Yesterday I received from amazon.com a copy of Cox's book _The Algebra of
Probable Inference_. (Thanks for the recommendation, Ben.)
In his preface Cox expresses his indebtedness to Keynes, and Keynes'
influence is obvious throughout. For this reason I was expecting to find
somewhere within the text a Keynesian-like attempt to rehabilitate the
Principle of Indifference.
However in this respect Cox breaks clearly from Keynes. Cox offers a
strong and clear argument against the principle, starting at the bottom of
page 31 and extending to about the middle of page 33 (in my paperback
edition).
Briefly, his argument is that the conditions necessary for applying the
principle of indifference are "exceptional" and "rare". They are present
only for example in such trivial cases as certain games of chance in which
the necessary conditions are "prescribed by the rules of the game or
result from the design of the equipment".
Cox offers a formal disproof of the principle in the case in which there
exist two mutually exclusive outcomes and nothing else is known. In such
situations the principle prescribes that we assign prior probabilities of
.5 to each outcome. Cox shows this to be absurd and unfounded, and writes
this about his own conclusion:
"This conclusion agrees with common sense and might perhaps have been
reached without formal argument, because the knowledge of a probability,
though it is knowledge of a particular and limited kind, is still
knowledge, and it would be surprising if it could be derived from the
truism, which is the expression of complete ignorance, asserting nothing."
Indeed!
-gts
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