Every interval-valued metric can be naturally interpreted in fuzzy terms

In situations when we do not have a good idea of what decision is the best, a natural idea is to look at similar situations in the past and see what worked in the past and what did not. From this viewpoint, it is important to design and study different measures of similarity. An efficient similarity measure must reflect our intuitive understanding of what is similar and what is not. Thus, when a general mathematical notion of a similarity measure is proposed, a reasonable question is how natural are measures satisfying this definition? One of the known measures of similarity are related to metrics (distance functions). It is known that every metric can be naturally interpreted in fuzzy terms. In this paper, we extend this interpretation to a more realistic case of interval-valued metrics and similarity measures.