Abstract
Rough set theory, initiated by Pawlak, is a mathematical tool in dealing with inexact and incomplete information. Numerical characterizations of rough sets such as accuracy measure, roughness measure, etc. which aim to quantify the imprecision of a rough set caused by its boundary region, have been extensively studied in the literature. However, very few of them are explored from the viewpoint of rough logic, which, however, will help to establish a kind of graded reasoning mechanism under the framework of rough logic. For this purpose, we introduce a kind of numerical approach to the study of rough logic in this paper. More precisely, we propose the notions of accuracy degree and roughness degree for each formula in rough logic, with the aim of measuring the extent to which any formula is accurate (or definable) and rough, respectively. Then, we investigate their properties in detail. One of the obtained results is that the sets of accuracy degree and roughness degree coincide with the set of rational numbers in [0, 1], and hence they are both dense in [0, 1].
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