On the theory of biased bisection operations and their inverses

A theory of fundamental measurement involving multiple operations is presented. The operations are bisection, and two inverses of the bisection operation called right and left displacement, analogous to “displacement” of an interval “up” or “down” a scale. Following Pfanzagl, “biased” operations are permitted, and linear, reflexive representations are presented for the biased operations. Results are obtained on the constructability of unbiased scaling operations, and several methods of constructing standard sequences of equally spaced stimuli using biased operations, are described. A striking property of biased linear representation models is derived, showing that if bias is present, indefinite iterations of one or the other of the two displacement operations will converge to asymptote. Preliminary data support this surprising convergence property.