Symmetry, direct measurement, and Torgerson's conjecture

Two forms of direct measurement are considered in the article: a strong form in which ratio productions named by number words are interpreted veridically as the numerical ratios they name; and a weak form in which the ratio productions named by number words may have interpretations as ratios that are different from numerical ratios they name. Both forms assume that the responses to instructions to produce ratios are represented numerically by ratios, and thus the word “ratio”—and supposedly the participants concept associated with it—is being “directly” represented. The strong form additionally “directly represents” the number mentioned in the instruction by itself. The article provides an axiomatic theory for the numerical representations produced by both forms. This theory eliminates the need for assuming anything is being “directly represented,” allowing for a purely behavioral approach to ratio production data. It isolates two critical axioms for empirical testing. An measurement-theoretic explanation is provided for the puzzling empirical phenomenon that subjects do not distinguish between ratios and differences in a variety of direct measurement tasks.