Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-03T00:33:59.182Z Has data issue: false hasContentIssue false
  • English
  • Français

Probability Functions: The Matter of Their Recursive Definability

Published online by Cambridge University Press:  01 April 2022

Hugues Leblanc  and
Peter Roeper
Hugues Leblanc
Affiliation:
Department of Philosophy, Temple University
Peter Roeper
Affiliation:
Department of Philosophy, The Australian National University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the extent to which probability functions are recursively definable. It proves, in particular, that the (absolute) probability of a statement A is recursively definable from a certain point on, to wit: from the (absolute) probabilities of certain atomic components and conjunctions of atomic components of A on, but to no further extent. And it proves that, generally, the probability of a statement A relative to a statement B is recursively definable from a certain point on, to wit: from the probabilities relative to that very B of certain atomic components and conjunctions of atomic components of A, but again to no further extent. These and other results are extended to the less studied case where A and B are compounded from atomic statements by means of “∀” as well as “~” and “&”. The absolute probability functions considered are those of Kolmogorov and Carnap, and the relative ones are those of Kolmogorov, Carnap, Rényi, and Popper.

Type
Research Article
Information
Philosophy of Science , Volume 59 , Issue 3 , September 1992 , pp. 372 - 388
Copyright
Copyright © 1992 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

†

The paper, of which a French version is concurrently appearing in the Canadian journal Dialogue, was presented at the 1990 Meeting of the Society for Exact Philosophy at Florida State University, and at the 1991 Meeting of the Canadian Philosophical Association at Queen's University, Kingston, Ontario. Thanks are extended to Steven K. Thomason, who suggested that our results be phrased in terms of recursive definability. Hugues Leblanc held a Grant in Aid of Research from Temple University while the paper was written.

‡

Send reprint requests to the authors, Department of Philosophy, The Faculties, The Australian National University, Canberra, ACT 2601, AUSTRALIA.

References

Carnap, R. (1950), Logical Foundations of Probability. Chicago: University of Chicago Press.Google Scholar
Carnap, R. (1952), The Continuum of Inductive Methods. Chicago: University of Chicago Press.Google Scholar
Kolmogorov, A. N. ([1933] 1950), Foundations of the Theory of Probability. Translated by N. Morrison. (Originally published as “Grundbegriffe der Wahrscheinlichkeitsrechnung”, Ergebnisse der Mathematik 2: 1–61.) New York: Chelsea.Google Scholar
Leblanc, H. (1983), “Alternatives to Standard First-Order Semantics”, in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. 1, Elements of Classical Logic. Dordrecht: Reidel, pp. 189–274.CrossRefGoogle Scholar
Leblanc, H. and Roeper, P. (1986), “Absolute Probability Functions: A Recursive and Autonomous Account”, in Leinfellner, W. and Wuketits, F. M. (eds.), The Tasks of Contemporary Philosophy. Vienna: Hölder-Pichler-Tempsky, pp. 39–49.Google Scholar
Leblanc, H. and Roeper, P. (1989), “On Relativizing Kolmogorov's Absolute Probability Functions”. Notre Dame Journal of Formal Logic 30: 485–512.CrossRefGoogle Scholar
Leblanc, H. and Roeper, P. (1990), “What are Absolute Probabilities a Function of?”, in Dunn, J. M. and Gupta, A. (eds.), Truth or Consequences: Essays in Honor of N. D. Belnap. Dordrecht: Kluwer, pp. 307–325.CrossRefGoogle Scholar
Popper, K. R. (1959), The Logic of Scientific Discovery. New York: Basic Books.Google Scholar
Rényi, A. (1955), “On a New Axiomatic Theory of Probability”. Acta Mathematica Academiae Scientiarum Hungaricae 6: 286–335.CrossRefGoogle Scholar
Rényi, A. (1964), “Sur les Espaces Simples des Probablités Conditionnelles”. Annales de l'Institut Poincaré, N Série, Section B: 3–21.Google Scholar
Roeper, P. and Leblanc, H. (1991), “Indiscernibility and Identity in Probability Theory”. Notre Dame Journal of Formal Logic 32: 1–46.Google Scholar
Wittgenstein, L. (1922), Tractatus Logico-Philosophicus. London: Routledge & Kegan Paul.Google Scholar