Recently, E G. Ashby and J. T. Townsend ( 1986) introduced a theory of perceptual identification, the General Recognition Theory (GRT), that rigorously defined previously obscure notions of perceptual independence of stimulus dimensions. Also, they established many theoretical relationships between these different types of independence whereby experimenters could obtain support for (or falsification of) one type or another. Ashby and Townsend's Theorem 4 stated several assumptions, including optimality, are sufficient for a strong form of perceptual independence. It is shown here, by counterexample, that this theorem is false. A review of the basic concepts of GRT is provided, with a discussion of the aforementioned limitation and its implications. An amended version of Theorem 4 is offered. Simulation results illustrate optimistic prospects for testing for independence in lieu of the counterexample.