Dialectical Rough Sets, Parthood and Figures of Opposition

In one perspective, the central problem pursued in this research is that of the inverse problem in the context of general rough sets. The problem is about the existence of rough basis for given approximations in a context. Granular operator spaces were recently introduced by the present author as an optimal framework for anti-chain based algebraic semantics of general rough sets and the inverse problem. In the framework, various subtypes of crisp and non crisp objects are identifiable that may be missed in more restrictive formalism. This is also because in the latter cases the concept of complementation and negation are taken for granted. This opens the door for a general approach to dialectical rough sets building on previous work of the present author and figures of opposition. In this paper dialectical rough logics are developed from a semantic perspective, concept of dialectical predicates is formalized, connection with dialethias and glutty negation established, parthood analyzed and studied from the point of view of classical and dialectical figures of opposition. Potential semantics through dialectical counting based on these figures are proposed building on earlier work by the present author. Her methods become more geometrical and encompass parthood as a primary relation (as opposed to roughly equivalent objects) for algebraic semantics. Dialectical counting strategies over antichains (a specific form of dialectical structure) for semantics are also proposed. [This paper is scheduled to appear as two separate papers (because of length) with some overlap and enhancements.]