A Logic for Reasoning about Similarity
Beata Konikowska · Studies in fuzziness and soft computing · 1998
A similarity relation is a reflexive and symmetric, but in general not transitive binary relation between objects. Similarity can be regarded as a relative notion parametrised by the set of classification attributes used as a basis for determining similarity or dissimilairty of objects. In the paper we present a polymodal formal language for reasoning about such a relative notion of similarity. For each subset of a given set of attributes, we have two modalities, corresponding semantically to so-called upper and lower approximations of a set of objects with respect to that set of attributes; intuitively, the latter approximations could be described as the interior and completion of a set of objects with respect to the similarity relation generated by the considered set of attributes, respectively. Formulae of the language evaluate to sets of objects, and a formula is said to be true if it evaluates to the whole universe of the model. The language is given a sound and complete deduction system in Rasiowa-Sikorski style: it consists of fundamental sequences of formulae which represent axioms of the system, and decomposition rules for sequences of formulae which represent inference rules. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.