Rough Sets and Vague Sets Zbigniew Bonikowski 1 and Urszula Wybraniec-Skardowska 2 1 Institute of Mathematics and Informatics, University of Opole, Oleska 48, 45-052 Opole, Poland zbonik@math.uni.opole.pl 2 Autonomous Section of Applied Logic, Pozna´ n School of Banking, Department in Chorz´ow, Poland uws@uni.opole.pl Abstract. The subject-matter of the consideration touches the prob- lem of vagueness. The notion of the rough set, originated by Zdzislaw Pawlak, was constructed under the influence of vague information and methods of shaping systems of notions leading to conceptualization and representation of vague knowledge, so also systems of their scopes as some vague sets. This paper outlines some direction of searching for a solution to this problem. In the paper, in connection to the notion of the rough set, the notion of a vague set is introduced. Some operations on these sets and their properties are discussed. The considerations intend to take into account a classical approach to reasoning, based on vague premises, and suggest finding a logic of vague sentences as a non-classical logic in which all counterparts of tautologies of classical logic are laws. 1 Introduction Logicians and philosophers have been interested in the problem area of vague knowledge for a long time, looking for some logical bases of a theory of vague notions (terms) constituting such knowledge. Recently it has become the sub- ject of investigations of computer scientists interested in the problems of AI, in particular, in problems of reasoning on the basis of incomplete or vague infor- mation and applications of computers to support and represent such reasoning in the computer memory. Significant results obtained by computer scientists in the scope of imprecision and vagueness: the Zadeh’s fuzzy set theory [20], the Shafer’s theory of evidence [17] and the Pawlak’s rough sets theory [14] greatly contributed to actualization and intensification of research into vagueness. The present paper proposes a new approach to vagueness and considers the problem of denotations of vague notions (terms) from the logical and computer sciences perspective. It yields logical foundations to a theory of vague notions (terms) and should be an essential contribution to that problem. The paper consists of four sections. In Section 2, we introduce the notion of unit information (unit knowledge) and vague information (vague knowledge). The main notion of the vague set, inspired by the Pawlak’s notion of a rough set is defined in Section 3. In Section 4 some operations on vague sets and their M. Kryszkiewicz et al. (Eds.): RSEISP 2007, LNAI 4585, pp. 122–132, 2007. c  Springer-Verlag Berlin Heidelberg 2007