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Its not that people don't want to admit their weakness in logic. Rather it is because people hate to change their minds.
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Then there's
confirmation bias. We are hard wired to sort data into already established patterns. New information usually isn't even recognized as a challenge to one's beliefs, it is quickly interpreted to conform to what we already think.
It's not (usually) that people are too proud or lazy to change their minds. The new information simply is not recognized as requiring a new set of deductions.
This is one reason rationalism is overrated (though it's the best we have).