It is self-contradictory and fundamentally incoherent to invoke the universal, invariant, objective laws of logic to oppose objective knowledge. An autonomous subjectivist or relativist cannot justify the standards of rationality to which he appeals. His attempt to destroy rationality and knowledge is self-refuting.
He appeals to those constructs to argue his case to those who ARE bound by them, who DO believe in them. Yes, he would readily admit that if he cannot know anything, then he cannot know that he cannot know anything. But he's not trying to convince himself. He's trying to convince someone who actually believes he CAN know something, and that logic IS binding.