Abstract
In this paper we present a tableau calculus for BL, basic fuzzy logic introduced by Petr Hájek in his monograph Metamathematics of Fuzzy Logic. We show that it is sound and complete with respect to continuous t-norms, and demonstrate the refutational procedure and the search for models procedure on a selected example. The idea of the calculus is based on the decomposition theorem for a continuous t-norm, by which this operation is shown to be equivalent to the ordinal sum of a family of t-norms defined on countably many intervals.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Annellis, I.: From semantic tableaux to Smullyan trees: the history of the falsifiability tree method. Modern Logic 1(1), 36–69 (1990)
Bova, S., Montagna, F.: Proof Search in Hájek’s Basic Logic. ACM Transactions on Computational Logic 9(3), Article 21 (June 2008)
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)
Cintula, P., Hájek, P., Noguera, C.: Handbook of Mathematical Fuzzy Logic, vols. 1 & 2. College Publications (2011)
Fitting, M.: Introduction. In: D’Agostino, M., Gabbay, D.M., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods, pp. 1–44. Kluwer Academic Publishers (1999)
Hájek, P.: Basic fuzzy logic and BL-algebras. Soft Computing 2, 124–128 (1998)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers (1998)
Kułacka, A., Pattinson, D., Schröder L.: Syntactic Labelled Tableaux for Lukasiewicz Fuzzy ALC. In: IJCAI (2013)
Metcalfe, G., Olivetti, N., Gabbay, D.: Proof Theory for Fuzzy Logics. Springer (2009)
Montana, F.: Generating the variety of BL-algebras. Soft Computing 9, 869–874 (2005)
Mostert, P.S., Shields, A.L.: On the structure of semigroups on a compact manifold with boundary. Annals of Mathematics 65, 117–144 (1957)
Olivetti, N.: Tableaux for Łukasiewicz Infinite-valued Logic. Studia Logica 73, 81–111 (2003)
Orłowska, E., Golińska-Pilarek, J.: Dual Tableaux: Foundations, Methodology, Case Studies. Springer (2011)
Tarski, A.: A decision method for elementary algebra and geometry. University of California Press, Berkeley (1951)
Vetterlein, T.: Analytic Calculi for Logics of Ordinal Multiples of Standard t-Norms. Journal of Logic and Computation 18(1), 35–57 (2007)
Vidal, A., Bou, F., Godo, L.: An SMT-Based Solver for Continuous t-norm Based Logics. In: Hüllermeier, E., Link, S., Fober, T., Seeger, B. (eds.) SUM 2012. LNCS, vol. 7520, pp. 633–640. Springer, Heidelberg (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Kułlacka, A. (2014). Tableau Calculus for Basic Fuzzy Logic BL. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2014. Communications in Computer and Information Science, vol 442. Springer, Cham. https://doi.org/10.1007/978-3-319-08795-5_34
Download citation
DOI: https://doi.org/10.1007/978-3-319-08795-5_34
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08794-8
Online ISBN: 978-3-319-08795-5
eBook Packages: Computer ScienceComputer Science (R0)