"true or false: This sentence is false."
The sentence is both true and false. In answering true, the sentence is false, therefore, the answer false. In answering false, the sentence is true, therefore becoming false. You could be lying to me that the sentence is false, in which case the sentence becomes true. The sentence is false. However, you could be telling the truth, in which case the sentence is false. The sentence is true. If the sentence is in fact false, then the sentence is true. The sentence is neither true nor false, therefore having no answer. Again, if the question has no answer, the question does not exist.