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Substructural fuzzy logics

Published online by Cambridge University Press:  12 March 2014

George Metcalfe  and
Franco Montagna
George Metcalfe
Affiliation:
Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville TN 37240, USA. E-mail: metcalfe@logic.at
Franco Montagna
Affiliation:
Department of Mathematics, University of Siena, Via Del Capitano 15, 53100 Siena, Italy. E-mail: montagna@unisi.it
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Abstract

Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) V ((B → A)∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MIX and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 834 - 864
Copyright
Copyright © Association for Symbolic Logic 2007

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References

[1] Avron, A., A constructive analysis of RM, this Journal, vol. 52 (1987), no. 4, pp. 939–951.Google Scholar
[2] Avron, A., Hypersequents. logical consequence and intermediate logics for concurrency, Annals of Mathematics and Artificial Intelligence, vol. 4 (1991), no. 3–4, pp. 225–248.CrossRefGoogle Scholar
[3] Baaz, M., Ciabattoni, A., and Montagna, F., Analytic calculi for monoidal t-norm based logic, Fundamenta Informaticae, vol. 59 (2004), no. 4, pp. 315–332.Google Scholar
[4] Baaz, M. and Zach, R., Hypersequents and the proof theory of intuitionistic fuzzy logic, Proceedings of CSL 2000, LNCS, Springer-Verlag, 2000, pp. 187–201.Google Scholar
[5] Ciabattoni, A., Esteva, F., and Godo, L., T-norm based logics with n-contraction, Neural Network World, vol. 12 (2002), no. 5, pp. 441–453.Google Scholar
[6] Cintula, P., Weakly implicative (fuzzy) logics I: Basic properties, Archive for Mathematical Logic, vol.45 (2006), pp. 673–704.CrossRefGoogle Scholar
[7] De Baets, B., Idempotent uninorms, European Journal of Operational Research, vol. 118 (1999), pp. 631–642.CrossRefGoogle Scholar
[8] Esteva, F., Gispert, J., Godo, L., and Montagna, F., On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic, Studio Logica, vol. 71 (2002), no. 2, pp. 199–226.CrossRefGoogle Scholar
[9] Esteva, F. and Godo, L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems, vol. 124 (2001), pp. 271–288.CrossRefGoogle Scholar
[10] Gabbay, D. and Metcalfe, G., Logics based on [0, 1)-continuous uninorms, Archive for Mathematical Logic, vol. 46 (2007), pp. 425–449.CrossRefGoogle Scholar
[11] Hájek, P., Metamathematics of fuzzy logic, Kluwer, Dordrecht, 1998.CrossRefGoogle Scholar
[12] Jenei, S. and Montagna, F., A proof of standard completeness for Esteva and Godo's MTL logic, Studia Logica, vol. 70 (2002), no. 2, pp. 183–192.CrossRefGoogle Scholar
[13] Metcalfe, G., Olivetti, N., and Gabbay, D., Analytic proof calculi for product logics, Archive for Mathematical Logic, vol. 43 (2004), no. 7, pp. 859–889.CrossRefGoogle Scholar
[14] Metcalfe, G., Sequent and hypersequent calculi for abelian and Łukasiewicz logics, ACM Transactions on Computational Logic, vol. 6 (2005), no. 3, pp. 578–613.CrossRefGoogle Scholar
[15] Ono, H. and Komori, Y., Logics without the contraction rule, this Journal, vol. 50 (1985), pp. 169–201.Google Scholar
[16] Pottinger, G., Uniform, cut-free formulations of T, S4 and S5 (abstract), this Journal, vol. 48 (1983), no. 3, p. 900.Google Scholar
[17] Restall, G., An introduction to substructural logics, Routledge, London, 1999.Google Scholar
[18] Takeuti, G. and Titani, T., Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, this Journal, vol. 49 (1984), no. 3, pp. 851–866.Google Scholar
[19] Tsinakis, C. and Blount, K., The structure of residuated lattices, International Journal of Algebra and Computation, vol. 13 (2003), no, 4, pp. 437–461.Google Scholar
[20] Yager, R. and Rybalov, A., Uninorm aggregation operators. Fuzzy Sets and Systems, vol. 80 (1996), pp. 111–120.CrossRefGoogle Scholar