100%

Given that probability is a formalization of subjective degree of belief, it would reasonable to consider absolute subjective certainty to correspond to a probability of 100%. Likewise, being absolutely certain that something is false would correspond to assigning it a probability of 0%.

According to Bayes’ theorem, if something has a probability of 100%, that must remain unchanged no matter what evidence is observed, as long as that evidence has a finite probability of being observed. If the probability of the evidence being observed is 0%, then Bayes’ formula results in a division by zero. This happens because a probability of 0% should mean that it is impossible for this evidence to come up, and indicates that one was simply wrong to claim that there was no chance of this, and a different probability should have been assigned.

The fact that logical consistency requires a probability of 100% to remain permanently fixed, no matter what happens, implies that it is generally a bad idea to claim such certainty, even in cases where you have absolute objective certainty such as mathematical demonstration. Thus in the previously cited anecdote about prime numbers, if SquallMage claimed to be absolutely certain that 51 was a prime number, he should never admit that it is not, not even after dividing it by 3 and getting 17. Instead, he should claim that there is a mistake in the derivation showing that it is not prime. Since this is absurd, it follows that in fact he should never have assigned a 100% probability to the claim that the number was prime. And since there was subjectively probably not much difference between 41 and 51 for him at the time, with respect to the claim, neither should he have claimed a 100% probability that 41 was prime.

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