Probabilistic implications

A new family of implication operators, called probabilistic implications, is discussed. The suggested implications are based on conditional copulas and make a bridge between probability theory and fuzzy logic. It is shown that probabilistic fuzzy implications have some interesting properties, especially those connected with the dependence structure of the underlying environment. Moreover, some other approaches for combining probability and theory of fuzzy implications are considered. In particular, it is shown that in probabilistic environment S-implications coincide with QL-implications.