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

Abstract

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.

On the T-conditionality of T-power based implications

“…Fuzzy implications are mainly used in fuzzy logic systems to model fuzzy conditionals, and to perform fuzzy inference processes through generalized modus ponens and generalized modus tollens rules [4,6,7,10,32]. It is well known that the generalized DOI: 10.14736/kyb-2022-1-0043 modus ponens can be implemented by a scheme enabled by a functional inequality called the law of T -conditionality (T -conditionality for short) [3], also known as modus ponens inequality [26]. Hence, the T -conditionality for many kinds of implication functions has been extensively studied (see for instance, [1,16,17,23,24,26,27,28]).…”

“…It is well known that the generalized DOI: 10.14736/kyb-2022-1-0043 modus ponens can be implemented by a scheme enabled by a functional inequality called the law of T -conditionality (T -conditionality for short) [3], also known as modus ponens inequality [26]. Hence, the T -conditionality for many kinds of implication functions has been extensively studied (see for instance, [1,16,17,23,24,26,27,28]). Recently, Li et al [16] have studied the T -conditionality for T -power based implications.…”