You're sidestepping the whole point of cousin_it, which is that your mathematical knowledge is certainly enough to determine whether the millionth digit of pi is odd or even. One of these two statements is a trivial consequence of the Peano axioms and some well-known representation of pi as an infinite series. It's just that its being a trivial consequence is witnessed by a very long (and also trivial) proof which you're not aware of, and you don't know which of the two statements is backed by such a long proof.
I don't think I'm sidestepping the issue. The point of cousin_it's comment, as I understood it, was that assigning probabilities to "logically uncertain" statements results in inconsistencies. What I tried to show is that for probabilistic assignments to be consistent, it is only necessary to be logically omniscient at propositional calculus, not at full-power PA. And this is an important difference, because propositional calculus is decidable.
This post is a long answer to this comment by cousin_it:
I'd like to attempt to formally define logical uncertainty in terms of probability. Don't know if what results is in any way novel or useful, but.
Let X be a finite set of true statements of some formal system F extending propositional calculus, like Peano Arithmetic. X is supposed to represent a set of logical/mathematical beliefs of some finite reasoning agent.
Given any X, we can define its "Obvious Logical Closure" OLC(X), an infinite set of statements producible from X by applying the rules and axioms of propositional calculus. An important property of OLC(X) is that it is decidable: for any statement S it is possible to find out whether S is true (S∈OLC(X)), false ("~S"∈OLC(X)), or uncertain (neither).
We can now define the "conditional" probability P(*|X) as a function from {the statements of F} to [0,1] satisfying the axioms:
Axiom 1: Known true statements have probability 1:
P(S|X)=1 iff S∈OLC(X)
Axiom 2: The probability of a disjunction of mutually exclusive statements is equal to the sum of their probabilities:
"~(A∧B)"∈OLC(X) implies P("A∨B"|X) = P(A|X) + P(B|X)
From these axioms we can get all the expected behavior of the probabilities:
P("~S"|X) = 1 - P(S|X)
P(S|X)=0 iff "~S"∈OLC(X)
0 < P(S|X) < 1 iff S∉OLC(X) and "~S"∉OLC(X)
"A=>B"∈OLC(X) implies P(A|X)≤P(B|X)
"A<=>B"∈OLC(X) implies P(A|X)=P(B|X)
etc.
This is still insufficient to calculate an actual probability value for any uncertain statement. Additional principles are required. For example, the Consistency Desideratum of Jaynes: "equivalent states of knowledge must be represented by the same probability values".
Definition: two statements A and B are indistinguishable relative to X iff there exists an isomorphism between OLC(X∪{A}) and OLC(X∪{B}), which is identity on X, and which maps A to B.
[Isomorphism here is a 1-1 function f preserving all logical operations: f(A∨B)=f(A)∨f(B), f(~~A)=~~f(A), etc.]
Axiom 3: If A and B are indistinguishable relative to X, then P(A|X) = P(B|X).
Proposition: Let X be the set of statements representing my current mathematical knowledge, translated into F. Then the statements "millionth digit of PI is odd" and "millionth digit of PI is even" are indistinguishable relative to X.
Corollary: P(millionth digit of PI is odd | my current mathematical knowledge) = 1/2.