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:

- https://stats.stackexchange.com/q/190187/297721
- https://stats.stackexchange.com/q/190184/297721
- https://stats.stackexchange.com/q/189757/297721

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?