Extensive Measurement When Concatenation Is Restricted and Maximal Elements May Exist1

Theories of fundamental measurement begin with purported qualitative laws about observable relations among certain entities, and using these laws a numerical representation is constructed which reflects their formal structure. The best known example, extensive measurement, arose from an analysis of such familiar physical measures as mass and length. In measuring mass, the classical theory supposes that we can ascertain which of two entities has the greater mass by comparing them, for example, on an equalarm pan balance in a vacuum-the one in the pan that drops having, by definition, the greater mass. If the comparison of x with y shows that x has more than or an amount equivalent to y of the attribute being measured, we write xRy. Furthermore, the theory assumes that new entities can be generated from old ones by placing two or more of the old ones on a single pan. The latter combining operation is called "concatenation," and the new entity generated by concatenating x with y is denoted by xoy. Various plausible and, to some degree, empirically true assumptions about o, R, and their interconnections are stated (for a summary of them, see Definition 2 and the discussion following it), from which it is shown that we can assign to each entity x a real number q ( x ) in such a way that the function p, has the following three properties. First, it is order preserving (monotonic),