REPRESENTING APPROXIMATE ORDERING AND EQUIVALENCE RELATIONS

The central question considered is: given appropriate precisations of the ideas of an empirical system's approximately satisfying laws of measurement with error at most ε{lunate} (for some ε{lunate} ≥ 0), and of a real-valued function over its domain providing an approximate representation of its basic operations and relations with error at most δ, can it be shown that satisfaction of the laws with 'sufficiently small' error insures numerical representability with arbitrarily small error? Positive answers are given in the cases of ordinal and nominal measurement, together with some indications of the sizes of the errors involved. Problems of extending the theory to more complex types of measurement are discussed, some open problems and conjectures are formulated, and a relation between the 'approximate representation' and 'stochastic choice model' approaches to measurement with fallible data is established. © 1979.