Metric Spaces Under Interval Uncertainty: Towards an Adequate Definition

In many practical situations, we only know the bounds on the distances. A natural question is: knowing these bounds, can we check whether there exists a metric whose distance always lie within these bounds – or such a metric is not possible and thus, the bounds are inconsistent. In this paper, we provide an answer to this question. We also describe possible applications of this result to a description of opposite notions in commonsense reasoning. 1 An AI Problem and the Resulting Mathematical Problem Starting point: commonsense negation vs. formal negation. Negation and opposites are an important part of our reasoning. Thus, to better understand human reasoning, it is desirable to analyze how we use negation. The standard way to describe negation is to use mathematical logic. In mathematical logic, negation has a very precise meaning: a negation ¬S of a statement S is true if and only if the statement S is false. This formal logical notion of a negation corresponds to the notion of a complement to a set: a complement −S is the set of all the objects that do not belong to the original set S. Similarly, in fuzzy logic (see, e.g., [7, 8, 10]): – once we have a fuzzy set, i.e., a function μP that assigns, to each object x, a degree μ(x) to which this object satisfies a given imprecise property P (e.g., is small), – then the negation is usually defined as a membership function