Logic of approximate entailment in quasimetric and in metric spaces

It is known that a quasimetric space can be represented by means of a metric space; the points of the former space become closed subsets of the latter one, and the role of the quasimetric is assumed by the Hausdorff quasidistance. In this paper, we show that, in a slightly more special context, a sharpened version of this representation theorem holds. Namely, we assume a quasimetric to fulfil separability in the original sense due to Wilson. Then any quasimetric space can be represented by means of a metric space such that distinct points are assigned disjoint closed subsets. This result is tailored to the solution of an open problem from the area of approximate reasoning. Following the lines of E. Ruspini's work, the Logic of Approximate Entailment ([Formula: see text]) is based on a graded version of the classical entailment relation. We present a proof calculus for [Formula: see text] and show its completeness with regard to finite theories.