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DONNAbrian comments on The dangers of zero and one - Less Wrong
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Interesting discussion but I suspect an important distinction may be required between logic and probability theory. Logic is a special case of probability theory where values are restricted to only 0 and 1, that is to 0% and 100% probability. Within logic you may arrive at certain conclusions but generally within probability theory conclusions are not certain but rather assigned a degree of plausibility.
If logic provides, in some contexts, a valid method of reasoning then conclusions arrived at will be either 0% or 100% true. Denying that 100% confidence is ever rational seems to be equivalent to denying that logic ever applies to anything.
It is certainly true that many phenomena are better described by probability than by logic but can we deny logic any validity. I understand mathematical proofs as being within the realm of logic where things may often be determined as being either true or false. For instance Euclid is credited with first proving that there is no largest prime. I believe most mathematicians accept this as a true statement and that most would agree that 53 is easily proven to be prime.
When you prove something in mathematics, at very least you implicitly assume you have made no mistakes anywhere, are not hallucinating, etc. Your "real" subjective degree of belief in some mathematical proposition, on the other hand, must take all these things into account.
For practical purposes the probability of hallucinations etc. may be very small and so you can usually ignore them. But the OP is right to demonstrate that in some cases this is a bad approximation to make.
Deductive logic is just the special limiting case of probability theory where you allow yourself the luxury of an idealised box of thought isolated from "real world" small probabilities.
edit: Perhaps I could say it a different way. It may be reasonable for certain conditional probabilities to be zero or one, so long as they are conditioned on enough assumptions, e.g. P("51 is a prime" given "I did my math correctly, I am not hallucinating, the external world is real, etc...")=1 might be achievable. But if you try to remove the conditional on all that other stuff you cannot keep this certainty.