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.

中文翻译：

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准度量和度量空间中的近似蕴涵逻辑。


众所周知，准度量空间可以用度量空间来表示；前一个空间的点成为后一个空间的闭子集，并且准度量的作用由豪斯多夫准距离承担。在本文中，我们证明，在稍微特殊的背景下，该表示定理的尖锐版本成立。也就是说，我们假设一个准度量来满足威尔逊原始意义上的可分离性。然后，任何准度量空间都可以通过度量空间来表示，使得不同的点被分配不相交的封闭子集。该结果是针对近似推理领域的开放问题的解决方案而定制的。遵循 E. Ruspini 的著作，近似蕴涵逻辑（[公式：参见文本]）基于经典蕴涵关系的分级版本。我们提出了[公式：见文本]的证明演算，并证明了其关于有限理论的完整性。