The "Raven paradox" was used as a starting point to the famous article "Natural Kinds" by W.V.O. Quine; it is one of the two articles by Quine that set the anthology Naturalizing Epistemology in motion, as mentioned in my article immediately previous to this one at http://lesswrong.com/r/discussion/lw/kp1/from_natural_or_naturalized_to_social_epistemology/
It seems to have motivated Quine's perhaps throwing up his hands on formal methods of epistemology, and suggesting we "settle for psychology" (not sure if he used that phrase -- if not, it's a commonly used characterization of his position).
At least part of the trouble seems to be that he proposes non-black non-ravens isn't a natural kind. Non-ravens would seem to be all "things" that aren't ravens, but consider what an incoherent concept that is. Do "things" include every atom in the universe? For quite a lot of "things" (atoms included, I think) the quality of blackness makes no sense.
So maybe there are around 100,000,000 ravens in the world, and as I examine Ravens and find N black ones and no non-black ones, I can say N down, 100,000,000-N to go, and that might seem like progress. Whereas when I pick one atom (does it have a color?), one H2O molecule, one green leaf, and one blue eye of newt, I have no meaningful concept of how many more "non-ravens" there are to sample.
Now if very hypothetically, ravens belonged to a genus with just one other species, also having 100,000,000 members, and the whole universe of ravenoids was frozen in time instead of multiplying and dying as we tried to sample them, we might say upon selecting one non-black non-raven, "That's one bit of evidence that doesn't contradict my hypothesis, and when I've sampled the whole 200,000,000 in the ravenoid universe with no contradiction of the hypothesis and a number all black ravens, I can say the hypothesis is true. A black non-raven also doesn't contradict the hypotheses and is also "one more down" and goes towards the ultimately complete sampling of the 200,000,000 entities during which we hope that every raven we find will be black.
I.e. our intuition, if we have one, that {{the equivalent logical proposition "All non-black non-ravens" really should have an analogous method for gathering evidence}} might be less ridiculous if only "non-black non-ravens" actually meant something coherent.
For what it's worth there is also a 48 page 2010 article "How Bayesian Confirmation Theory Handles the Paradox of the Ravens" by Branden Fitelson and James Hawthorne (fitelson.org/ravens.pdf -- actually it's only 29 pages in this PDF due to different layout I suppose.). I've been meaning to read it, but think I'll have to work my way up to it.
The raven paradox, originated by Carl Gustav Hempel, is an apparent absurdity of inductive reasoning. Consider the hypothesis:
H1: All ravens are black.
Inductively, one might expect that seeing many black ravens and no non-black ones is evidence for this hypothesis. As you see more black ravens, you may even find it more and more likely.
Logically, a statement is equivalent to its contrapositive (where you negate both things and flip the order). Thus if "if it is a raven, it is black" is true, so is:
H1': If it is not black, it is not a raven.
Take a moment to double-check this.
Inductively, just like with H1, one would expect that seeing many non-black non-ravens is evidence for this hypothesis. As you see more and more examples, you may even find it more and more likely. Thus a yellow banana is evidence for the hypothesis "all ravens are black."
Since this is silly, there is an apparent problem with induction.
Resolution
Consider the following two possible states of the world:
Suppose that these are your two hypotheses, and you observe a yellow banana (drawing from some fixed distribution over things). Q: What does this tell you about one hypothesis versus another? A: It tells you bananas-all about the number of black ravens.
One might contrast this with a hypothesis where there is one less banana, and one more yellow raven, by some sort of spontaneous generation.
Observations of both black ravens and yellow bananas cause us to prefer 1 over 3, now!
The moral of the story is that the amount of evidence that an observation provides is not just about whether it whether it is consistent with the "active" hypothesis - it is about the difference in likelihood between when the hypothesis is true versus when it's false.
This is a pretty straightforward moral - it's a widely known pillar of statistical reasoning. But its absence in the raven paradox takes a bit of effort to see. This is because we're using an implicit model of the problem (driven by some combination of outside knowledge and framing effects) where nonblack ravens replace black ravens, but don't replace bananas. The logical statements H1 and H1' are not alone enough to tell how you should update upon seeing new evidence. Or to put it another way, the version of induction that drives the raven paradox is in fact wrong, but probability theory implies a bigger version.
(Technical note: In the hypotheses above, the exact number of yellow bananas does not have to be the same for observing a yellow banana to provide no evidence - what has to be the same is the measure of yellow bananas in the probability distribution we're drawing from. Talking about "99 ravens" is more understandable, but what differentiates our hypotheses are really the likelihoods of observing different events [there's our moral again]. This becomes particularly important when extending the argument to infinite numbers of ravens - infinities or no infinities, when you make an observation you're still drawing from some distribution.)