Comparison of Rough-Set and Interval-Set Models for Uncertain Reasoning

In the rough-set model, a set is represented by a pair of ordinary sets called the lower and upper approximations. In the interval-set model, a pair of sets is referred to as the lower and upper bounds which define a family of sets. A significant difference between these models lies in the definition and interpretation of their extended set-theoretic operators. The operators in the rough-set model are not truth-functional, while the operators in the interval-set model are truth-functional. Within the framework of possible-worlds analysis, we show that the rough-set model corresponds to the modal logic system S5, while the interval-set model corresponds to Kleene’s three-valued logic system K3. It is argued that these two models extend set theory in the same manner as the logic systems S5 and K3 extend standard propositional logic. Their relationships to probabilistic reasoning are also examined.