Human knowledge is not deductively closed. Uncertainty can arise from that just as much as from lack of brute facts. (When a Harvard graduate was reported to have thought that the earth is farther from the sun in the winter than in the summer and that that's why seasons happen, I commented to someone that that was stupid because he ought to have known that when it's winter in the northern hemisphere, it's summer in the southern hemisphere and vice-versa. I was told that I was making the mistake of expected a person's knowledge to be deductively closed. I don't altogether agree, since those who make fun of Sarah Palin for (allegedly) not knowing that Africa is a continent were not really expecting all humans to be omniscient.; it's similar to that.)
But the conventional mathematical theory of probability is deductively closed. As a means of expressing uncertainty, its closure can be useful because in telling us how much uncertainty is justified, it can make us aware of logical connections we might have missed.
However, let's suppose you're wondering if some number with an almost unimaginably large number of digits is prime. (Say it's $5963$, which, unbeknownst to all humans, can be factored as $67\times89$.) After many years of work with supercomputers, you've ruled out the possibility of its divisibility by all primes up to and including $17$, which is the largest prime not exceeding its cube root. If I'm not mistaken, there are ways to estimate the sum of primes bigger than that but not bigger than the square root of the number to be factored (In the present example, $1/19 + 1/23 + 1/29 + 1/31 + 1/37 + 1/41 + 1/43 + 1/47 + 1/53 + 1/59 + 1/61 + 1/67 + 1/71 + 1/73 \approx 0.354$.) It seems reasonable to consider this a sort of upper bound on the probability that the large number considered is composite. But in a deductively closed system, that probability is $0$ or $1$, and if it's $1$, then $0.354$ is not an upper bound on it.
Is there any mathematically precise theory of probability without deductive closure?