Further extensions of a theorem of dimensional analysis

Abstract

This paper extends the Luce extension of the dimensional analysis theorem to cover ordinal and log interval scales. It shows (1) that no complex of dimensionally distinct interval or log interval scales maps into an ordinal scale; (2) that a complex of dimensionally distinct ratio scales maps into a log interval scale only by the power function f( x 1, x 2,…) = K Π i=1 n x i a i , where K > 0 and a i ≠ 0 are arbitrary, and the function A exp f( x 1, x 2,…), where A > 0 is arbitrary; and (3) a complex of ratio scales maps into an ordinal scale by any increasing monotonic function of f( x 1, x 2,…), and by no other function. Adjoined to the Luce results, these complete the analysis of all combinations of “independent” and “dependent” scales of the ratio, interval, log interval, and ordinal type.