# Logical uncertainty

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 the idea of logical uncertainty is that it gives non-zero probability to false statements. By the Principle of Explosion, a false statement implies anything. If I am asked to bet $100 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 this 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. For example, it follows that I will win $1 billion. Therefore the value of this bet is $500,000,050, which is 0.5 * $1,000,000 plus the usual $50 value on a $100 bet at 1-to-1 odds. 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.