[sent to analytic]
This is a language game, not a paradox. The indicators "true" or "false" have different units of analysis. One deployment of "true" refers to the sentence, not the thing being reported in it. The other does the opposite. So if you put the sentence in quotes, the "paradox" disappears. (One wonders if it is not syntactical rather than logical paradox). Hence, "this statement is false" asserts that there is a statement, X, that is inaccurate. Saying it is true that "this statement is false" merely says that it is correct to say that X is inaccurate. If it is correct to say that X is inaccurate, that does not make X accurate, it only makes the statement accurate.
Now, I think there are two other games being played here. I think "this statement is false" is being meant (in the game) not to pick out any particular statement, X, but to refer to itself? Is that the problem? So the deployment "this statement is false" means something like "I true a false." This tries to make the speech act like a performative utterance. But even here, however, one can make sense. If one confirms that something is false (like a detective), then one, in a sense, can "true the false" (confirm falsity). But what one clearly cannot do is "true the non-true" where the sense of true is an operator, not an indicator. That would be like saying "a is non-a" or something. I think the key to this game is that it deploys truth as an operator when the brain would naturally see it as an indicator. As an operator, two falses = truth. As an indicator, two falses merely mean two falses.