Dialectical Rough Sets, Parthood and Figures of Opposition-1
This work addresses foundational issues in rough set theory for researchers in formal logic and algebraic semantics, though it appears incremental as it builds on the author's previous frameworks.
The paper tackles the inverse problem in general rough sets by developing dialectical rough logics and formalizing dialectical predicates, establishing connections with dialetheias and glutty negation, and analyzing parthood from the perspective of classical and dialectical figures of opposition.
In one perspective, the main theme of this research revolves around the inverse problem in the context of general rough sets that concerns the existence of rough basis for given approximations in a context. Granular operator spaces and variants 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 sub-types of crisp and non-crisp objects are identifiable that may be missed in more restrictive formalism. This is also because in the latter cases concepts of complementation and negation are taken for granted - while in reality they have a complicated dialectical basis. This motivates 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 invented from a semantic perspective, a concept of dialectical predicates is formalised, connection with dialetheias and glutty negation are established, parthood analyzed and studied from the viewpoint of classical and dialectical figures of opposition 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.