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Volume 49 • Number 3

July 2012



An Endless Hierarchy of Probabilities

by Jeanne Peijnenburg and David Atkinson

Suppose q is some proposition, and let

P(q) = v0 (1)

be the proposition that the probability of q is v0.1 How can one know that (1) is true? One cannot know it for sure, for all that may be asserted is a further probabilistic statement like

P(P(q) = v0) = v1, (2)

which states that the probability that (1) is true is v1. But the claim (2) is also subject to some further statement of an even higher probability:

P(P(P(q) = v0) = v1) = v2, (3)

and so on. Thus, an infinite regress emerges of probabilities of probabilities, and the question arises as to whether this regress is vicious or harmless.

Radical probabilists would like to claim that it is harmless, but Nicholas Rescher (2010), in his scholarly and very stimulating Infinite Regress: The Theory and History of Varieties of Change, argues that it is vicious. He believes that an infinite hierarchy of probabilities makes it impossible to know anything about the probability of the original proposition q:

unless some claims are going to be categorically validated and not just adjudged probabilistically, the radically probabilistic epistemology envisioned here is going to be beyond the prospect of implementation. . . . If you can indeed be certain of nothing, then how can you be sure of your probability assessments. If all you ever have is a nonterminatingly regressive claim of the format . . . the probability is .9 that (the probability is .9 that (the probability of q is .9)) then in the face of such a regress, you would know effectively nothing about the condition of q. After all, without a categorically established factual basis of some sort, there is no way of assessing probabilities. But if these requisites themselves are never categorical but only probabilistic, then we are propelled into a vitiating regress of presuppositions.

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ISSN: 2152-1123