Nozick's theory is quite cool. A simple version of it is as follows...
S knows that P iff:
- P is true
- S believes that P
- If P were false, then S would not believe that P
- If P were true, then S would believe that P
To briefly explain 'possible worlds': We live in one possible world, let's call it "actuality". But you can imagine that things could have been different, in many (indeed, infinitely many) ways. When we conceive of an alternative possibility (say, that I'm sitting watching TV now instead of typing this), then that is an alternative "possible world". The greater the discrepancy from reality, the more "distant" that possible world is from actuality.
So, to explain Nozick's subjunctive conditionals:
a) in the closest possible worlds where P is false (unlike actuality), S no longer believes that P.
b) in all other close possible worlds where P is also true, S does believe P. (How close? Nozick suggests all those worlds that are closer than the first not-P world.)
Note that we are not concerned with extremely distant worlds - that is, it is possible to "know" something even if you haven't ruled out some (extremely unlikely) alternative possibilities.
Now, when this theory is applied to skeptical scenarios, we get the following results:
1) I know that I have two hands
but 2) I don't know that I'm not a (handless) brain-in-a-vat (BIV).
To explain why:
1) I have a true belief that I have two hands. In the closest possible worlds where I do not have two hands, it is because I lost them in some sort of accident, which I of course would be well aware of, and so I would then not believe that I have two hands. Furthermore, in all close worlds where I do have two hands, I also believe that I do. All the conditions are fulfilled, so I have knowledge of this fact.
2) I have a true belief that I am not a BIV. But in the closest possible world where this is false (i.e. where I am a BIV), I nevertheless would still believe that I was not one (since I would have had all the same conscious experiences as I have now). The third condition for knowledge is NOT met, so I do NOT know that I'm not a BIV.
Nozick embraces these results, but surely the contradictory nature of these abominable conjunctions counts against the theory. Fortunately, DeRose found a way to iron out the inconsistencies, by combining Nozick's theory with Contextualism.
Contextualism says that the epistemic standards required for knowledge will vary, and are context-dependent. The basic idea is that according to normal standards we know that we have hands (and thus DO know that we're not BIVs), but according to higher standards we don't (and don't know that we're not BIVs).
DeRose's key insight was to embrace Nozick's conditional theory and its 'possible worlds', but to drop the strict requirements, and instead allow the number of possible worlds under consideration to vary according to context. The strength of S's epistemic position (with regard to belief P) depends on how well S's belief in close possible worlds matches the truth in those respective worlds. The more possible worlds there are where S would have an accurate belief about P, the stronger S's epistemic position is (here and now) with regard to P.
He offers a visual metaphor here which I found illustrative: Imagine our world ("actuality") surrounded by all the close and distant possible worlds, modelled in a sort of 3-d space. Now imagine a sphere ballooning out from the centre (i.e. the actual world), but stopping the moment it comes to a world where S fails one of the two subjunctive conditionals previously mentioned (i.e. #a- S believes P when it is false, or #b- S doesn't believe it when it is true). Equivalently: S believes that P iff P is true in that possible world.
Now, the size of this sphere describes the strength of S's epistemic position. If small, then S can only claim to know P according to very low standards, whereas if the sphere is very large, then S's knowledge is correspondingly strong (after all, it means that S would be very unlikely to have formed a false belief here... things would have had to go very differently (since it is a very distant world) for S to have been mistaken). By restricting comparative knowledge claims to similar epistemic contexts (i.e. sphere-sizes), you avoid the contradictions entailed by Nozick's theory.
So, to apply this new & improved theory to the skeptical scenarios:
1) I have very strong knowledge that I have two hands (because you would have to go out to a VERY distant world before I had a mistaken belief about this!)
likewise, 2) I have very strong knowledge that I am not a BIV. How is this? Well, once again, you have to go out a very long way before you can find a possible world where I would have a mistaken belief about it.
So why do we find the skeptic's argument so convincing? Why do we feel that we don't know that we're not BIVs?
DeRose's answer is that mentioning skeptical scenarios raises the standards of knowledge so high that we cannot meet them. So (according to those extreme standards), the skeptic is right... our "sphere of knowledge" is not large enough to extend all the way out to those (extremely distant) possible worlds.
Just one thing remains to be explained: how does this 'raising of standards' occur?
DeRose posits the Rule of Sensitivity, which effectively says that when knowledge about P is mentioned, we tend to raise the standards of knowledge (if need be, and for an appropriate duration) so that S's belief in that particular P becomes sensitive. That is, the sphere of knowledge must extend all the way to the first possible world where P is false (like Nozick originally suggested).
That means that whenever anyone mentions BIVs, suddenly the standards for knowledge are raised to this absurdly large sphere, which must extend out all the way to that (extremely distant) possible world where S is a brain in a vat! It is no surprise that we do not have knowledge according to these excessive standards.
But, thankfully, our knowledge in more normal contexts is preserved. For most of our beliefs which we think we 'know', our sphere of knowledge extends out quite a fair distance, so we do have knowledge, according to contexts where the required standards are more moderate (i.e. smaller than the size of our sphere).
Thus the Skeptic is refuted, without any bizarre side-effects. If anyone knows of (or can think of) any objections to this solution, I would be very curious to hear them...?