Because the present questioner refers to my reply to Question 5536, I'll chime in here to clarify what I said there.
My point was about the fundamentality of LNC. I wrote, "I can't see how there could be any law more fundamental than the law of non-contradiction (LNC)." I gave the following reason: "Let F be any such law. If the claim 'F is more fundamental than LNC' is meaningful (and it may not be), then it conflicts with the claim 'F isn't more fundamental than LNC' -- but that reasoning, of course, depends on LNC." So that's why no law could be more fundamental than LNC, because LNC would need to be true before (in the sense of logical priority) the claim that some other law is more fundamental would even make sense.
If someone can make sense of the claim that some law is more fundamental than LNC, I'm all ears.