## Mathematical foundations of Universal Fechnerian Scaling

The main idea of Fechner’s original theory (Fechner, 1860, 1877, 1887) can be described as follows (see Figure 9.1). If stimuli are represented by real numbers (measuring stimulus intensities, or their spatial or temporal extents), the subjective distance from a stimulus a to a stimulus b > a is computed by cumulating from a to b, through all intermediate values, a measure of dissimilarity of every stimulus x from its “immediate” neighbors on the right. A modern rendering of Fechner’s theory (Dzhafarov, 2001) de•nes the dissimilarity between x and x + dx as

D x x dx c x x dx, ,+( ) = +( ) − γ 12 , where γ (x, y) is a psychometric function

γ (x, y) = Pr [y is judged to be greater than x]

with no “constant error” (i.e., γ (x, x) = 1/2), and c is a constant allowed to vary from one stimulus continuum to another. Assuming that γ (x, y) is differentiable, and putting

D x x dx

dx

x y

y F x

, ,+( ) =

∂ ( ) ∂ = ( )

γ ,

the Fechnerian distance from a to b ≥ a becomes

G a b F x dx a

,( ) = ( )∫ .