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.