What I do want to claim, perhaps, is that you don’t really understand something unless you can translate it into a transparent-context description.
Versions of this have appeared in philosophy before. The classical group of objections is over-identification. Naively, you would have to say that inertial mass and gravitational mass mean the same, as do “p is prime” and ”
But this is the tame version of the problem, because it’s already treating individual predicates in an otherwise extensional language. If you want to translate opaque messes into into extensional expressions, you’re operating on the level of sentences. The extension of a sentence is typically its truth value, which would mean there are only two of them: true and false. This leads into problems like Quines indeterminacy of translation. Modality doesn’t help: Expressions become more determined the more their truth-value, or belief in them, is observed to vary.
And while it doesn’t relate to the formal apparatus here, I’d also link my older and somewhat more galactic answer to similar problems.
I didn’t see the discussion, but it seems odd to me that a debate-aligner would grant the premise here—isn’t the whole idea of debate that informative responses do exist in very specific parts of the dialogue-tree? I don’t think the empirical case is as damning as you make it sound:
Mathematics mostly operates by the Scholastic Method. It has parts with easily demonstrable implications, but also other parts which are quite far from that, and it’s mostly fine.
And people do fail to recognise the right answer, justified with the right reasons, at times—but that is also not always the best argument (to them) for that right answer. You might have needed to present it differently, or attack the reasons people don’t accept the arguments first, or...etc. We don’t demand you do this stuff to get Science Genius Credits afterwards, but that also means you can find examples of people failing to appreciate the Perfect Science Genius, without that necessarily bearing on judging debates of Science-and-Pedagogy Geniuses.
I still don’t really expect this to work, but more so on priors.