relations between more than two ordered systems, then except for a single strictly increasing function, everything is related as powers of ratios of scale values. The only way the unknown function was eliminated was to insist that one of the ratios had a special status. In accounting for the regularity found with single stimulus procedures, he was forced to assume an implicit standard stimulus with respect to which ratios are computed. This formulation of Shepard ( 1978, 1981) earlier ideas is called relation theory.
Krantz ( 1972) concluded: "The principal reason for favoring relation theory over the others is that it gives a satisfactory account of generalization (iv): that magnitude estimates predict cross-modality matches, independent of the choice of the moduli in the cross-modality matching" (p. 174). His theory has the failing that ratios are preserved only up to an unknown function.
The present theory, which identifies stimulus ratios with translations in the qualitative structure describing the stimulus domains, meets all his criteria listed at the beginning (see the section "Krantz's Five Empirical Generalizations") without introducing a free function. It does this by introducing the idea of preserving stimulus ratios in the form of translation consistency, which is an empirically testable property. This property embodies the intuitive idea that the psychology of the situation (i.e., the matching relation) should be completely consistent with the physics of the situation (i.e., the two qualitative relational structures being matched). Should translation consistency fail, then the laws of matching simply cannot be formulated in a manner similar to the laws of physics. Put another way, the psychophysics of matching would not be an extension of physics.
The fact that approximate power functions are found empirically for certain attributes having to do with stimulus intensity suggests that translation consistency may be valid there, which has led to a search for sources of bias to account for discrepancies from power functions. What does not fit at all well into this view of psychophysics is the evidence that the exponents are subject to easy manipulation ( King & Lockhead, 1981).
Preparation of this chapter was supported in part by National Science Foundation Grant IRI-8996149 to the University of California at Irvine. The research on which it is based is Luce ( 1990)