If the system can evaluate the truthfulness of a statement, and it can generate statements, then it can create arbitrary numbers of candidate statements and filter for truthfulness.
[Author] This is what I meant. That even if it doesn't work at a single layer, you could build a "verifier" stage whose approach is similar to this verification process and it can make the decision.
Looks like you take the "alternatively" fork in the original statemet - is it not a tautology?