Similarity and dissimilarity measures between fuzzy sets: A formal relational study

The paper deals with the well-known notion of (dis)similarity measures between fuzzy sets. We provide three separate lists of axioms that fit with the respective notions of “general comparison measure”, “similarity measure” and “dissimilarity measure”. Then we review some of the most important axiomatic definitions of (dis)similarity measures in the literature, by referring to the axioms in those lists satisfied by each specific definition. This common framework will make our study about the formal relationships among different axiomatic definitions much easier: some of them, which are apparently different, do in fact share many commonalities. We provide a self-contained picture of these relationships, by providing formal results and counterexamples that reflect which of the (dis)similarity definitions in the literature are connected by implication relations and which of them are not. We finalize the paper with an in-depth study about the notion of “duality” between similarity and dissimilarity measures as well as with some concluding remarks.