Resisting pious preaching about admitting when you are wrong or being willing to change your mind when presented with new contradictory facts is rational. Mere admission is concession to opposed views which may be just as wrong. There's a better way to update your positions...
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Generally, when you are wrong in shallow ways (getting a solid fact wrong), it's not hard to admit it. Your natural resistance around deeper wrongness is because of your (correct) suspicion that there's probably something there that should force updates on ALL views.
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If you're wrong in a way that forces you to update a position, but your opponent is able to stay put and gloat-derp, and it's not something shallow, you're both probably missing something. As the one who's experienced the undermining first, you get to uncover what.
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Flipping the discipline, when you see your opponent slip up on a fact and forced to update, resist the temptation to treat it as an unforced error and an opening for you to "win" a point. You have the luxury of not being forced to understand, which is precisely why you should try
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