Ratios and differences in perceptual comparison: A reexamination of Torgerson’s conjecture

Abstract How do we compare stimuli that vary in magnitude? According to a well-known conjecture by Torgerson (1961), observers perceive only a single relation between stimuli, that is either a ratio or difference, but which one cannot be determined empirically. Previous research has used direct scaling procedures in which observers have judged ratios and differences numerically, but with mixed results. We used a novel behavioral task in which observers learned to produce non-symbolic ratios and differences by feedback and without explicit instruction. In two sets of experiments, observers viewed pairs of stimuli that varied in brightness, number of dots, or circle areas, and responded by clicking along a bar. Feedback was provided based on either the stimulus ratios or differences and the response location. Observers produced ratios and differences accurately, with average individual correlations of r = . 94 and .95 across experiments, respectively. Regressions showed that responding was controlled jointly by ratios and differences, with the untrained relation predicting significant variance in approximately half of individual cases. Results were further supported by rank-order analyses, non-metric scaling analyses, and Monte Carlo simulations. Although Torgerson’s conjecture was originally proposed in explicit tasks in which observers were required to use their mathematical knowledge, our results show that the perceptual system automatically computes both differences and ratios when comparing stimuli in a non-symbolic, implicit learning task. Thus Torgerson’s conjecture, as applied to our task, is fundamentally false. Evidence for two operations rather than one suggests that the perceptual system might represent elements of an algebraic field, which could support adaptive behavior such as spatial navigation that appears to be computationally complex.