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    Contradictions are incoherent nonsense. Not just from intuition, but from ambiguity in application. What does being a Wulture, and also a non-Wulture even mean? Can you differentiaite such an object from anything?
    – Michael Carey
    Commented May 5 at 23:29
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    You might be interested: philosophy.stackexchange.com/questions/61357/…
    – Allure
    Commented May 6 at 4:03
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    I believe if you forbid either $$P \rightarrow P \vee Q$$ or $$(P \vee Q) \wedge \neg P \rightarrow Q$$, you can come up with a paraconsistent logic, though it is significantly weaker than standard propositional logic
    – Kaia
    Commented May 6 at 22:56
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    "So, can somebody please give me a justification of the law of non-contradiction which doesn't just rely on intuition?" It does not seem like your doubts have anything specifically to do with the law of non-contradiction. You can equally well ask for a justification of Modus Ponens (from "A" and "if A, then B" infer "B") which "doesn't just rely on intuition", or a justification of disjunction introduction (from "A" infer "A or B") which "does not just rely on intuition", etc. So it seems a bit misleading to focus on non-contradiction when really your doubt is about justifying logical laws.
    – Adam Přenosil
    Commented May 7 at 2:58
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    the set either contains all vultures, including white vultures, OR it excludes things which are white, including white vultures. white vultures are either in the set or they aren't. if you can say that some things are both in and not in the set then your system of logic is not going to be very useful. however I don't think there is a justification for the law of non-contradiction; it is axiomatic.
    – Dave Cousineau
    Commented May 7 at 20:13