> The basic constraint is that if you want to prove something based on purely logical statements, you need to express the problem precisely enough for pure logic, that is without any wiggle room open to vauge human interpretation, and the notions you use have precise meaning. If not, your problem can't be solved by pure logic, but only by more wishy washy handwavy arguments, from which you can't derive any insights with logical certainty.

I think when you say "pure", you actually mean "formal" or "symbolic" or "mathematical". Many believe that there is much more to logic than a mere formalism – although that itself is a major point of contention in the philosophy of logic.

> You might be able to derive philosophical or empirical meaning from an interpretation of some non-precise statement, but you shouldn't hope that such an interpretation is a valid mathematical theorem.

Berry's paradox may not be a mathematical paradox – but that doesn't show it is not a paradox. And, even if it is not a mathematical paradox, it nonetheless made a valuable contribution to mathematics, by inspiring the construction of formal analogues, some of which turned out not to be paradoxical (but nonetheless interesting). However, I don't think it necessarily follows, that just because those formal analogues turned out not to be actual paradoxes, that its natural language version cannot be a real paradox. And, some argue there are formal analogues which are genuine paradoxes – see for example https://ojs.victoria.ac.nz/ajl/article/download/4972/4634/73...