I only call syllogisms about probabilities valid if they follow from Bayes' Theorem. You permit yourself a meta-probability distribution over the probabilities and call a syllogism valid if it is Cyan::valid on average w.r.t. to your meta-distribution.

But you're permitting yourself the same thing! Whenever you apply the Bayes Theorem, you're asserting a probability distribution to hold, even though that might not be the true generating distribution of the phenomenon. You would reject the construction of such as scenario (where your inference is way off) as a "counterexample" or somehow showing the invalidity of updates performed under the Bayes theorem. And why? Because that distribution is the best probability estimate, on average, for scenarios in which you occupy that epistemic state.

All I'm saying is that the same situation holds with respect to undefined tokens. Given that you don't know what D and H are, and given the two premises, your best estimate of P(H|D) is low. Can you find cases where it isn't low? Sure, but not on average. Can you find cases where it necessarily isn't low? Sure, but they involve moving to a different epistemic state.

No, a finite interval is not sufficient. You really need to specify the invariant measure to use maxent in the continuous case

Wrong:

The uniform distribution on the interval [a,b] is the maximum entropy distribution among all continuous distributions which are supported in the interval [a, b] (which means that the probability density is 0 outside of the interval).