The liar's paradox is fairly common knowledge where the sentence "this sentence is false" cannot be true or false. But there also the lesser known sentence "this sentence is true" which can be either true or false with no way to prove either case.
I'm not quite sure if I understand it...
@Sphinx If we assume "this sentence is false" (or equivalently "this sentence is a lie") is false then according to itself it must be false that it is false and so it is true. But we assumed it is false so there's a contradiction if we don't allow a sentence to be both true and false.
If we assume "this sentence is false" is true then according to the sentence it is false. So there's still a contradiction.
On the other hand "this sentence is true" if assumed to be true will lead to no contradiction since if it is true it is true that it is true. But we could also assume it is false and in which case it will be false that it is true so it will be false without any contradiction.
These sentences are interesting because they give examples where a truth value cannot be assigned to a statement despite the statement not quite being nonsense. At first it might seem like the self reference is the cause of the issue, but self-referential sentences can have truth values without issue like "This sentence has five words." The issue stems from some other feature of the sentence.