Professor Halpern has acknowledged that he did not produce
a counterexample to a theorem of Cox. I thank him for his posting of last
Friday night.
Professor Halpern briefly observes that the assumption which was
crucial in 1946 is no longer found in 1978. It was by then superfluous
since Cox, like Polya before him, had come to explain the probability rules
as relationships among variables. The range and density of potential values
for Cox's belief-variable are plainly disclosed in 1978.
Now that the false counterexample incident is behind us, I would
like to mention a more persistent problem.
There remain serious difficulties within the secondary literature
about Cox, chiefly that his work came to be misinterpreted as legislation
for the obligatory use of single-distribution probabilistic belief models.
The real Cox espoused other views. Many of the popular non-Bayesian calculi
can be modeled and understood within the scope of Cox's notion of
reasonable belief. Trying to comprehend Cox filtered through some of his
disciples is like studying Aristotle by reading Ayn Rand.
Best wishes.
Paul