# zbMATH — the first resource for mathematics

Algebraic and proof-theoretic characterizations of truth stressers for MTL and its extensions. (English) Zbl 1190.03026
Truth stressers, also known as modifiers in approximate reasoning applications of fuzzy logics, are modeled here as a kind of modalities. This generalizes an idea offered by P. Hájek [Fuzzy Sets Syst. 124, No. 3, 329–333 (2001; Zbl 0997.03028)] for the particular truth stresser “very true”.
The authors extend the Hilbert-type calculus of monoidal t-norm logic MTL, as well as of suitable related mathematical fuzzy logics, with such operators, develop adequate algebraic semantics (the truth stressers correspond to kinds of interior operators in residuated lattices) and give also equivalent hypersequent calculi. And they get decidability results via the discussion of the finite embeddability property.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 03B45 Modal logic (including the logic of norms) 03B50 Many-valued logic 03F52 Proof-theoretic aspects of linear logic and other substructural logics 03G25 Other algebras related to logic
Full Text:
##### References:
  Avron, A., A constructive analysis of RM, Journal of symbolic logic, 52, 4, 939-951, (1987) · Zbl 0639.03017  M. Baaz, Infinite-valued Gödel logics with 0-1-projections and relativizations, in: Proc. GÖDEL’96, Lecture Notes in Logic, Vol. 6, Springer, Berlin, 1996, pp. 23-33. · Zbl 0862.03015  F. Bou, F. Esteva, L. Godo, R. Rodríguez, On the minimum many-valued logic over a finite residuated lattice, Journal of Logic and Computation, to appear. · Zbl 1252.03040  Ciabattoni, A.; Esteva, F.; Godo, L., T-norm based logics with n-contraction, Neural network world, 12, 5, 441-453, (2002)  A. Ciabattoni, N. Galatos, K. Terui, From axioms to analytic rules in nonclassical logics, in: Proc. LICS, IEEE, 2008, 2008, pp. 229-240.  Ciabattoni, A.; Metcalfe, G., Density elimination, Theoretical computer science, 403, 328-346, (2008) · Zbl 1206.03027  Cox, D.; Little, J.; O’Shea, D., Ideals, varieties and algorithms, (1996), Springer Berlin  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, 2, 199-226, (2002) · Zbl 1011.03015  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  Girard, J., Linear logic, Theoretical computer science, 50, 1-102, (1987) · Zbl 0625.03037  Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Dordrecht · Zbl 0937.03030  Hájek, P., On very true, Fuzzy sets and systems, 124, 329-334, (2001) · Zbl 0997.03028  Höhle, U., Commutative residuated $$\ell$$-monoids, (), 53-106 · Zbl 0838.06012  Jenei, S.; Montagna, F., A proof of standard completeness for esteva and Godo’s MTL logic, Studia logica, 70, 2, 183-192, (2002) · Zbl 0997.03027  Jipsen, P.; Tsinakis, C., A survey of residuated lattices, (), 19-56 · Zbl 1070.06005  Metcalfe, G.; Montagna, F., Substructural fuzzy logics, Journal of symbolic logic, 72, 3, 834-864, (2007) · Zbl 1139.03017  G. Metcalfe, N. Olivetti, Proof systems for a Gödel modal logic, in: Proc. TABLEAUX 2009, Lecture Notes in Artificial Intelligence, Vol. 5607, Springer, Berlin, 2009, pp. 265-279. · Zbl 1260.03053  Metcalfe, G.; Olivetti, N.; Gabbay, D., Proof theory for fuzzy logics, Applied logic, Vol. 36, (2008), Springer Berlin  Montagna, F., Storage operators and multiplicative quantifiers in many-valued logics, Journal of logic and computation, 14, 2, 299-322, (2004) · Zbl 1061.03027  Novák, V., On fuzzy type theory, Fuzzy sets and systems, 149, 2, 235-273, (2005) · Zbl 1068.03019  Ono, H., Closure operators and complete embeddings of residuated lattices, Studia logica, 74, 3, 427-440, (2003) · Zbl 1030.03046  Restall, G., Modalities in substructural logics, Logique et analyse, 35, 303-321, (1992)  Zadeh, L.A., Fuzzy logics and approximate reasoning, Synthese, 30, 407-428, (1975) · Zbl 0319.02016
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.