Extensive Measurement in Semiorders

Abstract

In both axiomatic theories and the practice of extensive measurement, it is assumed that a series of replicas of any given object can be found. The replicas give rise to a standard series, the “multiples” of the given object. The numerical value assigned to any object is determined, approximately, by comparisons with members of a suitable standard series.This prescription introduces unspecified errors, if the comparison process is somewhat insensitive, so that “replicas” are not really equivalent. In this paper, it is assumed that the comparison process leads only to a semiorder, which allows for such insensitivity. It is shown that, nevertheless, extensive measurement can be carried out, provided that a certain set of (plausible) axioms is valid. Approximate measures, and their limits of error, can be derived from finite sets of semiorder observations. These approximate measures converge to ratio-scale exact measurement.