Metrics from Fuzzy Implications and Their Application

There have been a few works studying metrics obtained from fuzzy logic connectives such as t-norms and copulas which are either commutative, monotonically increasing, or associative. In this work, we define a distance function generated from a non-associative, non-commutative, and non-monotonic fuzzy logic connective, viz., a fuzzy implication. We consider fuzzy implication as a relation on [0, 1] and give a way to obtain metrics from \(S_\textbf{LK}\) - transitive relations that turn out to be monometrics w.r.t. the betweenness relation obtained from the underlying total order on [0, 1]. We also give some sufficient conditions under which certain families of fuzzy implications yield a metric. Our study, on the one hand highlights the usefulness of S-transitive fuzzy relations as much as T-transitive fuzzy relations, and on the other hand, illustrates emphatically the need for fuzzy logic operations on non-linear posets.