An Endless Hierarchy
of Probabilities
by Jeanne Peijnenburg and David Atkinson
Suppose q is some proposition, and let
P(q) = v_{0} (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) = v_{0}) = v_{1}, (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) = v_{0}) = v_{1}) = v_{2}, (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.

