Laws of Contraposition and Law of Importation for Probabilistic Implications and Probabilistic S-implications

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. © Springer International Publishing Switzerland 2014.