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On contra-symmetry and MPT conditionality in fuzzy logic

2005; Wiley; Volume: 20; Issue: 3 Linguagem: Inglês

10.1002/int.20068

1098-111X

Enric Trillas, C. Alsina, Eloy Renedo, Ana Pradera,

Fuzzy Logic and Control Systems

International Journal of Intelligent SystemsVolume 20, Issue 3 p. 313-326 On contra-symmetry and MPT conditionality in fuzzy logic E. Trillas, E. Trillas [email protected] Departamento Inteligencia Artificial, Universidad Politécnica de Madrid, 28660, Boadilla del Monte, Madrid, SpainSearch for more papers by this authorC. Alsina, C. Alsina [email protected] Secció de Matemàtiques i Informàtica, Departament d'Estructures a l'Arquitectura, Universitat Politècnica de Catalunya, Diagonal, 649, 08028 Barcelona, SpainSearch for more papers by this authorE. Renedo, E. Renedo [email protected] Departamento Inteligencia Artificial, Universidad Politécnica de Madrid, 28660, Boadilla del Monte, Madrid, SpainSearch for more papers by this authorA. Pradera, A. Pradera [email protected] Departamento Informática, Estadística y Telemática, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, SpainSearch for more papers by this author E. Trillas, E. Trillas [email protected] Departamento Inteligencia Artificial, Universidad Politécnica de Madrid, 28660, Boadilla del Monte, Madrid, SpainSearch for more papers by this authorC. Alsina, C. Alsina [email protected] Secció de Matemàtiques i Informàtica, Departament d'Estructures a l'Arquitectura, Universitat Politècnica de Catalunya, Diagonal, 649, 08028 Barcelona, SpainSearch for more papers by this authorE. Renedo, E. Renedo [email protected] Departamento Inteligencia Artificial, Universidad Politécnica de Madrid, 28660, Boadilla del Monte, Madrid, SpainSearch for more papers by this authorA. Pradera, A. Pradera [email protected] Departamento Informática, Estadística y Telemática, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, SpainSearch for more papers by this author First published: 19 January 2005 https://doi.org/10.1002/int.20068Citations: 24AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract This article deals with the N-contrapositive symmetry of fuzzy implication operators J verifying either Modus Ponens or Modus Tollens inequalities, in a similar and complementary framework to the one in which Fodor ("Contrapositive symmetry of fuzzy implications." Fuzzy Set Syst 1995;69:141–156) did begin with the subject in fuzzy logic, that is, with the verification of J(a, b) = J(N(b), N(a)) for all a, b in [0,1] and some strong-negation function N. This property corresponds to the classical p → q = ¬q → ¬p. The aim of this article is to study that property in relation to either Modus Ponens or Modus Tollens meta-rules of inference when the functions J are taken among those that belong to the usual families of implications in fuzzy logic. That is, the contra-positive of S implications, R implications, Q implications, and Mamdani–Larsen operators, verifying either Modus Ponens or Modus Tollens inequalities or both, the conditionality's aspect on which lies the complementarity with Fodor. Within this study new types of implication functions are introduced and analyzed. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 313–326, 2005. References 1 Birkhoff G. Lattice theory. Providence, RI: American Mathematical Society; 1973. 2 Megill ND, Pavičić M. Orthomodular lattices and a quantum algebra. Int J Theor Phys 2001; 40: 1387– 1410. 3 Elkan C. The paradoxical success of fuzzy logic. IEEE Expert 1994; 9: 3– 8. 4 Trillas E, Alsina C, Renedo E. On the law (a.b′)′ = b + a′.b′ in De Morgan algebras and orthomodular lattices. Softcomputing 2003; 8(1): 71– 73. 5 Schweizer B, Sklar A. Probabilistic metric spaces. New York: Elsevier, North-Holland; 1983. 6 Klement EP, Mesiar R, Pap E. Triangular norms. Dordrecht: Kluwer Academic Publishers; 2000. 7 Trillas E. Sobre funciones de negación en la teoría de conjuntos difusos. Stochastica 1979;III:47–60; reprinted in English: On negation functions in fuzzy set theory. In: S Barro, A Bugarín, A Sobrino., editors. Advances in fuzzy logic. Spain: Universidade de Santiago de Compostela; 1998. pp 31– 45. 8 Trillas E, Alsina C. On the joint verification of Modus Ponens and Modus Tollens in fuzzy logic. In: Proc Int Conf in Fuzzy Logic and Technology, EUSFLAT-2001, De Montfort University, Leicester, UK, September 2001. pp 257– 260. 9 Trillas E, Alsina C, Pradera A. On MPT-implication functions for fuzzy logic. Rev R Acad Cien Serie A Mat 2004; 98(1): 259– 271. 10 Gottwald S. Fuzzy sets and fuzzy logic. Braunschweig/Wiesbaden: Vieweg; 1993. 11 Klir GJ, Yuan B. Fuzzy sets and fuzzy logic. Upper Saddle River, NJ: Prentice Hall PTR; 1995. 12 Fodor JC. Contrapositive symmetry of fuzzy implications. Fuzzy Set Syst 1995; 69: 141– 156. 13 Guadarrama S, Trillas E, Gutiérrez J, Fernández F. A step towards conceptually improving Tagaki–Sugeno approximation. In: Proc Int Conf on Information Processing and Management of Uncertainty in Knowledge-based Systems, IPMU 2002, vol II, Annecy, France, July 2002. pp 1789– 1794. 14 del Campo C, Trillas E. On Mamdani–Larsen's type fuzzy implications. 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