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.
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.