On the Use of Conjunctors With a Neutral Element in the Modus Ponens Inequality

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 I to 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 I are 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.