Possibility and Necessity Evaluations based on Ordinal Comparability

Subjective uncertainty is one of the most essential subjects for evaluation that was the reason Fuzzy theory was proposed. Related research areas are widely spread such as decision making, data analysis, information retrieval, psychology, and human computer interaction and so on. As a basis for various researches mathematical interpretation of evaluation methods occupies an important position. In Possibility theory, possibility and necessity evaluations have been utilized as ways for evaluating a vague achievement in an ambiguous situation. However, mathematical interpretations for these evaluations have been insufficient. In this article, possibility and necessity evaluations are examined based on the ordinal comparability in social choice theory. Equivalent conditions to the evaluations are deduced so that axiomatic interpretations for the evaluations become possible, and order relations related to the two evaluations are deduced. Keywords— evaluation, measure, necessity, ordinal comparability, ordinal utility, possibility.