$$-norm based fuzzy logics with truth-constants: Distinguished semantics and completeness properties, Annals of Pure and Applied Logic (2009) (submitted) · Zbl 1222.03027 [29] Francesc Esteva, Lluís Godo, Carles Noguera, On expansions of $$t$$-norm based logics with truth-constants, Fuzzy Sets and Systems (2009) (in press) · Zbl 1222.03027 [30] Flaminio, Tommaso, Strong non-standard completeness for fuzzy logics, Soft computing, 12, 4, 321-333, (2007) · Zbl 1132.03332 [31] Flaminio, Tommaso; Marchioni, Enrico, $$t$$-norm based logics with an independent involutive negation, Fuzzy sets and systems, 157, 4, 3125-3144, (2006) · Zbl 1114.03015 [32] Galatos, Nikolaos; Jipsen, Peter; Kowalski, Tomasz; Ono, Hiroakira, () [33] Gödel, Kurt, Zum intuitionistischen aussagenkalkül, Anzieger akademie der wissenschaften wien, 69, 65-66, (1932) · JFM 58.1001.03 [34] Gottwald, Siegfried, () [35] Hájek, Petr, Basic fuzzy logic and BL-algebras, Soft computing, 2, 3, 124-128, (1998) [36] Hájek, Petr, () [37] Hájek, Petr, Observations on the monoidal $$t$$-norm logic, Fuzzy sets and systems, 132, 1, 107-112, (2002) · Zbl 1012.03035 [38] Hájek, Petr, Arithmetical complexity of fuzzy predicate logics — A survey, Soft computing, 9, 12, 935-941, (2005) · Zbl 1093.03012 [39] Hájek, Petr; Cintula, Petr, On theories and models in fuzzy predicate logics, Journal of symbolic logic, 71, 3, 863-880, (2006) · Zbl 1111.03030 [40] Petr Hájek, Petr Cintula, Triangular norm predicate fuzzy logics, Fuzzy Sets and Systems (2009) (in press) · Zbl 1200.03020 [41] Hájek, Petr; Godo, Lluís; Esteva, Francesc, A complete many-valued logic with product conjunction, Archive for mathematical logic, 35, 3, 191-208, (1996) · Zbl 0848.03005 [42] Rostislav Horčík, Algebraic properties of fuzzy logics, Ph.D. Thesis, Czech Technical University, Faculty of Electrical Egineering, Prague, 2005 [43] Horčík, Rostislav, Standard completeness theorem for $$\operatorname{\Pi}$$MTL, Archive for mathematical logic, 44, 4, 413-424, (2005) · Zbl 1071.03013 [44] Horčík, Rostislav, Decidability of cancellative extension of monoidal $$t$$-norm logic, Logic journal of the interest group of pure and applied logic, 14, 827-843, (2006) · Zbl 1114.03018 [45] Horčík, Rostislav, On the failure of standard completeness in $$\operatorname{\Pi}$$MTL for infinite theories, Fuzzy sets and systems, 158, 619-624, (2007) · Zbl 1117.03032 [46] Horčík, Rostislav; Cintula, Petr, Product łukasiewicz logic, Archive for mathematical logic, 43, 4, 477-503, (2004) · Zbl 1059.03011 [47] Jenei, Sándor; Montagna, Franco, A proof of standard completeness for esteva and godo’s logic MTL, Studia logica, 70, 2, 183-192, (2002) · Zbl 0997.03027 [48] Ling, C.H., Representation of associative functions, Publicationes mathematicae debrecen, 12, 189-212, (1965) · Zbl 0137.26401 [49] Łukasiewicz, Jan, O logice trójwartościowej (on three-valued logic), Ruch filozoficzny, 5, 170-171, (1920) [50] Łukasiewicz, Jan; Tarski, Alfred, Untersuchungen über den aussagenkalkül, Comptes rendus des Séances de la société des sciences et des lettres de varsovie, 23, iii, 30-50, (1930) · JFM 57.1319.01 [51] Metcalfe, George; Montagna, Franco, Substructural fuzzy logics, Journal