In Richard Carrier's fairly recent book Proving History, he argues for a particular epistemological approach that is (he proves it!) better than or equal to any other approach to history: Bayes' theorem. In one of the books he cites but seems not to have read, which is basically an argument that Bayesian reasoning is the optimal way for any person to update/re-evaluate their beliefs given what they believed before and what new information they are presented with, the author at one points gives as an example two people who are presented with the same information but who conclude opposite things because their starting beliefs differed so greatly. For Carrier, this completely defeats his entire argument, but for the purposes of this thread, it is what using "logic" means. When people refer to being logical they frequently conceptualize this in terms of rational and even in terms of "normal", at least in the sense that one typically doesn't hear paranoid schizophrenics referred to as "logical".
The problem is that logic is a reasoning process. If you believe that space aliens are trying to read your mind and tin foil hats can stop the thought rays from penetrating your skull, then it is entirely logical to wear tin foil hats. If you believe that religious thinking is poison and a danger to society than it is logical to want laws put in place to prevent the spread of religious thought. If you believe in a particular religion and that non-believers should be converted or killed, it is logical to convert or kill everyone you can.
I've used the example before, but as Kurt Gödel provides such a perfect illustration, I'll use it again. He actually had a clinical-level fear of being poisoned and would not eat food prepared by someone other than his wife. So when she was hospitalized due to illness, he died of starvation. The greatest logician in the world (and likely of all time), with a brain that then as before vastly outstripped virtually all others in terms of analytical abilities, died of starvation surrounded by easily accessible food because he reasoned, entirely logically, that given people were trying to poison him and that he could only trust his wife, he could not eat food his wife didn't prepare. And he died.
That's logic. It's great when we can use it to reason from what we hope is a pretty decent starting place about the world around us, but there's nothing illogical about any conclusion as long as particular premises are accepted. As logic cannot tell you whether a premise is true (except perhaps for premises that are necessarily true, but these don't matter for any practical purposes), it provides only a basis for reasoning, not rationality (defined as what would generally be considered within the bounds of "normal" mentality/worldview).
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