Re: Cox's Theorem
Joseph Halpern (halpern@cs.cornell.edu)
Fri, 21 May 1999 22:25:31 -0400 (EDT)
A long time ago (March 1, to be exact), Paul Snow posted the message
below. I've been tied up with teaching and other things, so haven't had
a chance to respond until now. Very briefly, I'm embarrassed to admit
that Paul is quite right that Cox did indeed make the assumption that I
accused him of not making in his 1946 paper (although he does not make
this assumption explicitly in his later 1978 paper). However, I believe
that this observation has absolutely no effect on either the correctness
of the technical results in my paper nor on their interpretation.
Roughly speaking, Cox's Theorem says that under some minimal assumptions
on a notion of belief, it is forced to be isomorphic to a probability
measure. There is no question that there are assumptions that are
strong enough to force this to be true. In my paper, I showed by
example that a certain collection of assumptions -- which are the ones I
thought that Cox was making (it is hard to dig out of Cox's papers
exactly what assumptions he is making) and certainly are at least as
strong as those made in many other papers that prove Cox-style theorems
-- are not strong enough to force a notion of belief to be isomorphic to
a probability measure. I've written a short note that states some
positive results, providing various sets of assumptions under which a
Cox-style theorem is true. Unfortunately, none of these sets of assumptions
seem to me to be particularly compelling. For those who are interested,
the note is available on my web page at
http://www.cs.cornell.edu/home/halpern/papers/cox1.ps or .../cox1.pdf.
Cheers,
Joe
----------------------
Greetings :-
Recently, Steve Minton circulated an announcement on this list
on the occasion of _JAIR_'s publication of Professor Halpern's article which
claims a counterexample to Cox's Theorem. Many readers will be familiar
with the controversy from Professor Halpern's AAAI 1996 paper.
Cox made the assumption which has been identified as sufficient to
defeat the counterexample. It appears on page 6, second column, immediately
after the introduction of the crucial equation (8) in Cox's 1946 paper. Some
further remarks on the subject, including the prominence of this assumption
in Cox's thought and the use he made of it in later work, can be found in
the other paper cited below.
I post because a colleague has run into some criticism from other
scholars for relying upon Cox's Theorem while its correctness appears to be
in dispute. I am also familiar with an earlier, similar situation. It is
unproductive that a true and competently demonstrated fact of mathematics
should any longer remain under an unwarranted cloud.
Paul
R.T. Cox, Probability, frequency, and reasonable expectation, _American
Journal of Physics_ 14(1), 1-13, January-February 1946.
P. Snow, On the correctness and reasonableness of Cox's Theorem for finite
domains, _Computational Intelligence_ 14(3), 452-459, August 1998.