This project will investigate the fundamental laws governing perceptual discrimination and will further develop the theory of Generalized Fechnerian Scaling. The term discrimination refers to the ability to decide whether two objects are the same or different (in all respects, in a specified respect, or in the sense of belonging to one and the same category). This is one of the basic cognitive abilities of living organisms and one of the basic requirements of artificial intelligent systems. The principle of Regular Minimality has been proposed as a fundamental property of same-different judgments. It says, in essence, that stimulus Y is least discriminable from stimulus X if and only if X is least discriminable from Y. This intuitively plausible and empirically corroborated principle is the cornerstone of the theory of Generalized Fechnerian Scaling, in which subjective distances among stimuli (i.e., distances "from the point of view" of a perceiver, be it a human observer, group of people, a technical system, or a "paper-and-pencil" computational procedure) are computed from discrimination probabilities. Until recently, however, the logic of this computation was posited to be different for continuous stimulus spaces, such as a space of colors, and for discrete spaces, such as a space of color names. One aim of the project is to remedy this discrepancy by developing a computational logic which would apply to all possible stimulus spaces ("Universal Fechnerian Scaling"). This aim is achieved through a new mathematical concept, Dissimilarity Function, whose axiomatic theory generalizes that of metric spaces. In addition to Regular Minimality, discrimination probabilities typically exhibit another property, called Nonconstant Self-Dissimilarity. If X is least discriminable from Y, and so is A from B, the discriminability of X from Y is not generally the same as that of A from B. The conjunction of Regular Minimality with Nonconstant Self-Dissimilarity has important consequences for understanding same-different comparisons. Thus, this conjunction is incompatible with the hypothesis that the subjective distance between X and Y is monotonically related to the probability with which X is discriminated from Y (the hypothesis underlying, among other things, the widely used techniques of Multidimensional Scaling, when applied to discrimination probabilities). Regular Minimality and Nonconstant Self-Dissimilarity are also shown to be incompatible with the widely used well-behaved Thurstonian-type models, according to which the decision whether X and Y are different or the same is based on random images of X and Y, with each stimulus affecting its image's distribution "sufficiently smoothly." The Thurstonian-type models can, however, approximate Regular Minimality, which makes it critical to determine the limits of precision with which Regular Minimality holds in empirical data. Available data do not answer this question, and this project aims at investigating it by means of specially designed adjustment/matching procedures. A related aim of the project is to investigate, analytically and by means of computer simulations, whether the conventional variants of Thurstonian-type models which can approximate Regular Minimality within limits of experimental error can generate realistic discrimination probability functions.
Generalized Fechnerian Scaling and theory of same-different judgments have applications in educational assessment and professional training, in applied statistics, in the construction of artificial cognitive systems and perceptual aids, and in the analysis of large-scale polls of public opinion and consumer surveys. The project is interdisciplinary in its nature, bridging behavioral, social, mathematical, and computer sciences. The project is also international. The investigators are from different countries, and other collaborations will be bolstered by a series of symposia and workshops organized at international conferences. Topics related to this project will be used in courses taught by the investigators at Purdue and Oldenburg Universities. An effort will be made to attract undergraduate students to participate in the project, with a special emphasis on the involvement of women and minorities.
