Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T14:36:04.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensive Measurement Without an Order Relation

Published online by Cambridge University Press:  14 March 2022

Eric W. Holman
Eric W. Holman*
Affiliation:
University of California, Los Angeles
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper states two sets of axioms sufficient for extensive measurement. The first set, like previously published axioms, requires that each of the objects measured must be classifiable as either greater than, or less than, or indifferent to each other object. The second set, however, requires only that any two objects be classifiable as either indifferent or different, and does not need any information about which object is greater. Each set of axioms produces an extensive scale with the usual properties of additivity and uniqueness except for unit. Moreover, the axioms imply Weber's Law: whether two objects are indifferent depends only upon the ratio of their scale values.

Type
Research Article
Information
Philosophy of Science , Volume 41 , Issue 4 , December 1974 , pp. 361 - 373
Copyright
Copyright © 1974 by The Philosophy of Science Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

This work was supported by Grant GB-13588X from the National Science Foundation.

References

REFERENCES

[1] Fishburn, P. C. “Intransitive Indifference with Unequal Indifference Intervals.” Journal of Mathematical Psychology 7 (1970): 144–149.CrossRefGoogle Scholar
[2] Krantz, D. H. “Extensive Measurement in Semiorders.” Philosophy of Science 34 (1967): 348–362.10.1086/288173CrossRefGoogle Scholar
[3] Krantz, D. H. “Color Measurement and Color Theory—I. Representation Theorem for Grassman Structures.” Michigan Mathematical Psychology Program, MMPP 71–3. Department of Psychology, University of Michigan, 1971.Google Scholar
[4] Luce, R. D. “Three Axiom Systems for Additive Semiordered Structures.” SIAM Journal on Applied Mathematics 25 (1973): 41–53.10.1137/0125008CrossRefGoogle Scholar
[5] Miller, G. A. “Sensitivity to Changes in the Intensity of White Noise and its Relation to Masking and Loudness.” Journal of the Acoustical Society of America 19 (1947): 609–619.10.1121/1.1916528CrossRefGoogle Scholar
[6] Pólya, G. and Szegö, G. Aufgaben und Lehrsätze aus der Analysis. Berlin: Springer-Verlag, 1925.Google Scholar
[7] Roberts, F. S. “On Nontransitive Indifference.” Journal of Mathematical Psychology 7 (1970): 243–258.10.1016/0022-2496(70)90047-7CrossRefGoogle Scholar
[8] Suppes, P. “A Set of Independent Axioms for Extensive Quantities.” Portugaliae Mathematica 10 (1951): 163–172.Google Scholar