Rough Set Rudiments by Zdzislaw Pawlak and Andrzej Skowron.
From the basic philosophy section:
The rough set philosophy is founded on the assumption that with every object of the universe of discourse we associate some information (data, knowledge). For example, if objects are patients suffering from a certain disease, symptoms of the disease form information about patients. Objects characterized by the same information are indiscernible (similar) in view of the available information about them. The indiscernibility relation generated in this way is the mathematical basis for rough set theory.
Any set of all indiscernible (similar) objects is called an elementary set, and forms a basic granule (atom) of knowledge about the universe. Any union of some elementary sets is referred to as crisp (precise) set – otherwise the set is rough (imprecise, vague).
Consequently each rough set has boundary-line cases, i.e., objects which cannot be with certainty classified neither as members of the set nor of its complement. Obviously crisp sets have no boundary-line elements at all. That means that boundary-line cases cannot be properly classified by employing the available knowledge.
Thus, the assumption that objects can be “seen” only through the information available about them leads to the view that knowledge has granular structure. Due to the granularity of knowledge some objects of interest cannot be discerned and appear as the same (or similar). As a consequence vague concepts, in contrast to precise concepts, cannot be characterized in terms of information about their elements. Therefore, in the proposed approach, we assume that any vague concept is replaced by a pair of precise concepts – called the lower and the upper approximation of the vague concept. The lower approximation consists of all the objects which surely belong to the concept and the upper approximation contains all objects which possibly belong to the concept. Obviously, the difference between the upper and lower approximation constitutes the boundary region of the vague concept. Approximations are two basic operations in rough set theory.
Suspect that the normal case is “rough” sets, with “crisp” sets being an artifice of our views of the world.
This summary is a bit dated so I will use it as a basis for an update with citations to later materials.