Hostname: page-component-7dd5485656-wxk4p Total loading time: 0 Render date: 2025-10-30T22:15:18.919Z Has data issue: false hasContentIssue false
  • English
  • Français

A Probabilistic Theory of Extensive Measurement

Published online by Cambridge University Press:  01 April 2022

Jean-Claude Falmagne
Jean-Claude Falmagne*
Affiliation:
New York University
Get access Rights & Permissions [Opens in a new window]

Abstract

Algebraic theories for extensive measurement are traditionally framed in terms of a binary relation ≲ and a concatenation (x,y) → xy. For situations in which the data is “noisy,” it is proposed here to consider each expression y ≲ x as symbolizing an event in a probability space. Denoting P(x,y) the probability of such an event, two theories are discussed corresponding to the two representing relations:with

Axiomatic analyses are given, and representation theorems are proven in detail.

Information

Type
Research Article
Information
Philosophy of Science , Volume 47 , Issue 2 , June 1980 , pp. 277 - 296
Copyright
Copyright © Philosophy of Science Association 1980

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.)

Article purchase

Temporarily unavailable

Footnotes

†

This work is supported by Grant No. BNS 77-16984 to New York University from the National Science Foundation. The author is grateful to the Department of Psychology and Social Relations at Harvard University for its hospitality. I also wish to thank H. Colonius and Z. Domotor for their comments on a previous draft.

References

Block, H. and Marschak, J. (1960), “Random orderings and stochastic theories of response.” In Olkin, I., (ed.), Contributions to Probability and Statistics. Stanford University Press.Google Scholar
Campbell, N. R. (1957), Physics: the elements. Cambridge: Cambridge University Press, 1920. Reprinted as Foundations of Science: the philosophy of theory and experiment. New York: Dover.Google Scholar
Cohen, M., and Falmagne, J. C. (1978), “Random Scales Representation of Binary Choice Probabilities: a counterexample to a conjecture of Marschak.” Mathematical Studies in Perception and Cognition. Technical Report. No 78–5 Dpt. of Psychology, New York University.Google Scholar
Doignon, J. P. and Falmagne, J. C. (1974), “Difference Measurement and Simple Scalability with Restricted Solvability,” Journal of Mathematical Psychology 11: 473–499CrossRefGoogle Scholar
Falmagne, J. C. (1971), “Bounded Versions of Holder's Theorem with Applications to Extensive Measurement,” Journal of Mathematical Psychology, 8: 495–507.CrossRefGoogle Scholar
Falmagne, J. C. (1975), “A Set of Independent Axioms for Positive Holder Systems,” Philosophy of Science 42: 137–151.CrossRefGoogle Scholar
Guilbaud, G. (1953), “Sur une difficulté de la théorie du risque,” Colloques Internationaux du Centre National de la Recherche Scientifique. (Econometric) 40: 19–25.Google Scholar
Krantz, D. H., Luce, R. D., Suppes, P. and Tversky, A. (1971), Foundations of Measurement. Volume I, Additive and Polynomial Representations. New York and London: Academic Press.Google Scholar
Luce, R. D., and Suppes, P, (1965), “Preference, utility, and subjective probability,” in Luce, R. D., Bush, R. R., & Galanter, E. (eds.), Handbook of Mathematical Psychology Volume 3. New York: Wiley.Google Scholar
McFadden, D., and Richter, M. K. (1971), “Revealed Stochastic Preference,” University of California, Department of Economics (unpublished).Google Scholar
Marschak, J. (1960), “Binary choice constraints on random utility indicators,” in K. Arrow, (ed.) Stanford Symposium on Mathematical Methods in the Social Sciences. Stanford, California: Stanford University Press.Google Scholar