According to Aristotelian logic, the statements “All crows are black” and “All non-black things are non-crows” are logically equivalent. Some take this to mean that observing a red apple or a white shoe — both non-black, non-crow entities — somehow supports the claim that all crows are black. This strange conclusion is known as Hempel’s Paradox or the Raven Paradox — sometimes jokingly called the Paradox of Indoor Ornithology. Despite the argument’s apparent innocence, it leads to deeply unintuitive consequences and has inspired considerable philosophical work.
But let’s put aside red apples and white shoes. What about actually observing a black crow? Surely that counts as evidence for the claim that all crows are black? Our intuition may say yes — but intuition can be misleading.
In his 1967 paper “The White Shoe is a Red Herring”, I.J. Good provides an interesting example. Imagine two worlds:
In World A, there are 1 million birds, of which 100 are black crows.
In World B, there are 2 million birds, including 200,000 black crows and 1.8 million white crows.
Now, suppose we observe a single black crow. From which world is it more likely to come? Surprisingly, probabilistic reasoning tells us it is 1,000 times more likely to have come from World B — the one with many white crows. Thus, observing a black crow actually favours the hypothesis that not all crows are black.
Good’s example highlights a subtle but crucial point: evidence is always relative to a hypothesis and its alternatives. Observing a black crow — even hundreds of them — proves nothing unless we define the competing hypotheses clearly. The same lesson applies to the famous Black Swan theory. Prior to their discovery in Australia, Europeans considered black swans mythical and impossible. The real problem wasn’t the observation of a black swan — it was the fact that no one had assigned it even a small prior probability.
This is precisely why Dennis V. Lindley, reviewing Nassim Nicholas Taleb’s The Black Swan, criticises the attempt to draw a line between “usual randomness” and “radical uncertainty”. Lindley wrote:
“Sorry, Taleb, but the calculus of probability is adequate for all cases of uncertainty and randomness,…”
The issue is not with randomness itself, but with our priors. If people had reasoned like this:
“I’ve only seen white swans, but I’ve also seen black sheep, and pigeons and parrots of many colours. So perhaps non-white swans exist somewhere”,
they would have assigned a low but non-zero prior probability to the existence of black swans — and would not have been epistemically devastated when one appeared.
The lesson is subtle: we need not assign a non-zero prior to every imaginable hypothesis (we don’t need to entertain the existence of Spider-Man), but we should be careful not to assign zero probability to plausible extensions of what we already know.