zbMATH — the first resource for mathematics

T-norm-based logics with an independent involutive negation. (English) Zbl 1114.03015
Summary: We investigate the addition of arbitrary independent involutive negations to t-norm-based logics. We deal with several extensions of MTL and establish general completeness results. Indeed, we show that, given any t-norm-based logic satisfying some basic properties, its extension by means of an involutive negation preserves algebraic and (finite) strong standard completeness. We deal with both propositional and predicate logics.

03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
Full Text: DOI
[1] Baaz, M., Infinite-valued Gödel logics with 0-1-projections and relativizations, (), 23-33 · Zbl 0862.03015
[2] Butnariu, E.; Klement, E.P.; Zafrany, S., On triangular norm-based propositional fuzzy logics, Fuzzy sets and systems, 69, 241-255, (1995) · Zbl 0844.03011
[3] Cignoli, R.; Esteva, F.; Godo, L.; Torrens, A., Basic fuzzy logic is the logic of continuous t-norms and their residua, Soft computing, 4, 106-112, (2000)
[4] Cintula, P., An alternative approach to the ł\(\operatorname{\Pi}\) logic, Neural network world, 11, 561-572, (2001)
[5] Cintula, P., Weakly implicative logics I. basic properties, Arch. math. logic, 45, 6, 673-704, (2006) · Zbl 1101.03015
[6] P. Cintula, E.P. Klement, R. Mesiar, M. Navara, Fuzzy logics with an additional involutive negation, submitted for publication. · Zbl 1189.03028
[7] Esteva, F.; Gispert, J.; Godo, L.; Montagna, F., On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic, Studia logica, 71, 199-226, (2002) · Zbl 1011.03015
[8] Esteva, F.; Godo, L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy sets and systems, 124, 271-288, (2001) · Zbl 0994.03017
[9] F. Esteva, L. Godo, J. Gispert, C. Noguera, Adding truth constants to logics of continuous t-norms: axiomatization and completeness results, 2006, submitted for publication. · Zbl 1117.03030
[10] Esteva, F.; Godo, L.; Hájek, P.; Navara, M., Residuated fuzzy logics with an involutive negation, Arch. math. logic, 39, 103-124, (2000) · Zbl 0965.03035
[11] Esteva, F.; Godo, L.; Montagna, F., The ł\(\operatorname{\Pi}\) and ł\(\operatorname{\Pi} \frac{1}{2}\) logics: two complete fuzzy logics joining łukasiewicz and product logic, Arch. math. logic, 40, 39-67, (2001) · Zbl 0966.03022
[12] T. Flaminio, E. Marchioni, Extending the Monoidal T-norm-based Logic with an independent involutive negation, in: Proc. of EUSFLAT’05, 2005, pp. 860-865.
[13] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0937.03030
[14] Z. Haniková, On the complexity of propositional logics with an involutive negation, in: Proc. of EUSFLAT’03, Zittau, Germany, 2003.
[15] Hekrdla, J.; Klement, E.P.; Navara, M., Two approaches to fuzzy propositional logics, J. multi.-valued logic soft comput., 9, 343-360, (2003) · Zbl 1043.03017
[16] Horčík, R., Standard completeness theorem for \(\Pi\)MTL, Arch. math. logic, 44, 4, 413-423, (2005) · Zbl 1071.03013
[17] Jenei, S.; Montagna, F., A proof of standard completeness for esteva and Godo’s logic MTL, Studia logica, 70, 183-192, (2002) · Zbl 0997.03027
[18] McKenzie, R.; McNulty, G.; Taylor, W., Algebras, lattices, varieties, vol. I, (1987), Wadsworth and Brooks/Cole Monterey, CA
[19] Montagna, F.; Ono, H., Kripke semantics, undecidability and standard completeness for esteva and Godo’s logic MTL\(\forall\), Studia logica, 71, 227-245, (2002) · Zbl 1013.03021
[20] Nguyen, H.T.; Walker, E.A., A first course in fuzzy logic, (2000), Chapman & Hall, CRC London, Boca Raton, FL · Zbl 0927.03001
[21] Szász, G., Théorie des treillis, (1971), Dunod Paris · Zbl 0208.28901
[22] Trillas, E., Sobre funciones de negación en teoría de conjuntos difusos, Stochastica, 3, 1, 47-59, (1979)
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.