Qualitative fuzzy system structures

The authors attempt to establish a measurement-theoretic foundation for fuzzy sets. A qualitative relational system called the fuzzy system structure is introduced. This forms the basis for examining the ordering and properties of a fuzzy system. More specifically, an empirical relation is used to characterize the relationship between two objects with respect to a fuzzy set. Another empirical relation is used to characterize the relationship between two fuzzy sets with respect to an object x. These two primitive relations together characterize a qualitative fuzzy system, by which the necessary and sufficient conditions that justify the use of the min-max numerical system are identified