| Abstract: | If ≥r and ≥d are two quaternary relations on an arbitrary set A, a ratio/difference representation for ≥r and ≥d is defined to be a function f that represents ≥r as an ordering of numerical ratios and ≥d as an ordering of numerical differences. Krantz, Luce, Suppes and Tversky (1971, Foundations of Measurement. New York, Academic Press) proposed an axiomatization of the ratio/difference representation, but their axiomatization contains an error. After describing a counterexample to their axiomatization, Theorem 1 of the present article shows that it actually implies a weaker result: if ≥r and ≥d are two quaternary retations satisfying the axiomatization proposed by Krantz et al. (1971), and if ≥r′ and ≥d′ are the relations that are inverse to ≥r and ≥d, respectively, then either there exists a ratio/difference representation for ≥r and ≥d, or there exists a ratio/difference representation for ≥r′ and ≥d′, but not both. Theorem 2 identifies a new condition which, when added to the axioms of Krantz et al. (1971), yields the existence of a ratio/difference representation for relations ≥r and ≥d. |
