Consider a skeptical hypothesis H, such as "I am a bodiless brain in a vat," and an ordinary knowledge claim O, such as "I have hands." Adapting the suggestions of Keith DeRose (1995, "Solving the Skeptical Problem," The Philosophical Review 104.1), we may claim the following:
(1) I don't know that ~H;
(2) If I know that O, then I know that ~H;
(3) I know that O.
From these three individually-plausible premises a contradiction appears to follow, and this motivates us to seek a solution to what we can term the "skeptical problem." DeRose's solution is to appeal to epistemic contextualism, which may allow us to affirm (1), (2) and (3) all at once without contradiction. In particular, an epistemic contextualist is free to suggest that the knowledge we have of O is a different sort of knowledge than that we seek for H. On this view, there is a context C1 in which we don't know that ~H, and a context C2 in which we do know that O. Thus the premises may be reformulated as follows:
(4) I don't know/C1 that ~H.
(5) If I know/C1 (know/C2) that O, then I know/C1 (know/C2) that ~H;
(6) I know/C2 that O.
Clearly, this interpretation allows us to affirm all three premises simultaneously without contradiction, hence solving the skeptical problem. So we may consider two remaining questions.
Is the contextualist interpretation of the skeptical problem correct?
At least we can be confident that some form of epistemic contextualism is true, as shown by DeRose's classic bank cases (1992, "Contextualism and Knowledge Attributions," Philosophy and Phenomenological Research 52.4) and other straightforward illustrations. So what we know in one context might be different than what we know in another context, and this makes it possible for us to simultaneously affirm for a fixed P what is expressed by utterances of the form "I know that P" and "I don't know that P," insofar as we may speak across contexts. However, just because sometimes what counts as knowledge can vary from context to context doesn't mean all knowledge is contextually varied. In particular, it is not necessarily the case, even given epistemic contextualism, that there exist distinct contexts C1 and C2 such that we can interpret (1)-(3) as equivalent to (4)-(6).
To investigate this, let us consider successively the context in which we decide that (3) is true and the context for deciding (1) is true. Regarding (3), our first instinct is to think of how silly we should sound were we ever to express genuine doubt for it. The idea here is that O is the sort of belief that people not only take for granted, but expect others to do the same. In this way, doubting O is likened, whether rightly or wrongly, to a kind of useless sophistry, which in turn is seen as a vice to be avoided. Even if we discover that can't logically justify O, our commitment to logical justification is weak enough and our commitment to O is strong enough that we all expect each other to remain nonetheless firmly and unshakably convicted of O. To doubt (3) is seen as betraying a doubt of O, and hence we deem it similarly intolerable.
The context, then, of (3) as we normally evaluate its truth appears to involve our widely-shared tendency not to question basic assumptions about the way the world works. In contrast, the context of (1) runs quite contrary to this tendency; for it consists in a direct challenge to those everyday assumptions. We conclude that (1) and (3) involve different contexts, and indeed since ~H is among the underlying assumptions of the context of (3), this quite clearly explains apparent inconsistency of (1)-(3). Thus the contextualist interpretation (4)-(6) of utterances (1)-(3) appears to be correct.
Do any further skeptical problems remain on the contextualist solution?
Philosopher John Koethe remarks of this contextualist interpretation that "as a response to scepticism it strikes me as a nonstarter" (2005, Scepticism, Knowledge, and Forms of Reasoning, p72). According to him, we must be able to affirm (3) given any "legitimate" context whatsoever. For he insists that the truth of (3) is "simply obvious," and to deny it even under the umbrella of something like C1 would be "just wrong." This makes C1 illegitimate as a context for knowledge claims, and hence (4) an unreasonable interpretation of (1).
Instead, he suggests that people tend to assume the allegedly fallacious principle that knowledge is incompatible with the possibility of the contrary. So, for instance, we might mistakenly suppose that since H is possible then we do not know that ~H. In this way, he implies that (1) only appears to be true if one holds to that principle, and on this view the natural solution to the skeptical problem is to deny (1) regardless of context.
On the other hand, I don't think Koethe's objection stands up to the contextualist interpretation when we consider that it offers a compelling explanation for why our intuition resists any denial of (3). In particular, even if it is appropriate to deny (3) under C1, our intuition persistently moves us to consider only C2, where (3) is clearly true; indeed this phenomenon seems evident upon self-reflection. So at the very least, my personal intuition is satisfied by the contextualist interpretation, and whatever remaining skeptical problems of the sort expounded in by Koethe have only to do with his own personal intuitions and those of others like him.
A stronger objection, in my judgment, to the contextualist solution, turns on the plausible principle that for any fixed contexts D and E, and proposition P,
(7) If it is not likely/D that P, then I do not know/E that P.
In other words, although we might be able to strengthen the likelihood of P by moving to a narrower context, something which is not likely at all in one context cannot be known in another. Furthermore, given a sufficiently broad context D, the following appears to be true:
(8) It is not likely/D that ~H,
and given probabilistic closure we also have
(9) If it is likely/D that O then it is likely/D that ~H.
It's easy to see that (6)-(9) are inconsistent, and so this constitutes a remaining skeptical problem which requires resolution. I shall not offer one here, except to say that while it might be intuitively tempting to deny (8), it seems to me instead that (8) is true but (7) is false.