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, because computation effort has not yet been expended.

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. So, 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 I am about to make 1 billion dollars (Principle of Explosion). Therefore the value of this bet is at very least $500,000,050, which is 0.5 * $1,000,000 plus the usual $50 that one would calculate on a $100 bet at 1-to-1 odds. This logic, of course, would lead to any desired expected value on the bet, an absurdity. Clearly one could in principle run a supercomputer to calculate this digit and resolve the matter, and no unlimited source of money would pop into existence.