The key here is "shown not to hold in finite domains". Basically, Cox' Theorem implicitly assumes that we have a wide enough variety of inputs to produce a whole continuum of "probabilities". But if we only have a small handful of possible inputs, then this can fail - that's what Halpern's counterexample shows.
I'd expect this to be an issue in practice mainly in two cases:
Basically, there have to be very few "bits of randomness" available.
As long as there's lots of bits of randomness, even if the domains are technically finite, reasonable smoothness assumptions should make Cox' Theorem work again. (As the number of bits of randomness increases, to use the loophole we need to have large discontinuous jumps closer and closer together in the functions used in the theorem.)
This fits neatly with evidence from another direction: information theory/minimum description length. Other than Cox, info theory/MDL is the other main foundation of probability which is "purely epistemic" - i.e. it doesn't talk about coherence or utilities. And the basic info theory foundation is weak in exactly the same place where Cox is weak: we need many bits of randomness in order for the use of "average message length" to be justified, and also for discretization error to be small. When there are few bits of randomness, information theory's theorems are less compelling.
Thanks for the reply!
This stuff is way over my head. What is the tldr version? Is the interpretation of probability that Professor Jaynes expounded in his book correct? Can I use the results that he derives in his book along with the interpretation of probability as extended logic?
Any references on bits of randomness, MDL, and the purely epistemic interpretation of probability?
If the purely epistemic interpretation of probability has these weaknesses, are there any other interpretations of probability which are more applicable in practice?
This counterexample saga reminds me of Lakato's "Proofs and Refutations". You have a result that is "essentially true" but you can still find some "counterexamples" to it by conveniently stressing the "obvious setting" in which the result was originally formulated. Note in any case that whereas Euler has been "refuted" he is still credited for his original V - E + F = 2 formula.
Please note that Halpern himself corrected his own first paper, which you cite. In a second paper (https://doi.org/10.1613/jair.644) he showed how to solve the difficulties he raised in his first paper. See also the work by Snow (https://doi.org/10.1111/0824-7935.00070), which again corrects Halpern's first paper and also some points by Paris (https://doi.org/10.1017/CBO9780511526596) and finds no serious difficulties with Cox's main points.
What implications does the paper “A Counter Example to Theorems of Cox and Fine” by J. Y. Halpern have for Cox’s theorem and probability theory as extended logic? This is the description of the paper:
“Cox's well-known theorem justifying the use of probability is shown not to hold in finite domains. The counterexample also suggests that Cox's assumptions are insufficient to prove the result even in infinite domains. The same counterexample is used to disprove a result of Fine on comparative conditional probability.”
Edit: You can access the paper here - https://arxiv.org/abs/1105.5450
A similar question seems have been posted (but not answered) here:
Why is Cox’s theorem being disputed? Are there any non-sequiturs in the proof that Professor Jaynes give for it in his book? If not, then how can it be disputed?