Generalized Quantifiers in the Context of Rough Set Semantics

Abstract

Looking back to Prof. Zadeh’s paradigm of Computing with Words (CWW) [28, 29, 30], one can notice that the initial attempt of such an endeavour was to set up a basic vocabulary of linguistic words, and fix their semantics based on fuzzy sets. Then a grammar was proposed to generate compound linguistic expressions based on the primitive ones, and simultaneously based on the semantic interpretations of those basic linguistic expressions a general scheme for the semantics of the rest of linguistic expressions were proposed. Sentences involving linguistic quantifiers and vague predicates constitute a fragment of natural language. In this paper, we choose this fragment of the natural language, and explore the semantics from the perspective of rough sets [13, 14, 16, 17, 18, 21]. We fix a set of basic crisp quantifiers, mainly of proportional kind. A set of vague quantifiers are proposed to lie in a close vicinity of those crisp quantifiers in the sense that a particular vague quantifier can be visualized as a blurred, may be called rough, image of a set of crisp quantifiers. Address for correspondence: Institute of Mathematics, University of Warsaw, 02 Banacha, 02-097 Warsaw, Poland ∗This work has been carried out during the ERCIM Alain Benseossan fellowship. †This work was partially supported by the Polish National Science Centre (NCN) grants DEC-2011/01/D/ST6/06981, DEC2012/05/B/ST6/03215, DEC-2013/09/B/ST6/01568 as well as by the Polish National Centre for Research and Development (NCBiR) under the grant O ROB/0010/03/001. 214 S. Dutta and A. Skowron / Generalized Quantifiers with Rough Set Semantics Semantics of the rest of the vague quantifiers can be obtained based on the subjective perception of the interrelations among the (vague) quantifiers.