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Suppose, he argues, that our real-world experience is limited to dice with fewer
than twenty sides, such as six-sided cubic or four-sided pyramidal dice. In that
case, we still would feel justified in assigning an equal probability p(i) = 1/20 to
each side i of a twenty-sided die such that, when it is thrown for the very first
time, we have a 1/20 probability that side i will show face up. He further posits
that the only justification we can muster for such an assumption will employ
some version of the PoI. However, I am quite able to articulate an alternative
justification: For in the case of the twenty-sided die, our intuition is grounded
not in a probabilistic abstraction, but in first-hand experience with and nthhand knowledge of physical systems, and the mathematical laws which govern
them. This position is not unique; it has been acknowledged, for example, by
Frank Ramsay, who noted that “we sometimes really assume a theory of the
world with laws and chances” (emphasis original).6 In particular for the case
of the twenty-sided die, we appeal to our understanding of symmetry, which is
determined, it seems to me, in large part if not entirely by real-world experience.
In other words, we know something about how falling objects behave, and this
knowledge is gleaned from applying induction to our past observations.
A defender of a PoI might observe that these two accounts are not mutually exclusive; that is, if one appeals to experience with symmetrical systems,
one may still require additional assumptions, among them the PoI. I sympathize with this notion, which perhaps is true in some limited sense. Indeed,
there are many assumptions required when acting in the world at all, which
when formalized often seem arcane and almost always have some initial appearance of arbitrariness. A classic example of this is the problem of induction, a
universally-agreeable solution to which remains indefinitely elusive. Yet we all
accept induction in some form, and it has made possible the generation an even
greater, in my judgment, body of enormously useful results than Collins claims
for his restricted PoI. We may in similar fashion consider codifying some form
of the PoI, and classing it, as Collins implies that we must, among those necessary assumptions for navigating our human experience. Yet while I would be
amenable to attempting such an exercise, I am forced to regard Collins’ restrictions on the PoI, for the reasons already given, as insufficient to this task. For
as we have seen, in pointing out that, for example, dice games may lack direct
empirical justification, he overlooks the usefulness of indirectly-related (e.g. the
relationship of experience with four- and six-sided dice to the predicted behavior of twenty-sided dice) similar past experience to inform assignments of prior
probability distributions. Furthermore, whatever principle we formulate must
withstand the direct arguments against the PoI, which I will present later in
this paper.
It is important at this point to keep in mind that Collins is aiming his effort
not at justifying the assumptions of statisticians, but rather generalizing those
assumptions into a single principle. Such a general rule will only be useful in
cases whereby assigning a uniform distribution is not already agreeably appro6 Ramsay,
Frank
Plumpton.
“Reasonable
Degree
of
Belief”
(1928),
Philosophical Papers (1990), ed.
D.H. Mellor, ISBN 0-521-37621-1, p97.