Now I have seen some interesting papers that make expanded probability theories that include 0 and 1 as logical falsehood and truth respectively. But that still does not include a special value for contradictions.

Except, contradictions really are the only way you can get to logical truth or falsehood; anything other than that necessarily relies on inductive reasoning at some point. So any probability theory employing those must use contradictions as a means for arriving at these values in the first place.

I do think that there's not much room for contradictions in probability theories trying to actually work in the real world, in the sense that any argument of the form A->(B & ~B) also has to rely on induction at some point; but it's still helpful to have an anchor where you can say that, if a certain relationship does exist, then a certain proposition is definitely true.

(This is not like saying that a proposition can have a probability of 0 or 1, because it must rely, at least somewhere down the line, on another proposition with a probability different from 0 and 1).

I think you and potato are talking about different things; potato's criticism is close to what's discussed in this post.

But for what it's worth, Bayesians with finite (and flawed) computational powers can meaningfully assign probabilities to mathematical statements, and update as they prove more theorems.

Um, you can derive from Komolgorov that (A & ~A) has prob zero. Very easily.

If (A & B) = nullset, then P(A & B) = 0.
(A & ~A) = nullset
so: P(A & ~A) = 0.

In other words, if two sets are disjoint, the probability of their intersection is zero. Any set and its compliment are disjoint, so the probability of their conjunction is zero.

It also helps if you think of the boolean variables as fractions of a dart board, and the probability as the area of that fraction. The formalism is perfectly isomorphic. Obviously, the intersection of any fraction and its compliment will have an area of zero.

See, I believe I have a proof that the probability of any contradiction is zero. So I am going to have to ask for a proof to change my mind, or a problem with my proof (which I doubt, it's very short, and I math often).

ISTM that you're using the word contradiction in a non-standard way: in the usual sense they are logical falsehoods. What do you actually mean? (ETA: I guess paradoxes such as “This sentence is false”?)