By "our particular logic" I mean the particular method we've learned for exploiting how the universe works to cause our discrete symbols to have consistent behavior that mostly models the universe. There's no requirement that logic be only represented as a finite sequence of symbols generated by replacement rules and variable substitution from a set of axioms; it's just what works best for us right now. There are almost certainly other (and probably better) representations of how the universe works that we haven't found yet. For instance it seems like it would be really useful to have a quantum logic that "just worked" by being made out of entangled particles and having rules that exploit quantum mechanics directly instead of having to simulate how the wavefunction behaves using our mathematics. They both might be able to fully embed the other but I think it's worth making a distinction between them.

Which thing Harry said? Harry said lots of smart things in HPMoR. And some stupid things, too.

The last thing Harry was quoted saying in the post, specifically.