H1: All ravens are black.
H1': If it is not black, it is not a raven.
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.
Question: H1 and H1' appear to be logically equivalent to:
H1'' There do not exist any things which are both not black, and a raven.
And this seems to have different implications in a finite universe and an infinite universe.
For instance, in a finite universe of 10,000 things, if you've found 99 yellow bananas and 1 black raven, there are 9,900 things which could potentially disprove H1''. If you then observe an additional 100 yellow bananas, there are now only 9,800 things that could potentially disprove H1'', so it would make sense that H1'' becomes a small amount more likely, since if all of the remaining untested things were yellow bananas, and you tested them all, at the point at which you tested the last thing you would be much more confident about H1'', and presumably that confidence grows as you get closer to testing the last thing as opposed to coming all at once at only the last thing.
But:
In a infinite universe of infinite things, if you've found 99 yellow bananas and 1 black raven, there are infinite things which could potentially disprove H1''. If you then observe an additional 100 yellow bananas, there are still an infinite number of things that could potentially disprove H1'', so H1 would not necessarily become a small amount more likely because of the argument I just gave since there is no 'last thing' to test.
When I looked at http://en.wikipedia.org/wiki/Raven_paradox , I'm not sure if anything I just said is any different from the Carnap approach, except that the Carnap approach described in the article does not appear to mention infinities, so I'm not sure if I'm making an error or not.
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.)