An axiomatization of the ratio/difference representation

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 .