Abstract
Recently, Grzegorzewski [5-7] introduced two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications. They are based on conditional copulas and make a bridge between probability theory and fuzzy logic. In the same article [7] author gives a motivation to his idea and indicates some interesting connections between new families of implications and the dependence structure of the underlying environment. In this paper the laws of contraposition and the law of importation are studied for these families of fuzzy implications.
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References
Baczyński, M., Beliakov, G., Bustince, H., Pradera, A. (eds.): Advances in Fuzzy Implication Functions. STUDFUZZ, vol. 300. Springer, Heidelberg (2013)
Baczyński, M., Jayaram, B.: Fuzzy Implications. STUDFUZZ, vol. 231. Springer, Heidelberg (2008)
Fodor, J.C.: Contrapositive symmetry of fuzzy implications. Fuzzy Sets and Systems 69, 141–156 (1995)
Fodor, J.C., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer, Dordrecht (1994)
Grzegorzewski, P.: Probabilistic implications. In: Proceedings of the 7th Conference on European Society for Fuzzy Logic and Technology (EUSFLAT-2011) and LFA-2011, pp. 254–258. Atlantis Press, Amsterdam (2011)
Grzegorzewski, P.: On the properties of probabilistic implications. In: Melo-Pinto, P., Couto, P., Serôdio, C., Fodor, J., De Baets, B. (eds.) Eurofuse 2011. AISC, vol. 107, pp. 67–78. Springer, Heidelberg (2011)
Grzegorzewski, P.: Probabilistic implications. Fuzzy Sets and Systems 226, 53–66 (2013)
Hlinĕná, D., Kalina, M., Král’, P.: Fuzzy preference modelling and some consequences for fuzzy implications and quasi-copulas. In: Proceeding of the 8th Conference of the European Society for Fuzy Logic and Technology (EUSFLAT 2013), pp. 260–265. Atlantis Press, Amsterdam (2013)
Jayaram, B.: On the law of importation (x ∧ y) → z ≡ (x → (y → z)) in fuzzy logic. IEEE Trans. Fuzzy Systems 16, 130–144 (2008)
Jenei, S.: A new approach for interpolation and extrapolation of compact fuzzy quantities. The one dimensional case. In: Klement, E.P., Stout, L.N. (eds.) Proc. 21st Linz Seminar on Fuzzy Set Theory, Linz, Austria, pp. 13–18 (2000)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000)
Nelsen, R.B.: An Introduction to Copulas. Springer, New York (1999)
Štěpnička, M., Jayaram, B.: On the suitability of the Bandler-Kohout subproduct as an inference mechanism. IEEE Trans. Fuzzy Systems 18, 285–298 (2010)
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Baczyński, M., Grzegorzewski, P., Niemyska, W. (2014). Laws of Contraposition and Law of Importation for Probabilistic Implications and Probabilistic S-implications. 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_17
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DOI: https://doi.org/10.1007/978-3-319-08795-5_17
Publisher Name: Springer, Cham
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