A “Fundamental” Axiomatization of Multiplicative Power Relations Among Three Variables

Abstract

Suppose that entities composed of two independent components are qualitatively ordered by a relation that satisfies the axioms of conjoint measurement. Suppose, in addition, that each component has a concatenation operation that, together either with the ordering induced on the component by the conjoint ordering or with its converse, satisfies the axioms of extensive measurement. Without further assumptions, nothing can be said about the relation between the numerical scales constructed from the two measurement theories except that they are strictly monotonic. An axiom is stated that relates the two types of measurement theories, seems to cover all cases of interest in physics, and is sufficient to establish that (the multiplicative form of) the conjoint measurement scales are power functions of the extensive measurement scales.