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The inference rule of Modus Ponens has been extensively investigated in the framework of approximate reasoning, especially for the case of t-norms. Recently, more general kinds of conjunctors have also been considered, like semi-copulas, copulas, and conjunctive uninorms. A common feature of all these kinds of conjunctors is the fact that they have a neutral element e∈ ]0,1]. This paper is devoted to the study of Modus Ponens for conjunctors with a neutral element with no additional conditions. Many properties are proved to be necessary for a fuzzy implication function Ito satisfy the Modus Ponens with respect to a conjunctor with neutral element e∈ ]0,1]. Although the most usual families of fuzzy implication functions do not satisfy all these properties, other possibilities for Iare presented showing many new examples and generalizing some already known results on this topic. Moreover, all fuzzy implication functions satisfying the Modus Ponens with respect to the least (and with respect to the greatest) conjunctor with neutral element e∈ ]0,1[ are characterized. The particular case of e= 1, that provides semi-copulas, is studied separately, retrieving many known results that can be easily derived from the current study.

Copulas have been deeply investigated because of their applications in many fields. From the theoretical point of view, a key point in this research lies in the search of new construction methods of parametrized families of copulas. This paper presents some construction methods based on fuzzy implication functions by reversing the construction methods of fuzzy implication functions from copulas presented by P. Grzegorzewski in some recent papers. Specifically, the PSI and SSI-construction methods of copulas are proposed which provide copulas from a given fuzzy implication function. In addition, the analysis of these construction methods of copulas lead to the characterization of the intersection of the probabilistic Sand survival S-implications with (S, N) and R-implications.