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Probability Functions: The Matter of Their Recursive Definability

Hugues Leblanc and Peter Roeper
Philosophy of Science
Vol. 59, No. 3 (Sep., 1992), pp. 372-388
Stable URL: http://www.jstor.org/stable/188154
Page Count: 17
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Probability Functions: The Matter of Their Recursive Definability
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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, Renyi, and Popper.

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