Psychophysics without physics: extension of Fechnerian scaling from continuous to discrete and discrete-continuous stimulus spaces

The computation of subjective (Fechnerian) distances from discrimination probabilities involves cumulation of appropriately transformed psychometric increments along smooth arcs (in continuous stimulus spaces) or chains of stimuli (in discrete spaces). In a space where any two stimuli that are each other's points of subjective equality are given identical physical labels, psychometric increments are positive differences ψ(x,y)-ψ(x,x) and ψ(y,x)-ψ(x,x), where x≠y and ψ is the probability of judging two stimuli different. In continuous stimulus spaces the appropriate monotone transformation of these increments (called overall psychometric transformation) is determined uniquely in the vicinity of zero, and its extension to larger values of its argument is immaterial. In discrete stimulus spaces, however, Fechnerian distances critically depend on this extension. We show that if overall psychometric transformation is assumed (A) to be the same for a sufficiently rich class of discrete stimulus spaces, (B) to ensure the validity of the Second Main Theorem of Fechnerian Scaling in this class of spaces, and (C) to agree in the vicinity of zero with one of the possible transformations in continuous spaces, then this transformation can only be identity. This result is generalized to the broad class of “discrete-continuous” stimulus spaces, of which continuous and discrete spaces are proper subclasses.