Consider this proposition:

N: Brian does not know that N.

Assume N is false. That is, I know that N. Knowledge is factive, so N. That contradicts our original assumption. So N must not be false. So it follows, at least classically, that N is true. So I don’t know N.

But I can follow the reasoning that showed N is true. And I accept that reasoning, so I believe N. And the reasoning justifies me in believing N. So I have a justified true belief that isn’t knowledge. So the JTB theory of knowledge fails.

My reasoning didn’t go via any false lemmas. It went via a false assumption, but making false assumptions for purposes of reductio is consistent with knowledge. So I have a JTB with no false lemmas, but no knowledge. So much for the JTB+No false lemmas.

I’m (generally) a competent logical reasoner. My belief in N, which is a true belief, was a product of my logical competence. Indeed, I formed the belief in N, rather than some alternative, because of that competence. So I should have Sosa-style animal knowledge of N. Indeed, I can reflectively, and aptly, endorse the claim that my belief in N is accurate because it was an exercise of competence. So I should have Sosa-style reflective knowledge that N. But I don’t; clearly I don’t know N.

It seems to me that pretty much any otherwise plausible theory of knowledge will fall this way. Whatever qualities or virtues a belief might have, short of knowledge, my belief in N has. But I don’t know N. Indeed, logic prevents me from knowing N. So any such theory must be false.

N also undermines various proposals people have relating knowledge to other things. Some people think knowledge is a norm of belief. But there seems to be nothing wrong with my believing N on the basis of the reasoning above, even though I don’t know N. So knowledge isn’t a norm of belief. Many people think knowledge is a norm of assertion. But I don’t see why I shouldn’t assert N. I have a deductive argument that it is true after all; I simply don’t know that it is true. So knowledge isn’t a norm of assertion.

I’m not sure whether N alone could knock out Williamson’s thesis that all and only evidence is knowledge, commonly known as E=K. But N’s good friend E can do the trick.

E: Brian’s evidence does not include E.

Assume E is false. Then my evidence includes E. Either evidence is factive or it isn’t. If it isn’t, then E=K is false for independent reasons. If it is, then it follows E is true, contradicting our assumption. So E is true. Since I can follow this argument competently, I know its conclusion is true. (Unlike the argument about N, logic doesn’t stop me knowing E is true.) So I know E, but E is, as it says, not part of my evidence. So E=K is false.

Note that this argument doesn’t touch the plausible view that “evidence is all and only our non-inferential knowledge”:http://www.philosophersdigest.com/philphen/fallibilism-epistemic-possibility-and-concessive-knowledge-attributions-trent-dougherty-and-patrick-rysiew. Even if I know E via that argument, it is clearly inferential knowledge. So while I can refute all theories of knowledge with self-referential propositions, I can’t refute all theories of evidence.