On the Possible Psychophysical Laws" Revisited: Remarks on Cross-Modal Matching

A nonrelational theory of cross-modal matching, including magnitude estimation, is proposed for the general class of 1-dimensional measurement structures that have real-unit (ratio and interval scale) representations. A key feature of these structures is that each point of the structure can be mapped into each other point by a translation, which is the structural analogue of ratio scale transformations of the representation. Let M denote a 1:1 matching relation between 2 (not necessarily distinct) unit structures. The major assumption is that for each translation T of one structure, there is a translation a, of the other such that if xMs, then r(x)Mar(s). This property is shown to be equivalent to a power law holding between the unit representations. A concept of similar relations is taken from dimensional analysis, and 2 matching relations are shown to be similar if and only if their power laws differ only in the unit (modulus), not the exponent. A relation R between pairs in each system is said to be a ratio relation relative to a matching relation M satisfying the above condition provided that (x, y)R(s, t) obtains if and only if for some translation T both r(x)Ms and r(y)Mt. In that case, R is unique and is represented by the ratio of the values assigned to x and y in their unit representation being equal to a power of the ratio of the values assigned to s and t in their unit representation; the exponent is that of M. This theory is related to an earlier article by the author and is compared with the relational theory of Shepard (1978, 1981) and Krantz (1972).