of symbolic logic, 72, 3, 834-864, (2007) · Zbl 1139.03017 [52] Montagna, Franco, Generating the variety of BL-algebras, Soft computing, 9, 12, 869-874, (2005) · Zbl 1093.03039 [53] Montagna, Franco, On the predicate logics of continuous $$t$$-norm BL-algebras, Archive for mathematical logic, 44, 97-114, (2005) · Zbl 1070.03013 [54] Montagna, Franco, Subreducts of MV-algebras with product and product residuation, Algebra universalis, 53, 1, 109-137, (2005) · Zbl 1086.06010 [55] Montagna, Franco; Noguera, Carles; Horčík, Rostislav, On weakly cancellative fuzzy logics, Journal of logic and computation, 16, 4, 423-450, (2006) · Zbl 1113.03021 [56] Montagna, Franco; Ono, Hiroakira, Kripke semantics, undecidability and standard completeness for esteva and godo’s logic MTL$$\forall$$, Studia logica, 71, 2, 227-245, (2002) · Zbl 1013.03021 [57] Mostert, Paul S.; Shields, Allen L., On the structure of semigroups on a compact manifold with boundary, The annals of mathematics, second series, 65, 117-143, (1957) · Zbl 0096.01203 [58] Carles Noguera, Algebraic study of axiomatic extensions of $$t$$-norm based fuzzy logics, Ph.D. Thesis, University of Barcelona, Barcelona, 2006 [59] Noguera, Carles; Esteva, Francesc; Gispert, Joan, On some varieties of MTL-algebras, Logic journal of the interest group of pure and applied logic, 13, 443-466, (2005) · Zbl 1078.03051 [60] Di Nola, Antonio; Esteva, Francesc; Garcia, Pere; Godo, Lluís; Sessa, Salvatore, Subvarieties of BL-algebras generated by single-component chains, Archive for mathematical logic, 41, 673-685, (2002) · Zbl 1023.03060 [61] Novák, Vilém; Perfilieva, Irina; Močkoř, Jiří, Mathematical principles of fuzzy logic, (2000), Kluwer Dordrecht · Zbl 0940.03028 [62] Ono, Hiroakira, Proof-theoretic methods in non-classical logic: an introduction, (), 207-254 · Zbl 0947.03073 [63] Hiroakira Ono, Logic without contraction rule and residuated lattices, unpublished [64] Ono, Hiroakira; Komori, Yuichi, Logic without the contraction rule, Journal of symbolic logic, 50, 169-201, (1985) · Zbl 0583.03018 [65] Pavelka, Jan, On fuzzy logic I, II, III, Zeitschrift für mathematische logik und grundlagen der Mathematik, 25, 45-52, (1979), 119-134, 447-464 · Zbl 0435.03020 [66] Mathiase E. Ragaz, Arithmetische Klassifikation von Formelmengen der unendlichwertigen Logik, Ph.D. Thesis, Swiss Federal Institute of Technology, Zürich, 1981 · Zbl 0516.03011 [67] Rasiowa, Helena, An algebraic approach to non-classical logics, (1974), North-Holland Amsterdam · Zbl 0299.02069 [68] Savický, Petr; Cignoli, Roberto; Esteva, Francesc; Godo, Lluís; Noguera, Carles, On product logic with truth constants, Journal of logic and computation, 16, 2, 205-225, (2006) · Zbl 1102.03030 [69] Wang, San-Min; Wang, Bao-Shu; Pei, Dao-Wu, A fuzzy logic for an ordinal sum $$t$$-norm, Fuzzy sets and systems, 149, 2, 297-307, (2005) · Zbl 1060.03043 [70] Wang, San-Min; Wang, Bao-Shu; Ren, Fang, NMł, a schematic extension of F. esteva and L. godo’s logic MTL, Fuzzy sets and systems, 149, 2, 285-295, (2005) · Zbl 1060.03044 [71] Zadeh, Lotfi A., Fuzzy sets, Information and control, 8, 3, 338-353, (1965) · Zbl 0139.24606
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.