On the bisection operation

Abstract

Given two perceived magnitudes of a single kind of sensory attribute, A and B, the operation of bisection is the selection of the perceived magnitude of this attribute, C, that differs from A as much as it differs from B (Plateau, 1872). With M A , M B , and M C denoting the measures of A, B, and C, respectively, Pfanzagl (1959) proposed thatwith w constant in the interval [0, 1], and showed that the bisection operation is compatible with this weighted arithmetic mean when it satisfies the following condition. Bisymmetry ConditionFor a given sensory attribute, let M 1 , M 2 , M 3 , and M 4 denote any measures of perceived magnitude sufficiently different from one another. Throughout this article, subscript numerals are used as labels. Bisections yield the measuresWith w unknown, Equations 2-5, 8, and 9 implyEquation 11 expresses the bisymmetry condition (Aczél, 1948). When w .5, Equations 2-10 implyThe empirical verification of all equalities in Equation 12 is believed to support the possibility that M C M A M B M C (w .5), making actual measurement possible (Adams & Fagot, 1975;Coombs, Dawes, & Tversky, 1970;Cross, 1964;Fagot & Stewart, 1970;Irtel, 2005;Pfanzagl, 1971;Raslear, Shurtleff, & Simmons, 1992). Gage's ConditionWhen the effects of the orders of presentation of A, B, and C are counterbalanced, Cross (1964, p. 5) showed that the bisymmetry condition is equivalent to the following condition proposed by Gage (1934a). Let M 1 and M 2 denote measures of two perceived magnitudes delimiting a sufficiently large sensory interval. Bisections yield the measuresEquations 13-16 imply On the bisection operation SERGIO CESARE MASIN AND ENRICO TOFFALINI University of Padua, Padua, ItalyFor two perceived sensory magnitudes, A and B, the bisection operation yields a perceived sensory magnitude, C, equally different from A and B. Available data do not resolve whether this operation is equivalent to a linear or a nonlinear mean. We tested these alternatives, using the evidence of functional measurement research that ratings are linear sensory measures. With A and B varied factorially, linear means predict that the curves relating rated C to rated A for each B are parallel straight lines, and nonlinear means predict that these curves are nonlinear and nonparallel. Using brightness and perceived size as sensory attributes, the present experiments confirm the first of these predictions, indicating that the bisection operation is equivalent to a linear mean.

Information integration study of Metelli's and Morinaga's theories of achromatic transparency

“…Studies in information integration theory have provided large evidence that ratings of perceived magnitude are linearly related to perceived magnitude when floor and ceiling effects are removed, typically by using two anchor stimuli: one larger than the largest experimental stimulus and one smaller that the smallest experimental stimulus (Anderson, 1996, pp. 94–98; Curtis & Fox, 1969; Masin & Toffalini, 2009; Weiss, 1972). According to this evidence the rated transparency of the disk is…”