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Paul_Gowder comments on Infinite Certainty - Less Wrong
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We can go even stronger than mathematical truths. How about the following statement?
~(P &~P)
I think it's safe to say that if anything is true, that statement (the flipping law of non-contradiction) is true. And it's the precondition for any other knowledge (for no other reason than if you deny it, you can prove anything). I mean, there are logics that permit contradictions, but then you're in a space that's completely alien to normal reasoning.
So that's *lots* stronger than 2+2=4. You can reason without 2+2=4. Maybe not very well, but you can do it.
So Eliezer, do you have a probability of 1 in the law of non-contradiction?
The truth of probability theory itself depends on non-contradiction, so I don't really think that probability is a valid framework for reasoning about the truth of fundamental logic, because if logic is suspect probability itself becomes suspect.