On the Search for a Measure to Compare Interval-Valued Fuzzy Sets

Multiple definitions have been put forward in the literature to measure the differences between two interval-valued fuzzy sets. However, in most cases, the outcome is just a real value, although an interval could be more appropriate in this environment. This is the starting point of this contribution. Thus, we revisit the axioms that a measure of the difference between two interval-valued fuzzy sets should satisfy, paying special attention to the condition of monotonicity in the sense that the closer the intervals are, the smaller the measure of difference between them is. Its formalisation leads to very different concepts: distances, divergences and dissimilarities. We have proven that distances and divergences lead to contradictory properties for this kind of sets. Therefore, we conclude that dissimilarities are the only appropriate measures to measure the difference between two interval-valued fuzzy sets when the outcome is an interval.