Graham Priest has this idea that there can be true contradictions that some contradictions are true. His view is not that all contradictions are true, but just some, and he thinks that we should only accept contradictions to be true if we have no other options in thinking. He, for example, thinks the liar paradox is a good example of a true contradiction. So, if you say, ‘this sentence is false’. If ‘this sentence is false’ is true, then it's false, but if ‘this sentence is false’ is false, then it’s true, so it's always contradictory.
I wanted to point this out because the principle of non-contradiction that we inherited is an artifact of the debates of Aristotle's time; he is often credited with at least the version of it that became such a bedrock of our inheritance. Yet it didn't always exist; in Socrates' day it was open to debate, and has recently become so again (as the interview goes on to note, most philosophical debate is back open again, some of it for the first time in ages).
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