A Note on Definability and Approximations

Definability and approximations are two important notions of the theory of rough sets. In many studies, one is used to define the other. There is a lack of an explicit interpretation of the physical meaning of definability. In this paper, the definability is used as a more primitive notion, interpreted in terms of formulas of a logic language. A set is definable if there is a formula that defines the set, i.e., the set consists of all those elements satisfying the formula. As a derived notion, the lower and upper approximations of a set are two definable sets that approximate the set from below and above, respectively. This formulation may be more natural, bringing new insights into our understanding of rough set approximations.