The Raven paradox is a good example of why we need Bayesian inference. The paradox is not a real paradox -- it doesn't imply true and false at the same time -- but it makes a strong case for a system that integrates degree of confidence through and through, as opposed to merely assigning true and false.
I'd go with Dave Cole. From the Wikipedia article: "when one finds a black raven, it is a much more important discovery than finding a non-black object simply because there are fewer black ravens than everything else"
Finding a black raven yields evidence that all ravens are black.
Finding a green apple also yields evidence that all ravens are black, but very weak, much weaker than finding a black raven. It offers the same weak (and in this case misleading) evidence that all ravens are pink.
Non-paradox, since each non-black object that is not a raven will infinitesimally increase your belief in the proposition, until you find a counterexample or exhaust the set of all non-black objects.
The reaseon that finding a black raven is more important is because the set of ravens is much smaller than the set of non-black objects, so it increases your certainty more.
The green apple does indeed offer the same infinitesimal evidence that all ravens are pink, but by going through the set of non-pink objects in the universe and eventually finding a raven, the proposition will be disproven.
So part of the "paradox" here is a conflation of "evidence" and "proof". The other part of the paradox lies in the fact that the evidence gained by seeing one green apple is SO tiny that it won't even register on the human brain's "is this evidence?" heuristic.
"The green apple does indeed offer the same infinitesimal evidence that all ravens are pink, but by going through the set of non-pink objects in the universe and eventually finding a raven, the proposition will be disproven."
I had residual doubts about these pink ravens, but that's a great way to put it. Now I really feel "on top" of the paradox.
What came to my mind when I first read the paradox was that the class of ravens is closed whereas the class of non ravens is open. We have a definition of raven that's independent of its colour, otherwise we wouldn't be able to tell whether a particular thing is a raven, and thus, whether its colour is evidence for anything related to ravens at all. However, we don't have a definition for all non raven things, so the number of non-ravens is not merely large, as the article suggests, it is in fact infinite as counting needs definition to tell one thing from another. Drawing conclusions by induction from one of an infinite number of instances seems meaningless to me.
But there is another interesting question that arises from the Bayes explanation at the bottom of the article. If we had all things in the universe available and sufficiently defined to tell what is one thing and what is another, and we found that there are indeed no non-black ravens, that would still leave open the question of whether there could possibly be non-black ravens in the future. They could be born right in this moment, so there would be a race condition between making the statement and the statement being falsified by the event of a non-black raven being born. And I think that's one of the problematic things about purely extensional logic advocated by philosophers like Quine.
"If we had all things in the universe available and sufficiently defined to tell what is one thing and what is another, and we found that there are indeed no non-black ravens, that would still leave open the question of whether there could possibly be non-black ravens in the future."
That's not a bug in Bayesian inference, that's a feature! It would be a mistake to ever assign a probability of 0.0 or 1.0 to any statement. It means you have infinite confidence in the statement, which is impossible unless you have a screwy prior distribution.
You also don't have to evaluate all of the data if you're giving a probability. You can just report your degree of confidence in the proposition based on the data you've evaluated so far, and this degree of confidence is called a probability.
I know the whole point of Bayesian statistics is that you actually need much less data to get a good prediction. But the edge case I was talking about is still interesting I think, if not as a criticism of Bayes.
Yes absolutely, but logic aspires to be useful in the world. Therefore it seems important whether all ravens are black because all white ones have been culled a minute ago, because the word raven is defined as a black bird with some additional features, or because the genetic pattern of ravens causes them to always be black. Each of those possibilities would have very different consequences for inductive reasoning.
I don't get why this logic violates intuition. The proposed solution in the article is fairly obvious and intuitive. Just because two logical statements are equivalent doesn't mean evidence for one gives evidence for the other:
"The origin of the paradox lies in the fact that the statements "all Ravens are black" and "all non-black things are non-ravens" are indeed logically equivalent, while the act of finding a black raven is not at all equivalent to finding a non-black non-raven. Confusion is common when these two notions are thought to be identical."
If two logical statements are equivalent, then yes indeed, evidence that implies one of them (in classical logic) implies the other one. The solution is indeed intuitive but is not formalizable in classical logic.
Classical logic does not, in general, jibe with the intuition: consider the case of the negative antecedent:
hmm, I see the point. But isn't that just the limitations of classical logic? Just seems like a question of semantics rather than a clash between logic and intuition, but maybe I don't get it
It is a clash between classical logic and intuition, which is the point -- it indicates a difficulty with applying classical logic, and encourages us to take up other avenues.
The statement that "my pet raven is black" is not even weak evidence that "all ravens are black." Rather, it's strong evidence that "there exist black ravens." That's the flaw I see.
