Title: Dialectical Rough Sets, Parthood and Figures of Opposition-1,
Abstract: 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.