The marriage of predicate The marriage of predicate logic and probability is indeed an ongoing problem. Most current work in Artificial Intelligence focuses on quantifying over probabilities. Partially due to neglectedness, and partially because we suspect that quantifying over probabilities is not going to end up a good approach, MIRI’s workshops on logical uncertainty have tended to attack the problem from the opposite direction, assigning probabilities to logical statements, e.g: http://intelligence.org/files/DefinabilityTruthDraft.pdf http://intelligence.org/files/Non-Omniscience.pdf http://arxiv.org/abs/1508.04145 But none of this is challenging the Kolmogorov axioms or denying Cox’s Theorem, because once you assign probabilities to logical formulas, those probabilities obey the standard probability axioms. It might be better to say that the theory of uncertain reasoning extends the theory of certain reasoning, if you object to the phraseology which says that probability extends logic. Another obvious example of an element of rationality that’s not contained in the Kolmogorov axioms for probability is, of course, the prior. But since logic never addressed priors either, this doesn’t mean that the theory of uncertain reasoning fails to extend the theory of certain reasoning. What you’re really talking about, I think, is a mixture of the problem of extending our priors to logical theories that might be true of the empirical world, and the problem of relating logical theories to the empirical world at all. If I knew that, given some logical set of beliefs, it was 60% likely for a coin to come up heads, and then I saw the coin come up tails, I would know what to do with that. The problem is all in going from the logical set of beliefs to the 60% probability of seeing the coin come up heads. But if you can’t follow that link, you don’t have a problem with “extending logic to probabilities” so much as you have a problem with “relating anything phrased in predicate calculus to an experimental theory that makes predictions”. Even if I could tell you the probability of the Goldbach Conjecture being a semantic tautology of second-order arithmetic after updating on observation of the first trillion examples being true, I might not have solved the problem of going from any set of axioms phrased in predicate calculus to an experimental prediction. You’re free to say of this that “probability can’t extend logic!” but I wouldn’t describe that as being the main issue. An AI-grade solution to this problem, I suspect, will tackle that issue head-on and give a naturalistic account of physical words that obey logical axioms, assign priors to those different observers, and try to locate observers or agents inside those worlds. But this is not the same problem as assigning a probability to the Goldbach Conjecture. Jaynes clearly didn’t get everything right. For example, Probability Theory: The Logic of Science discusses Jaynes’s disbelief in the Copenhagen interpretation (justified) but it’s clear that Jaynes has not understood the import of the very standard argument from Bell’s Theorem which rules out Jaynes’s suggested resolution. Even so, the part of the argument where probability extends logic and (the repaired forms of) Cox’s Theorem gives us reason to suspect we won’t find any other way to do it, seems quite clear and cogent to me; even more so when you consider that in decision theory we have no good alternatives to utility functions and utility functions want scalar quantitative weights on outcomes, so useful uncertainty has to be a scalar quantity. I think you would probably just be happier with the wording if we said “the theory of scalar uncertainty extends the theory of qualitative belief and disbelief, and there are no good alternatives to scalar probabilities when it comes to AI-grade theories of uncertainty”.