I'm going to be discussing an argument that I know Jason Stanley to have given, but I'm away from my copy of his book at the moment, so I can't cite it properly, or check and see who else has discussed it (or even whether it's original to Jason). I'll follow up if citation protocol ends up demanding it.
Here's a naive argument against 'knows' contextualism. (This isn't the Stanley argument I want to discuss; it's part of the set-up for it.) Assume contextualism. Now suppose you're in a nonskeptical context, and I'm in a skeptical one, and we're both talking about me and the proposition that p, a proposition with which I stand in a pretty strong epistemic relation -- one strong enough for your non-skeptical 'knows', but not for my skeptical 'knows'. You say: "Jonathan knows p." Now, according to this naive objection, I'm forced to say this:
(1) I don't know that p, but what you said was true.
This sounds like a crazy thing to say, under the circumstances, but it looks like contextualism predicts that it should be fine.
Of course, contextualism doesn't predict that (1) should be fine; the naive objection is naive. Contextualism avoids the felicity of this utterance by observing that it won't be assertable for me. What you said entails p (even your nonskeptical 'knows' is factive). Since I'm not in a context in which "I know that what you said is true" is true, I therefore can't assert that what you said is true. Indeed, (1) is Moore-paradoxical, or near enough, since it straightforwardly and transparently entails "I don't know that p, but p."
So there's a naive objection to contextualism and a good response on the contextualist's behalf. But Jason Stanley thinks the game isn't over yet, for he has a tweak on the objection to make it less naive in a way that he thinks will avoid the response. Take the same set-up as before; you're in a non-skeptical context and you say "Jonathan knows p," and I'm in a skeptical context where "I know p" is false. Now, Jason asks, why shouldn't I give this sentence?
(2) I don't know that p, but if p is true, then what you said is true.
Here, the second conjunct doesn't entail that p, so the sentence isn't Moore-paradoxical. Insofar as (2) also sounds crazy, we have a version of the objection that isn't susceptible to the quick response given above. (Does (2) sound crazy? Is it a natural enough sentence to generate clear intuitions? I don't know. Let's grant for the purpose of argument that it sucks to have to render this sentence assertible.) The contextualist, I contend, is not committed to the assertibility of (2). Although (2) is not Moore-paradoxical, because the second conjunct does not entail p, its infelicity can still be explained as a violation of the knowledge norm of assertion, since it's second conjunct will, in the relevant cases, be unknown.
Stanley's argument suggests that the contextualist is committed to my 'knowing' (that's skeptical-standard 'knowing' -- let's call it KNOWING and contrast it with low-standards knowing) the second conjunct of (2). But there is no such commitment. It may well be that, for all I KNOW, p is true but I don't know that p. Since I can't assert what I don't KNOW, (2)'s second conjunct is unassertable, and so therefore is (2) itself. Stanley seems to be assuming that any time I know that p, the only thing preventing me from KNOWING that I know p is failure to KNOW p. There is no reason to make this assumption, especially given the failure of the program of factorizing knowledge (in this case, knowledge) into truth and some independent conditions. So the contextualist is not forced to admit that (2) is felicitous; he has room to insist that it cannot be KNOWN, and so cannot be asserted.
This move is not ad hoc, for at least two reasons. First, that making this move allows the view to capture the intuitive datum that (2) is infelicitous is a legitimate motivation for making it. Second and more impressively, the move is supported by general principles involving quantification over relevant possibilities. I'm just going to apply Zagzebski's recipe for Gettier cases: for any non-p-entailing condition C, modifying C by making p accidentally true will produce a Gettier case vis-a-vis p. If in my context, "I know p" is false, then there is some skeptical possibility s, relevant in my context, and uneliminated by my evidence, in which not-p. Since "Jonathan knows p" is true in your context, s is not relevant for you. Now here's the key move. For any such s, there is another possibility, s*, saliently similar to s, in which p is accidentally true. And given plausible metasemantic principles about quantifier domain restriction, s* will be relevant if s is. That is to say, in any context in which a skeptical scenario is relevant -- in any context in which there are not-p worlds consistent with the evidence -- there is also a Getter scenario relevant: a scenario just like that not-p world, with just the same evidence, but where p is true. But these s* worlds are p worlds in which I don't know p -- they're Gettier worlds. Since they're relevant in my skeptical context and uneliminated, I can't, therefore, KNOW that if p, then I know p. (On a strict conditional account of indicatives, it won't even be true in my context that if p, I know p; but on any account, I at least won't KNOW this conditional, since the conditional must at least entail the material conditional, and we've established that an antecedent-world that is not a consequent-world is epistemically possible. That's enough to explain the infelicity of (2).)
An example will illustrate the intuitiveness of the constraint I am suggesting. You and I are both talking about me and the proposition that I will play cards tomorrow. I am planning to go to play poker in Atlantic City tomorrow, and we both know this. (And, as a matter of fact, let's stipulate, I will carry out this plan.) You say, in your nonskeptical context: "Jonathan knows that he will play cards tomorrow." But I'm in a skeptical context; "I know that I'll play cards tomorrow" is false in my mouth, because I'm in a context in which certain skeptical possibilities are relevant. Suppose I'm treating as relevant the possibility that the weather will be horrific, so that I'll decide not to go to Atlantic City after all. (To make your sentence plausible, we stipulate that this is a very unlikely possibility that you legitimately ignore.) So there are some relevant-for-me skeptical scenarios in which I have my actual evidence, but I don't play cards tomorrow.
But that there are such scenarios relevant for me requires that other scenarios saliently similar also be relevant to me. The s that undermined KNOWLEDGE was a case in which the weather is terrible and I stay home and didn't play cards; but that world is so very much like an s* in which the weather is terrible and I stay home and play cards with friends instead of at the casino. This s* possibility is a p world -- I play cards in it -- but if it obtained, I wouldn't know that p, since it represents a Gettier case.
So a contexutalist can and should think that (2), like (1), should be unassertable, even if he allows that it might be true.
One final note to add: I've been using this move to defend contextualism, but I think it might work about as well against a parallel argument against a Jason-style 'SSI' kind of view. I'm not sure. "I don't know that p, but if p, then S knows that p" might be explicable in the same kind of way: if I don't know p, then there is some s that undermines my knowledge of p. Can we infer from this that there is also some s*, constructed by modifying s to make it accidentally p, that undermines my knowledge of the conditional? I filled in this move for the contextualist with a principle about what determines the domains of restricted quantifiers; could an SSI proponent do something similar? I'm not sure.