mathematics Article On the Search for a Measure to Compare Interval-Valued Fuzzy Sets Susana Díaz-Vázquez * ,† , Emilio Torres-Manzanera † , Irene Díaz † and Susana Montes †   Citation: Díaz-Vázquez, S.; Torres-Manzanera, E.; Díaz, I.; Montes, S. On the Search for a Measure to Compare Interval-Valued Fuzzy Sets. Mathematics 2021, 9, 3157. https://doi.org/10.3390/math9243157 Academic Editor: Michael Voskoglou Received: 29 October 2021 Accepted: 1 December 2021 Published: 7 December 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Statistics and O. R. and Department of Computer Sciences, University of Oviedo, 33007 Oviedo, Spain; torres@uniovi.es (E.T.-M.); sirene@uniovi.es (I.D.); montes@uniovi.es (S.M.) * Correspondence: diazsusana@uniovi.es; Tel.: +34-985-10-33-80 † These authors contributed equally to this work. Abstract: 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. Keywords: interval-valued fuzzy set; interval order; difference; distance; divergence; dissimilarity 1. Introduction It us usually understood that knowledge of comparisons of objects, opinions, etc. are incomplete. A widely accepted theory (and methodology) to cope with imprecision is fuzzy sets theory, where elements are not necessarily in a set or out of it, but rather intermediate degrees of membership are allowed. In this context, the classical ways to contrast sets do not apply, and several measures for comparing fuzzy sets have been introduced and can be found in the literature. An in-depth study was carried out by Bouchon-Meunier et al. in 1996 [1]. After this, many other measures have been proposed. Some of them are constructive definitions, i.e., specific formulae (see, among many others, Refs. [2–5]) and others are based on axiomatic definitions (see, for example, Refs. [6–8]). The presence of imprecision in real-life situations has been a challenge even from a theoretical point of view. In order to cope with this handicap, different extensions of fuzzy sets have been proposed. Interval-valued fuzzy sets (IVFSs) are one of the most successful and challenging extensions. This generalization was introduced independently and almost simultaneously by Zadeh [9], Grattan-Guiness [10], Jahn [11], and Sambuc [12]. Interval-valued fuzzy sets are a useful tool. They are used to model situations where the “classical” fuzzy sets are not appropriate. This occurs in the case when an objective procedure to determine crisp membership degrees is not available. IVFSs show high potential in practical applications. They were used in medical diagnosis in thyrodian pathology (see Sambuc [12]), in approximate reasoning (see, for instance, the contributions of Bustince [13] and Gozalczany [14]) and Cornelis et al. [15] and Turksen [16] applied this theory in logic. Due to its potential utility, different notions and tools connected to this extension must be studied. In particular, our interest is focused on the measures of comparison of two interval-valued fuzzy sets, which have been studied in the last years. Some of them are based on comparing the degree of similarity between them (see, e.g., [17–20]). However, Mathematics 2021, 9, 3157. https://doi.org/10.3390/math9243157 https://www.mdpi.com/journal/mathematics