Comparing Dependencies in Probability Theory and General Rough Sets: Part-A
This work addresses foundational issues in mathematical logic and set theory, but it appears incremental as it builds on the author's prior research.
The paper tackles the problem of comparing dependence concepts between general rough sets and probability theory, demonstrating both positive and negative results that clarify the limitations of such comparisons in a logic-based framework.
The problem of comparing concepts of dependence in general rough sets with those in probability theory had been initiated by the present author in some of her recent papers. This problem relates to the identification of the limitations of translating between the methodologies and possibilities in the identification of concepts. Comparison of ideas of dependence in the approaches had been attempted from a set-valuation based minimalist perspective by the present author. The deviant probability framework has been the result of such an approach. Other Bayesian reasoning perspectives (involving numeric valuations) and frequentist approaches are also known. In this research, duality results are adapted to demonstrate the possibility of improved comparisons across implications between ontologically distinct concepts in a common logic-based framework by the present author. Both positive and negative results are proved that delimit possible comparisons in a clearer way by her.