Coarse Set Theory: A Mathematical Foundation for Coarse Ethics

In ethical decision-making, individuals are often evaluated based on generalized assessments rather than precise individual performance. This concept, known as Coarse Ethics (CE), has primarily been discussed in natural language without a formal mathematical foundation. This paper introduces Coarse Set Theory (CST) to establish a mathematical framework for CE. We define coarse sets using totally ordered sets and propose axioms that characterize the hierarchical relationships between elements and their groupings. Additionally, we introduce coarse-grained sets, which partition an underlying set into equivalence classes based on predefined criteria. We extend this framework by defining coarse mappings, which transform detailed individual data into coarser representations while maintaining essential structural properties. To measure the information loss, we employ Kullback-Leibler (KL) divergence, demonstrating how different coarse partitions affect the preservation of information. We illustrate how CST can be applied to real-world grading systems through theoretical formulations and empirical analysis. This study provides a rigorous foundation for CE, enabling a more systematic exploration of fairness, interpretability, and decision-making trade-offs.