# A Logic for Reasoning about Similarity

• Beata Konikowska
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 13)

## Abstract

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.

## Keywords

Normal Form Similarity Relation Proof System Deduction System Terminal Sequence

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1998

## Authors and Affiliations

• Beata Konikowska
• 1
1. 1.Institute of Computer SciencePolish Academy of SciencesWarsaw, Ordona 21Poland