Iterated Knowledge isn't Better Knowledge

Recent work in epistemology implicitly assumes that we can measure the quality or strength of someone’s knowledge (whether understood intuitively, or by its normative connections to action, inquiry, belief, or assertion) by the number of iterations it permits. I show that this idea is hopeless, because, even in set-ups that look maximally friendly, one can construct cases where someone goes from having available only a single iteration of knowledge that p to having arbitrarily many such iterations, without their knowledge that p becoming better in any way.