From three to one: Equivalence and characterization of material implications derived from co-copulas, probabilistic S-implications and survival S-implications

Since their introduction in 2012 and 2013 by P. Grzegorzewski, probabilistic S-implications and survival S-implications have attracted the efforts of some researchers due to the connection that they represent between probability theory and fuzzy logic. In this paper, the characterizations of these two families of fuzzy implication functions are presented jointly with the characterization of the material implications derived from co-copulas and the classical negation. Even more, it is proved that the three families are actually the same. Thus every result concerning one of these families can be straightforwardly rewritten in terms of the other two families.