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So, what's the "content" in your example? I don't see that the example sentence has any content and so I don't see how it's relevant. If one were to say, "It is false." the natural response would be, "Huh?" or "What's the 'it'." There's nothing there that can be false. it's the same with the sentence, "This sentence is false." (Or, for that matter, "This sentence is true.") In order for something to be true or false, there need be something referred to.
I understand the stakes here and the ultimate conclusions that Godel came to with a related inquiry, but I can't get past the fact that there needs to be some content for the sentence to be admitted to the true or false game.
The sentence/statement, "This sentence is false." has no content, no "aboutness", and thus can't participate in the true-false game. It's like a bishop on a chessboard. Why is this explanation wrong?