Logical uncertainty

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Logical uncertainty applies the rules of probability to logical facts which are not yet known to be true or false.

Is the googolth digit of pi odd? This is a fact that we know is definitely true or falls by the rules of logic, even though we do not know if it is true or false. The probability that the fact is true intuitively is 0.5. Formalizing this sort of probability is the field of logical uncertainty.

One basic problem with this idea is that it gives non-zero probability to false statements. If I am asked to bet on whether the googolth digit of pi is odd, I can reason as follows: There is 0.5 chance that it is odd. Let P represent the parity of the googolth digit (odd or even, I don't know which) and Q represent the parity which is not the parity of this digit. If Q, then anything follows. (By the Principle of Explosion, a false statement implies anything.) For example, Q implies that I will win $1 billion. Therefore the value of this bet is at least $500,000,000, which is 0.5 * $1,000,000. This logic, of course, leads to an absurdity. Only expenditure of finite computational stands between the uncertainty and the proof.


References

Questions of Reasoning Under Logical Uncertainty by Nate Soares and Benja Fallenstein.