Measurement: The Theory of Numerical Assignments

In this article we review some generalizations of classical theories of measurement for concatenation (e.g., mass or length) and conjoint structures (e.g., momentum of mass-velocity pairs or loudness of intensity-frequency pairs). The earlier results on additive representations are briefly surveyed. Generalizations to nonadditive structures are outlined, and their more complex uniqueness results are described. The latter leads to a definition of scale type in terms of the symmetries (automorphisms) of the underlying qualitative structure. The major result is that for any measurement onto the real numbers, only three possible scale types exist that are both rich in symmetries but not too redundant: ratio, interval, and another lying between them. The possible numerical representations for concatenation structures corresponding to these scale types are completely described. The interval scale case leads to a generalization of subjective expected-utility theory that copes with some empirical violations of the classical theory. Partial attempts to axiomatize concatenation structures of these three scale types are described. Such structures are of interest because they make clear that there is a rich class of nonadditive concatenation and conjoint structures with representations of the same scale types as those used in physics.