Dominance-based rough set approach as a proper way of handling graduality in rough set theory

Referring to some ideas of Leibniz, Frege, Boole and Łukasie-wicz, we represent fundamental concepts of rough set theory in terms of a generalization that permits to deal with the graduality of fuzzy sets. Our conjunction of rough sets and fuzzy sets is made using the Dominance-based Rough Set Approach (DRSA). DRSA have been proposed to take into account ordinal properties of data related to preferences. We show that DRSA is also relevant in case where preferences are not considered but a kind of monotonicity relating attribute values is meaningful for the analysis of data at hand. In general, monotonicity concerns relationship between different aspects of a phenomenon described by data, e.g.: "the larger the house, the higher its price" or "the more a tomato is red, the more it is ripe". The qualifiers, like "large house", "high price", "red" and "ripe", may be expressed either in terms of some measurement units, or in terms of degrees of membership to some fuzzy sets. In this perspective, the DRSA gives a very general framework in which the classical rough set approach based on indiscernibility relation can be considered as a particular case.