On the continued theme of the paradox, I saw mentioned last night in a television programme on mathematics the alleged paradox "This statement cannot be proved." And this apparently a tangent of what is called the Epimenides Paradox, which seems to go in something like its pure form:
This statement is false.
So do we have here what Borges described as a crack in the architecture of reason where we see that the world is false? The supposed logic going, "Well, if it is true that it is false, then it is true. Which means it is false. Which means it is true." And so on. A unsolvable paradox. Logic has been breached. We are free!
The vital essence of language in the form of reason, rather than say poetry, is that at every level what one says is reasonable- makes sense. And here, as should be immediately obvious, is where this paradox gets in trouble. "This statement is false." What statement? There is no statement here, simply the referring to one which does not appear. The sentence, without the accompanying statement, is linguistically meaningless. And so it is of course meaningless to describe a non-existent statement as true or false.
"My dog is black" is a statement which may be true or false, depending on the colour of my dog, and in this correct understanding of what a statement is, no paradoxes occur. The statement exists- refers to something. "This statement" is not however a statement, and so the supposed paradox is simply resting on a foundation of gibberish. And so, alas, no paradox.
Identically- "This statement can't be proved." What statement?
Even if, and here I am giving far more respect and time to this nonsense than it merits, "This statement is false" wasn't meaningless and actually made some kind of sense, the moment it is false it makes no more sense to apply logic to it. It is illogical to treat illogical statements as logical- you don't build and form deductions from a foundation of "Two plus two equals five." Logic doesn't apply. You don't apply rational conclusions to irrational statements. That is simply, and literally, insanity: take meaningless nonsense and run with it. Logic applies to the logical implications of true statements, not false ones.
So summing up, "This statement is false" is linguistic nonsense as it isn't a statement, but in any case a logical train should not follow illogical statements.