Disclaimer: I'm not well-read in philosophy or epistemology, and haven't studied the subject formally at all. If this argument is actually widely known and not original, I'd appreciate that being brought to my attention, and apologies in advance if I communicate my ideas poorly.

The argument attempts to prove a form of academic skepticism. It doesn't try to truly prove that knowledge is impossible but relies on the strategy outlined here in the SEP:

One way of understanding the so-called problem of the “Cartesian Circle” illustrates the Pyrrhonian point: Descartes is relying throughout the Meditations on his power of reasoning to remove the skeptical doubts that he raises, but to do so requires that he exempt at least some of the propositions obtained through reasoning from the doubts that he raised in the “First Meditation” about the epistemic reliability of our faculties. A possible Cartesian reply could be as simple as paraphrasing Luther: Here I stand, as a philosopher with confidence in reason, and as such I can do no other.

In other words, I am only trying to show that, according to our epistemological framework, nothing can be proven. I am not claiming that we can, in some non-contradictory manner, reliably know that we cannot know anything.

The argument itself is very much like vanilla Cartesian skepticism: you could be mistaken about anything! Even if you have a clever argument for why fallibility doesn't imply doubt, for why our axioms are irrefutable, that skepticism presupposes them or that it isn't even a coherent statement to doubt them, you could be mistaken in your reasoning.

Yes, I know what you're thinking, that we have to examine the skeptical argument and not just assume it - but it doesn't seem to me that one needs to 'break it down' at all, or try to define what it really means to claim something is mistaken. In fact, my whole argument fails if you do. I've never heard a philosopher claim that all the conclusions in his paper are infallible and that to dispute them would be insanity (how could a true infallible belief even be disputed?). So it seems like everybody accepts that our reasoning is inherently fallible. Even if you believe that some statements or axioms are infallible, your argument for why they are surely isn't.

Another way to examine it: assume one of two possibilities, the first being that your reasoning for how we can evade the problem of skepticism is valid, and the second that it is not valid. How do you tell the difference? We cannot even verify our seemingly infallible sensations, since our reasoning for why they are infallible is subject to error (I'm not saying that we can't have direct infallible contact with qualia, only that the rational mind has no epistemic access to it).

Dialectically, this takes the form of the skeptic responding to every argument by saying 'but you could be mistaken, how do you know that you aren't'. And if his opponent argues for why that isn't a good objection, the skeptic can say the same in response to that reasoning, and so on ad infinitum. What kind of meta-argument could put it to rest? As I said, it is a commonsense idea - everyone must accept that our cognition is fallible - so the skeptic's own premise doesn't have to undergo the same cross-examination. If we parrot his own response back at him, he shrugs his shoulders and says, "Yeah, I guess so. I could be wrong." It simply isn't a defeater for skepticism of this sort.

So, that's basically it. As I said, I'm looking for more on this if something exists. The argument seems intuitively very hard to escape, and seems just a little too easy, so it's possible that I'm missing something critical that will make me feel stupid when I realize it. Right now the only counters I can see would be showing why one cannot hold this sort of commonsense premise, or to accept the whole argument but argue that it isn't even relevant for building an epistemology (not sure how, but there are all sorts of far-out philosophical positions, so I'm holding out hope that there are things I haven't even thought of yet).