Aristotle indeed couldn’t prove it, that is, to derive it as a conclusion step by step from the evidence of the senses. His reasons for this are sound: 1) an attempt to do so has to rely on PNC already, and 2) we can’t assume infinite regress.

Asking a proof of PNC would imply proving non-contradiction by some means that assumes that contradictions do exist.

PNC doesn’t need a proof; it needs validation: a process of establishing an idea’s relationship to reality, whether through deductive reasoning, inductive reasoning, or sense perception.

> logic is [not] about what exists but rather a tool

If logic describes something that is not real, then our ideas and even institutions are detached from reality, and so some people claim a right to secede from “established truths” and place anyone who disagrees outside the circle of rational dialogue. That would be a harmless academic issue if the last two centuries weren’t a living record of that detachment playing out in politics, ideology, and culture.