Psychophysics without physics: a purely psychological theory of Fechnerian scaling in continuous stimulus spaces

The theory of Fechnerian scaling, as developed by the present authors, uses “same-different” discrimination probabilities defined on a stimulus set to derive from them a measure of local discriminability (of each stimulus from its neighbors), and by cumulating this measure along special paths in the stimulus space it obtains subjective (Fechnerian) distances among stimuli. Previously the theory has been developed for two kinds of stimulus spaces: (A) “continuous spaces”, that were understood as connected regions of Euclidean space (such as the amplitude-frequency space of tones, or the CIE color triangle), and (B) discrete stimulus spaces (such as alphabets or words). In the former case the theory is psycho-physical rather than purely psychological, in the sense that the resulting subjective distances are based not only on discrimination probabilities but also on certain properties provided by physical measurements of stimuli. Thus, the two-dimensionality of the amplitude-frequency space of tones, its vectorial structure, and its Euclidean topology are all physical properties, and Fechnerian computations make use of them. This is an unsatisfactory situation, as the definition of a subjective distance between two stimuli should not critically depend on how these stimuli are measured by physicists. The theory of Fechnerian scaling for discrete stimulus spaces is, in contrast, purely psychological: the discreteness of a stimulus space and all Fechnerian computations can be defined there entirely in terms of discrimination probabilities. In the present work we show how to construct Fechnerian scaling as a purely psychological theory for arcwise connected (intuitively, “continuous”) spaces of arbitrary nature, including spaces with infinite-dimensional or nondimensional physical descriptions (such as spaces of pictures or motions). As in the Euclidean special case, this general theory of Fechnerian scaling is based on the defining property of discrimination, called Regular Minimality, and on the idea of regular variation of psychometric differentials, with all previously derived main theorems of Fechnerian scaling remaining valid.