A "fundamental" Axiomatization of Multiplicative Power Relations among Three Variables*

Suppose that entities composed o f two independent components are qualitatively ordered by a relation that satisfies the axioms o f conjoint measurement. Suppose, i n 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 o f 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. A n axiom is stated that relates the two types o f measurement theories, seems to cover all cases o f interest in physics, and is sufficient to establish that (the multiplicative form o f ) the conjoint measurement scales are power functions o f the extensive measurement scale^.^