Pseudo-monometrics from fuzzy implications

Abstract

Several works have proposed the construction of distance functions using fuzzy logic connectives to proffer further applications of the corresponding connectives. In these works, the authors define a distance function using t-norms, t-conorms, copulas, or quasi-copulas, all of which are either associative, commutative or monotonic fuzzy logic connectives. In this work, we define a distance function, denoted dI, from a non-associative, non-commutative, and mixed-monotonic fuzzy logic connective, viz., a fuzzy implication I, and study the above distance function along two aspects. Firstly, we investigate the necessary and sufficient conditions for dI to be a metric, wherein the role played by a transitivity type functional inequality involving the considered fuzzy implication and the Łukasiewicz t-conorm is highlighted. In the recent past, monometrics w.r.t. a ternary relation, called the betweenness relation, have garnered a lot of attention for their important role in decision-making and penalty-based data aggregation. One of the major challenges herein is that of obtaining monometrics on a given betweenness set. Our second contribution in this work is in establishing the existence of pseudo-monometrics using dI, from whence it appears that fuzzy implications are a natural choice for obtaining pseudo-monometrics on a given betweenness set.