The reason logic doesn't work is because beliefs are generally not conclusions; they're assumptions. If two people don't have the same set of premises (assumptions), then all logical conclusions are irrelevant. However, most of the time, failing to convince people via logic happens because you simply don't understand what their assumptions actually are, and so you can't argue from them properly. If you could figure out how they concluded what they did, you can point out flaws that they'll agree with. Maybe.
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