R. Duncan Luce

The article first summarizes the assumptions of Luce (2004, 2008) for inherently binary (2-D) stimuli (e.g., the ears and eyes) that lead to a "p-additive," order-preserving psychophysical representation. Next, a somewhat parallel theory for unary (1-D) signals is developed for intensity attributes such as linear extent, vibration to finger, and money. The 3rd section studies the property of bisymmetry in these 2 cases. For the 2-D case and the nontrivial p-additive forms, Proposition 3 shows that bisymmetry implies commutativity of the presentations. Bisymmetry has been empirically well sustained, whereas commutativity has been rejected for loudness, brightness, and perceived contrast, thus implying that pure additivity must obtain in the 2-D context. By contrast, bisymmetry and commutativity are automatically satisfied by the p-additive 1-D theory. The 4th section explores the resulting complex of cross-modal predictions. For the additive 1-D case and the 2-D case, the predictions are power functions. For the nonadditive 1-D cases, other relations are predicted (see Table 2). Some parameter estimation issues are taken up in Appendices B and C.