EVALUATION SETS AND MAPPINGS: THE ORDER-THEORETIC ASPECT OF THE MEANING OF PROPERTIES

In this contribution, we study some aspects of the mathematical representation of the solutions of problems encountered in everyday life, called evaluation problems. These problems consist in having to check whether objects in a given universe satisfy one or more properties. First, evaluation problems under a single property are considered. It is argued that (partial-pre)order relations play a central role in this study. Three equivalent ways of representing the order-theoretic aspect of these solutions are presented, the last of which bears a close resemblance to the representations extant in the literature concerning fuzzy set theory. This resemblance is investigated in several examples. After this basic work, the more complicated study of evaluation problems under more than one property is undertaken. It is argued that this approach is necessary in order to investigate the relations between properties and of course the combinations of properties that lead to such logical operations as not, and, or, . . . The approach followed for a single property is generalized and the same order-theoretic methods are used to represent the solutions of problems of this kind. Using this approach, the link with fuzzy sets and their set theoretical operations is made through the study of property combinators, truth-functionality and combination functions. This link is studied in several examples.