The post seems to confuse the law of non-contradiction with the principle of explosion. To understand this point, it helps to know about minimal logic which is like intuitionistic logic but even weaker, as it treats ((false)) the same way as any other primitive predicate. Minimal logic rejects the principle of explosion as well as the law of the excluded middle (LEM, which the main post called TND).

The law of non-contradiction (LNC) is just . (In the main post this is called ECQ, which I believe is erroneous; ECQ should refer to the principle of explosion (especially the second form).) The principle of explosion is either or . These two forms are equivalent in minimal logic (due to the law of non-contradiction). As mentioned above, minimal logic has the law of non-contradiction, but not the principle of explosion, so this shows that they're not equivalent in every circumstance. Rejecting the principle of explosion (especially the second form) is the defining feature of paraconsistent logics (a class into which many logics fall). Some of these still have the validity of the law of non-contradiction. Anti-intuitionistic logic does not, because LNC is dual to LEM, which is invalid intuitionistically.

Ok, so I ended up taking a lot of time researching that nitpick so I could say it correctly. Anyway, I'm curious to see where this is going.