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Adding truth-constants to logics of continuous t-norms: axiomatization and completeness results. (English) Zbl 1117.03030
Summary: We study generic expansions of logics of continuous t-norms with truth-constants, taking advantage of previous results for \({\L}\)ukasiewicz logic and more recent results for Gödel logic and product logic. Indeed, we consider algebraic semantics for expansions of logics of continuous t-norms with a set of truth-constants \(\{\bar r\mid r \in C\}\), for a suitable countable \(C\subseteq [0,1]\), and provide a full description of completeness results when (i) the t-norm is a finite ordinal sum of Łukasiewicz, Gödel and product components, (ii) the set of truth-constants covers all the unit interval in the sense that each component of the t-norm contains at least one value of \(C\) different from the bounds of the component, and (iii) the truth-constants in Łukasiewicz components behave like rational numbers.

03B52 Fuzzy logic; logic of vagueness
Full Text: DOI
[1] Aglianó, P.; Montagna, F., Varieties of BL-algebras I: general properties, J. pure appl. algebra, 181, 105-129, (2003) · Zbl 1034.06009
[2] Blok, W.J.; Pigozzi, D., Algebraizable logics, Mem. amer. math. soc., 77, 396, (1989) · Zbl 0664.03042
[3] Burris, S.; Sankappanavar, H.P., A course in universal algebra, graduate texts in mathematics, (1981), Springer New York · Zbl 0478.08001
[4] 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)
[5] Cignoli, R.; Mundici, D., Partial isomorphisms on totally ordered abelian groups and Hájek’s completeness theorem for basic logic, Mult. valued logic, special issue dedicated to the memory of grigore moisil, 6, 89-94, (2001) · Zbl 1020.03019
[6] Cignoli, R.; Torrens, A., An algebraic analysis of product logic, Mult. valued logic, 5, 45-65, (2000) · Zbl 0962.03059
[7] P. Cintula, From fuzzy logic to fuzzy mathematics, Ph.D. Dissertation, Czech Technical University, Prague, Czech Republic, 2005. · Zbl 1086.06008
[8] Cintula, P., Short note: on the redundancy of axiom (A3) in BL and MTL, Soft comput.—A fusion found. methodol. appl., 9, 12, 942, (2005) · Zbl 1093.03011
[9] Czelakowski, J.; Dziobiak, W., Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class, Algebra universalis, 27, 128-149, (1990) · Zbl 0695.08016
[10] Di Nola, A., MV-algebras in the treatment of uncertainty, (), 123-131
[11] Dummett, M., A propositional calculus with denumerable matrix, J. symbolic logic, 24, 97-106, (1959) · Zbl 0089.24307
[12] 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
[13] Esteva, F.; Godo, L.; Montagna, F., Equational characterization of the subvarieties of BL generated by t-norm algebras, Studia logica, 76, 161-200, (2004) · Zbl 1045.03048
[14] Esteva, F.; Godo, L.; Noguera, C., On rational weak nilpotent minimum logics, J. mult.-valued logic soft comput., 12, 1-2, 9-32, (2006) · Zbl 1144.03020
[15] I.M.A. Ferrerim, On varieties and quasivarieties of hoops and their reducts, Ph.D. Dissertation, University of Illinois at Chicago, 1992.
[16] Gaitán, H., The number of simple one-generated bounded commutative BCK-chains, Math. jpn, 38, 3, 483-486, (1993) · Zbl 0777.06015
[17] Gottwald, S., A treatise on many-valued logics, studies in logic and computation, vol. 9, (2000), Research Studies Press Baldock
[18] Hájek, P., Metamathematics of fuzzy logic, trends in logic, vol. 4, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0937.03030
[19] Hájek, P.; Godo, L.; Esteva, F., A complete many-valued logic with product-conjunction, Arch. math. logic, 35, 191-208, (1996) · Zbl 0848.03005
[20] Hay, L., Axiomatization of the infinite-valued predicate calculus, J. symbolic logic, 28, 77-86, (1963) · Zbl 0127.00703
[21] Horčík, R.; Cintula, P., Product łukasiewicz logic, Arch. math. logic, 43, 477-503, (2004) · Zbl 1059.03011
[22] Ling, C.M., Representation of associative functions, Publ. math. debrecen, 12, 189-212, (1965) · Zbl 0137.26401
[23] F. Montagna, C. Noguera, R. Horčík, On weakly cancellative fuzzy logics, J. Logic Comput. 16 (4) (2006) 423-450. · Zbl 1113.03021
[24] Mostert, P.S.; Shields, A.L., On the structure of semigroups on a compact manifold with boundary, Ann. math., 65, 117-143, (1957) · Zbl 0096.01203
[25] Novák, V., On the syntactico-semantical completeness of first-order fuzzy logic, part I, syntax and semantics, Kybernetika, 26, 47-66, (1990) · Zbl 0705.03009
[26] Novák, V., On the syntactico-semantical completeness of first-order fuzzy logic, part II, main results, Kybernetika, 26, 134-154, (1990) · Zbl 0705.03010
[27] Novák, V.; Perfilieva, I.; Močkoř, J., Mathematical principles of fuzzy logic, (1999), Kluwer Academic Publishers Dordrecht · Zbl 0940.03028
[28] Pavelka, J., On fuzzy logic I, II, III, Z. math. logik grundl. math., 25, 45-52, (1979), 119-134, 447-464 · Zbl 0435.03020
[29] Savický, P.; Cignoli, R.; Esteva, F.; Godo, L.; Noguera, C., On product logic with truth-constants, J. logic comput., 16, 205-225, (2006) · Zbl 1102.03030
[30] Vojtáš, P., Fuzzy logic programming, Fuzzy sets and systems, 124, 361-370, (2001) · Zbl 1015.68036
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