Distance functions from fuzzy logic connectives: A state-of-the-art survey

Abstract

While fuzzy logic connectives were seen as generalisations of classical logic connectives, their utility has extended beyond their intended use and context. One interesting avenue of exploration that began almost 4 decades ago is to obtain metrics from fuzzy logic connectives. Not only was this a fertile approach for obtaining metrics with myriad properties but such studies have also thrown up some interesting insights. In this work, we present a state-of-the-art survey of the different works detailing the multitude of operators used to obtain these distance functions, the host of properties they satisfy, the novel contexts in which they have been employed, and the insightful commentary that they have provided on the underlying structures.

Recently, monometrics - distance functions compatible with the underlying order - have attracted scrutiny for their utility in the fields of rationalisation of ranking rules, penalty-based aggregation, and binary classification. In this work, adding to the survey, we examine if and when the existing distance functions yield a monometric. Further, by employing monotonic fuzzy logic connectives and fuzzy negations, we offer a construction of distance functions that always yield monometrics and helps us in providing a characterisation of symmetric monometrics on the unit interval. Our work showcases a close relationship between monometrics and fuzzy implications.