On Subjective Intensity and Its Measurement

Threshold functions and subjective estimations of ratios and differences are a few examples from a class of psychological functions that can be conceptualized as functions from a set of stimuli onto itself. This paper analyzes axiomatically the structural properties of a broad subset of such functions, with an emphasis on general properties of their psychological processing. Measurement-theoretic representation and uniqueness results are derived for Weber's Law, ratio magnitude estimation, and a version of Luce's Possible Psychophysical Laws. An explanation is also provided for the experimental findings of W. Torgerson and others that indicates a qualitative identity between subjective estimation of ratios and subjective estimation of differences. In many paradigms in psychophysics and other areas of psychology, a subject's behavior can often be idealized as sets of functions from a set of stimuli onto itself. Two examples from psychophysics are ( i ) threshold functions Fp on X , where 1 > p > .5 and for each x in X , Fp(x) is the stimulus such that if y has more of the physical attribute than Fp(x), then the proportion of time y is judged subjectively more intense than x is > p; and (ii) ratio estimation functions G, on X such that for each x and y in X, y = G,(x) if and only if the subject judges y as being r times as intense as x. Functions from X onto itself that result from subjects' responses to instructions are called behavioral, and this chapter presents an axiomatic treatment of behavioral functions that result from responses involving a subject's evaluation of subjective intensities of stimuli. One main result is a characterization of situations where for a nonempty set B of behavioral functions there exists a mapping of the stimuli into the positive reals such that each element of B is represented as a nlliltiplication by a positive real. This result is used to characterize axiomatically a generalization of the psychophysical power law. Another result provides a new perspective for Luce's (1959b) seminal research on possible psychophysical laws. As a motivation for the kind of modeling and theory developed in this chapter, consider the following important and puzzling empirical finding of Torgerson (1961). K e y words and phrases. Psychophysics, subjective intensity, ratio estimation, difference estimation, Weber's law, power law, psychophysical law. Acknowledgments. Research for this chapter was supported by the NSF grant SBR-9520107 to the Department of Cognitive Sciences, University of California, Irvine. Address for c ~ r r e ~ p ~ n d e n c e : Louis Narens, Department of Cognitive Sciences, University of California Irvine, lrvine, CA 92697. Email: [email protected]