I'm sorry to break it to Mark Goldblatt, but the law of noncontradiction (LNC) has never been proved. Goldblatt writes:
No human being has ever lived, no human society has ever existed, that did not accept and rely upon the validity of the laws of thought [LNC, the law of the excluded middle (LEM), identity, causality]; they are the foundation of reasoning and knowing.
Wrong. There is at least one person alive today who has produced quite workable and beautiful logical systems (within logic, that is, not in some parallel universe outside logic) that don't rely on LNC. He also puts LEM, another one of Goldblatt's laws, in trouble. He exists! Look, that's him in the picture!
The onus is on proponents of LNC to explain why it can't cope with simple sentences we say every day. And Gödel's Incompleteness Theorem. And transfinite sets. And...