Are All Non-Black Things Non-Ravens?

The Paradox of the Ravens is a famous puzzle for the idea that theories are confirmed by positive instances. Carl Hempel pointed out that, in first-order logic, the proposition that all ravens are black is logically equivalent to the proposition that all non-black things are non-ravens. On the further plausible assumption that a piece of evidence that confirms a hypothesis should confirm anything logically equivalent to the hypothesis, we get the result that any non-black non-raven -- for example, a white tennis shoe -- confirms the hypothesis that all ravens are black. This seems a very peculiar result. Perhaps Bayes' Theorem can give us some insight into it.

Consider a case similar to the ravens case but in which we can use some semi-real data. Suppose I visit a small campus -- perhaps my alma mater, St. Olaf College -- and after seeing large numbers of blond, blue-eyed students with names like Christiansen, Rolvaag, and Ytterboe, I hypothesize that all the students at the college are of Norwegian descent. I can try to confirm this by examining individual students to see whether they are of Norwegian descent. P(E|T) = 1, the prior probability, let's say is .5, and let's say that P(E|~T) = .9, because I am pretty sure that even if the students are not all Norwegian, most of them are, so that the chances that the next one I observe will be Norwegian are pretty good even if my hypothesis is wrong. So what effect does one more Norwegian student have on my assessment of my hypothesis? Plug the numbers in and see!

If your results matched mine, the probability didn't go up very much, because the odds that the next student would be Norwegian were pretty good even if they aren't all Norwegian. But there was some discernible movement in the positive direction.

Now consider attempting to confirm the hypothesis by finding non-Norwegian non-Oles. P(E|T) and P(T) are unchanged. But what is the probability that the next non-Norwegian will be a non-Ole if my theory is false? Well, there are six billion or so non-Norwegians in the world and only around 3,000 Oles, so we should expect that the odds that the next non-Norwegian will be a non-Ole are around 5,999,997,000/6,000,000,000 = .9999995. Plug those values in and see what happens!

You probably didn't see any change at all between the prior and posterior probabilities. Does this mean the probability did not go up at all? Or is it just that our calculator isn't precise enough to show the change? Let's try a calculator that gives us more decimal digits:

So we've confirmed the hypothesis by a very, very tiny amount! Is this perhaps the right way to understand the ravens paradox as well? It's true, as the model of confirmation by positive instances holds, that positive instances confirm a generalization (unless they are equally likely if the theory is false). And it is true that what confirms a generalization also confirms anything logically equivalent to that generalization. What is missing from the traditional confirmation-by-positive-instances account is any sense of the degree of confirmation. When we take this into account, we see that the suspicion that, for example, non-Norwegian non-Oles do not confirm the hypothesis that all Oles are Norwegians is partially vindicated. They do confirm the hypothesis a little, but it is so little that it is negligible.