The Raven Paradox

I cannot believe that anyone is still discussing this, but Sabine Hossenfelder did recently, as did UpAndAtom in mid-2019. Both present it as a serious issue for the idea of evidence and hence the scientific method.

The paradox is due to Carl Hempel, one of the many philosophers who circled round Rudolph Carnap and the Vienna School. They loved them some logic, and this really is.

Consider the hypothesis "All ravens are black". Evidence for this would be a black raven. A counter-example would be a white raven. So far so obvious. But "All ravens are black" is logically equivalent to "All non-black things are non-ravens". The evidence for that is a white tennis shoe and a red tomato. So on the principle that two logically equivalent statements should have the same evidence base, white tennis shoes are evidence for "All ravens are black". Which of course they aren't.

Which is supposed to be a paradox.

Which it is only if we stop to admire it for too long.

It isn't a paradox. It's a sign that our idea of what counts as evidence is nuanced enough to distinguish between statements that are equivalent in the predicate calculus. Nothing says that logical equivalence trumps all other forms of equivalence or lays waste to all other distinctions. Unless you're the kind of person who hung out with the guys at Carnap's Bar and Grill.

The Raven Paradox is a useful edge case: a theory of evidence should not fall foul of it.

Notice that to a falsificationist, there is no problem here. Confirmations don't count, only falsifications. White shoes do not refute the raven hypothesis, but falsificationists do not count the number of refutations, as inductivists do count confirmations. One refutation is too many, and a hundred confirmations are too few. (Ahem.) Notice also that the only things that falsify "All non-black things are non-ravens" are also ravens of any non-black colour. So the positive and its contra-positive have the same counter-examples. Just another logical superiority of falsificationism. But I digress.

Another approach is to notice that white shoes also confirm the claim that "All ravens are green", or indeed any other colour. We might say that if a piece of evidence confirms an hypothesis H(black) and also H(green), H(purple), H(puce) and so on, it is in some sense trivial with respect to that set of hypotheses. It's not what we are really looking for, which is that every time we see a raven, it is reassuringly black. This is an attempt to capture the necessary quality of relevance that evidence must have. It is not perfect, but it's a start. I'll leave the lads at Carnap's Bar and Grill to debate the details.

Instead of trying to resolve the paradox, we should ask: how did we get here? What are we assuming that creates the paradox? Is it true? What are the other assumptions we might have in their place? Who says that "equivalent with respect to the predicate calculus" is the relevant equivalence? Why not "equivalent with respect to the legal concept of material relevance"?

Which would send the ravens flying.