In your scheme you have P=1/2 for anything nontrivial and its negation that's not already in X. It just so happens that this looks reasonable in case of the oddity of a digit of pi, but that's merely a coincidence (e.g. take A="a millionth digit of pi is 3" rather than "...odd").
No, a statement and its negation are distinguishable, unless indeed you maliciously hide them under quantifiers and throw away the intermediate proof steps.
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.