Logic for Rough Truth

Article type: Research Article

Authors: Banerjee, Mohua

Affiliations: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208 016, India. E-mail: mohua@iitk.ac.in

Abstract: Pawlak had proposed the notion of rough truth in 1987 [16]. The article takes a fresh look at this "soft" truth, and presents a formal system L_R, that is shown to be sound and complete with respect to a semantics determined by this notion. L_R is based on the modal logic S5. Notable is the rough consequence relation defining L_R (a first version introduced in [9]), and rough consistency (also introduced in [9]), used to prove the completeness result. The former is defined in order to be able to derive roughly true propositions from roughly true premisses in an information system. The motivation for the latter stems from the observation that a proposition and its negation may well be roughly true together. A characterization of L_R-consequence shows that the paraconsistent discussive logic J of Jaśkowski is equivalent to L_R. So, L_R, developed from a totally independent angle, viz. that of rough set theory, gives an alternative formulation to this well-studied logic. It is further observed that pre-rough logic [3] and 3-valued Łukasiewicz logic are embeddable into L_R.

Keywords: Rough sets, Modal logic S5, Consequence relation, Consistency

Journal: Fundamenta Informaticae, vol. 71, no. 2-3, pp. 139-151, 2006