Triadic judgment models and Weber's law

Let a, b, c, with a {greater than or slanted equal to} b {greater than or slanted equal to} c, be positive real numbers indicating the intensities of physical stimuli in a psychophysical experiment; let P_abc be the probability that b is judged to be more similar to a ("closer to") a than to c. This paper investigates the following representation and its subcases for triadic judgments{A formula is presented}where u is a real-valued, strictly increasing, continuous function and F is continuous, strictly decreasing in the first variable, and strictly increasing in the second variable. In addition to elucidating the connections between the representation and models of discrimination and bisection paradigms, this paper examines the mathematical consequences of Weber's law on subcases of the representation, demonstrating that the resulting analytic forms of these triadic models are very limited. The results constitute partial solutions to questions raised in Falmagne [(1985). Elements of psychophysical theory. New York: Oxford University Press] concerning the impact of Weber's law on probabilistic measurement models for triadic judgments.