Coarse Set Theory for AI Ethics and Decision-Making: A Mathematical Framework for Granular Evaluations

As artificial intelligence (AI) systems become increasingly embedded in ethically sensitive domains such as education, healthcare, and transportation, the need to balance accuracy and interpretability in decision-making has become a central concern. Coarse Ethics (CE) is a theoretical framework that justifies coarse-grained evaluations, such as letter grades or warning labels, as ethically appropriate under cognitive and contextual constraints. However, CE has lacked mathematical formalization. This paper introduces Coarse Set Theory (CST), a novel mathematical framework that models coarse-grained decision-making using totally ordered structures and coarse partitions. CST defines hierarchical relations among sets and uses information-theoretic tools, such as Kullback-Leibler Divergence, to quantify the trade-off between simplification and information loss. We demonstrate CST through applications in educational grading and explainable AI (XAI), showing how it enables more transparent and context-sensitive evaluations. By grounding coarse evaluations in set theory and probabilistic reasoning, CST contributes to the ethical design of interpretable AI systems. This work bridges formal methods and human-centered ethics, offering a principled approach to balancing comprehensibility, fairness, and informational integrity in AI-driven decisions.