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Extensive Measurement in Semiorders

Published online by Cambridge University Press:  14 March 2022

David H. Krantz
David H. Krantz*
Affiliation:
University of Michigan
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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.

Type
Research Article
Information
Philosophy of Science , Volume 34 , Issue 4 , December 1967 , pp. 348 - 362
Copyright
Copyright © Philosophy of Science Association 1967

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Footnotes

1

The research reported here was supported by grant GB 4947 from NSF to The University of Michigan. Preparation of this publication was aided by NIH 5 T1 GM 1231.

2

I wish to thank R. Duncan Luce for suggesting the investigation of extensive measurement in semiorders, and Amos Tversky for his criticisms of an earlier draft of this paper.

References

REFERENCES

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