Some properties of fuzzy implications based on copulas

In 2011 Grzegorzewski introduced two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications. They are based on copulas and make a bridge between probability theory and fuzzy logic. Another family of fuzzy conditional implication operators was proposed by Dolati et al. in 2013. In this paper we consider some properties of these three classes of fuzzy implications like the law of contrapositions and the law of importation. Moreover, we examine intersections of these families of implications with R-implications, (S, N)-implications, QL-operations and Yager’s f- and g-generated implications.