On Some Issues in the Foundation of Rough Sets: the Problem of Definition

In this presentation we shall raise the following issues connected with the discourse of rough set theory: 1. indistinguishability 2. definition of a rough set 3. language dependency, the problem of referent 4. vagueness 5. abstraction 6. application-friendliness We shall argue that existing practices prevailing in rough set research fall short in addressing most of the issues. After almost fifty years of rough-set studies it may be interesting, if not a necessity, to look into some basic aspects of its foundation. Indistinguishability At the base level of all kinds of rough sets lies an indiscernibility relation (ind) which is at least reflexive and symmetric. This relation gives rise to the granulation G of the universe. That is, clustering of the objects: G a , the granule corresponding to the object a is G a = {x : a ind x}. This is true also for covering-based rough sets since from a covering C at first the granules viz. One can consider elements belonging to the same granule as mutually indiscernible. While defining lower and upper approximations the granules are used differently. In fact any covering is itself a granulation. With the help of this, working granules N(a), N a , P a etc. are formed. This indiscernibility is at the philosophical root of rough set theory. We talk about one individual but think about every one in the granule which it belongs to. This duality prevails althrough. Some objects are indistinguishable though not identical-like quantam objects (Logical and Philosophical Remarks on Quasi-Set Theory, Da Costa, Krause). Also this indistinguishability lies at the root of the Sorites type characteristic feature of vagueness, about which we shall discuss also. With the help of the basic granules, approximations of a concept or set, a second-order entity is envisaged, and this is the main entry point to the application (or use) of granulation. A set 'A' is approximated by a pair G(A) and G(A) where usually G(A) ⊆ A ⊆ G(A), but not always. Then a second layer of indiscernibility arises, that of the concepts. Two concepts A and B are indistinguishable if and only if they admit the same approximations with respect to a given granulation. At this level also a dichotomy prevails; concepts A and B are not the same, yet there is no means to distinguish them. Again, a name has different referents. In the conceptual framework of rough set theory …