Intensions and extensions of granules: A two-component treatment

The concept of a granule (of knowledge) originated from Zadeh, where granules appeared as references to words (phrases) of a natural (or an artificial) language. According to Zadeh's program, “a granule is a collection of objects drawn together by similarity or functionality and considered therefore as a whole”. Pawlak's original theory of rough sets and its different generalizations have a common property: all systems rely on a given background knowledge represented by the system of base sets. Since the members of a base set have to be treated similarly, base sets can be considered as granules. The background knowledge has a conceptual structure, and it contains information that does not appear on the level of base granules, so such information cannot be taken into consideration in approximations. A new problem arises: is there any possibility of constructing a system modeling the background knowledge better? A two-component treatment can be a solution to this problem. After giving the formal language of granules involving the tools for approximations, a logical calculus containing approximation operators is introduced. Then, a two-component semantics (treating intensions and extensions of granule expressions) is defined. The authors show the connection between the logical calculus and the two-component semantics.