I'm not sure if I'm engaging in some sort of circular logical trap but I don't really think "this sentence is false" is all that logically problematic. But it would be helpful if someone could fix up my thinking on this if I've misstepped.
"This sentence is false" is correct that it is false but we find it problematic because we conflate correctness with truth. Undergoing a proof by contradiction, in my mind, is a kind of cross-examination of hypothetical correctness with regards to what a statement purports. But when we engage in such an indirect logical proof we are dealing with the sentence in two different ways, and then conflating them.
- The sentence as the sentence
- The sentence as a claimant of truth
The sentence as the sentence, with a specific (semantic) relationship to truth and falsity, is correct insofar as we can't prove it incorrect. Whereas the sentence as a claimant of truth by our instantiation of it through cross-examination, has a specific (logical) relationship to truth and falsity.
We can say the semantic sentence has a relationship to itself which is correct in that it says, without anything to contradict it, that it is false - as in not true, and not making a logical claim to truth.
Thus, this sentence is correct that it is false, because we cannot disprove its correctness through a proof by contradiction regarding its hypothetical truth or falsity, and therefore in a sense the paradox is simply false - semantically, and true - logically.
Apologies if I'm wasting people's time with this question. Please help me smooth over this difficulty.