The Raven and the Shoe

The hypothesis that all ravens are black is logically equivalent to the statement that all non black things are non ravens, and this is supported by the observation of a white shoe.

This is a paraphrasing of a famous paradox due to Hempel. There is a lot of fairly impenetrable discussion on the Wikipedia page about this paradox, some of which I believe to be incorrect, and so I include a readable resolution based on a Bayesian perspective here, and I also relate this issue to our ability to know the truth about physical laws.

There are two issues: One is that “is supported by” is not necessarily true without knowledge of alternative hypotheses. For example consider a situation with two bags: one, called A, containing 1 black stone and 999 white stones, and another bag, B, containing 500 black stones and 500 white stones, and the problem is: “given that we picked a black stone, which bag did it come from?” If we were actually considering hypothesis A in isolation, a black stone is an example of a valid pick from bag A and so this evidence seems to support it (compared with picking, say, a green stone). However when considering both of the competing hypotheses A and B, the fact we picked black actually is evidence against A and for B being the one we picked from, because a black stone is very unlikely to be picked from bag A. Therefore the notion of evidence supporting something (improving our belief that a hypothesis is true) is very different depending on what alternatives are available and becomes almost meaningless when there is only one hypothesis. This argument is due to Good.

Secondly, in the Bayesian world, the white shoe can indeed be evidence for all non black things being non ravens, but the amount of evidence is so vanishingly small in view of the fact that there are an infinity of objects which are non black things as to be actually zero, in line with our intuition of this being ridiculous. It is only when we have exhaustively tested all non black things for ravenness that we would have amounted enough evidence to validate the hypothesis. In fact this is the core problem in scientific hypothesis testing because, while we can amass evidence, we cannot completely validate a scientific truth; instead, we can only prove it to be incorrect by finding a counter-example.

In fact if we made 1000 observations of the planets and feel that this validates the law of gravity to a reasonable level of certainty, I might counter that this position is quite surprising considering that there remain an infinity of other possible observations left to check. Even with no more observations, the current law could be rendered uncertain tomorrow if another radically different theory was invented which gave precisely the same predictions over the current data set. Then there would be competing theories which could only be separated by gathering more data.