Dominance-based rough set approach to reasoning about ordinal data

Dominance-based Rough Set Approach (DRSA) has been proposed by the authors to handle background knowledge about ordinal evaluations of objects from a universe, and about monotonic relationships between these evaluations, e.g. "the larger the mass and the smaller the distance, the larger the gravity" or "the greater the debt of a firm, the greater its risk of failure". Such a knowledge is typical for data describing various phenomena, and for data concerning multiple criteria decision making or decision under uncertainty. It appears that the Indiscernibility-based Rough Set Approach (IRSA) proposed by Pawlak involves a primitive idea of monotonicity related to a scale with only two values: "presence" and "absence" of a property. This is why IRSA can be considered as a particular case of DRSA. Monotonicity gains importance when the binary scale, including only "presence" and "absence" of a property, becomes finer and permits to express the presence of a property to certain degree. This observation leads to very natural fuzzy generalization of the rough set concept via DRSA. It exploits only ordinal properties of membership degrees and monotonic relationships between them, without using any fuzzy connective. We show, moreover, that this generalization is a natural continuation of the ideas given by Leibniz, Frege, Boole, Łukasiewicz and Pawlak. Finally, the fuzzy rough approximations taking into account monotonic relationships between memberships to different sets can be applied to case-based reasoning. In this perspective, we propose to consider monotonicity of the type: "the more similar is y to x, the more credible is that y belongs to the same set as x".