The measurement of membership by comparisons

Suppose we want to use a particular algebraic operation (for example, the drastic product t-norm) for representing an operation on fuzzy sets (for example, the intersection). This algebraic operation will be used with membership degrees, i.e. most of the time, numbers between 0 and 1. If we want that the results of our calculations make sense, we must be sure that the algebraic operation is compatible with the nature of the membership degrees, which is determined by the technique used to measure them. But this technique must, in turn, be compatible with the structural properties of the knowledge we want to represent. This paper addresses these issues within the framework of measurement theory. We provide sound theoretical foundations for the measurement of membership on ordinal and interval scales. But we also show that the level of measurement (ordinal, interval or even ratio) is not critical for the choice of a particular algebraic operation. Within measurement theory, whatever the scale, only the max and the min can be used for representing the intersection and the union. We also present some results about the complementation.