Information Systems, Similarity Relations and Modal Logics

In the paper we study two types of information systems: ontological and logical. Systems of ontological type are Property systems and Attribute systems, where the information is represented in terms of the ontological concepts of object, property and attribute. Systems of logical type are Consequence systems and Bi-consequence systems where the information is represented by a collection of sentences, equipped with some inference mechanism. We prove that each Consequence system can be embedded into a certain Property system. The similar results hold for Bi-consequence systems and Attribute systems. These representation theorems are used to give an abstract characterization (by means of a finite set of first-order sentences) of some information relations in Property systems and Attribute systems, including various kinds of similarity relations. Several modal logics with modalities corresponding to some collections of information relations are introduced and their “query meaning” is discussed. One of the main results of the paper are the completeness theorems for the introduced modal logics with respect to their standard semantics.