Properties of the probabilistic implications and S-implications

Recently, Grzegorzewski (2011) 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 his previous articles author gave a motivation to his idea and indicates some interesting connections between new families of multivalued implications and the dependence structure of the underlying environment. In this paper the laws of contraposition, the law of importation and T-conditionality are studied for these families of fuzzy implications. Furthermore, we discuss the intersections of both new families of implications with R-implications, (S,N)-implications and QL-operations.