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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
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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.

Information

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

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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.

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