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Dialetheism is the view that some contradictions (i.e. p and not-p) are true. The argument for this is based on the liar's paradox:
This sentence is false.
If you follow the logic through, you get the conclusion that it is both true and false. It requires some changes to Frege-Russell-style classical logic to be coherent, but it allows one to solve almost all paradoxes in one philosophical move. For example, you can have naive set comprehension principles