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A contradiction often indicates that a premise of the argument is wrong (like taking a wrong turn, I suppose) which means that you need to make a u-turn and go back to where you erred and correct it. This can mean that the entire argument is false (if the premise was originally held) or is partial (if the premise was dialectically established over the course of the argument).
Paradoxes can also reveal the inaccuracy of intuition or reveal non-dualistic truths.
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