Yes, it does. The paradox is that if you accept “A and you cannot prove A” as a premise, and logically derive A, then you arrive at a contradiction: A is true and you can prove it. By the usual laws of logic, this means that you should refute the premise. That is, conclude that the judge is lying.
But if the judge is lying, then what basis do you have for proving that A is true?
Of course, this whole argument may not apply to someone who uses a different basis for reasoning than classical logic with the Law of Excluded Middle.
Yes, it does. The paradox is that if you accept “A and you cannot prove A” as a premise, and logically derive A, then you arrive at a contradiction: A is true and you can prove it. By the usual laws of logic, this means that you should refute the premise. That is, conclude that the judge is lying.
But if the judge is lying, then what basis do you have for proving that A is true?
Of course, this whole argument may not apply to someone who uses a different basis for reasoning than classical logic with the Law of Excluded Middle.