Three-way decisions with evaluative linguistic expressions

The theory of three-way decisions (3WD) requires dividing a finite, non-empty universe into three disjoint sets called positive, negative, and boundary regions. Three types of decisions are then made on the objects in each region: acceptance, rejection, and abstention (or non-commitment), respectively. Until today, a large number of 3WD extensions and applications have been proposed; some of the most recent ones also include aspects of linguistics. In this article, we first propose an innovative linguistic interpretation of three-way decisions, where the positive, negative, and boundary regions are constructed by means of the so-called evaluative linguistic expressions. These are expressions of natural language, such as small, medium, very short, quite roughly strong, extremely good, etc., and they are described within a logical theory based on the formal system of higher-order fuzzy logic. Furthermore, in line with our linguistic 3WD approach, we introduce the novel notion of linguistic rough sets, thus contributing to the development of Rough Set Theory. Finally, we connect the theory of linguistic three-way decisions with the standard 3WD model based on probabilistic rough sets, establishing conditions under which the two approaches coincide. Our results highlight connections between two different research areas: three-way decisions and the theory of evaluative linguistic expressions.