Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

You spot a pink flamingo and wonder to yourself whether all flamingos are pink. What would it take to confirm or disprove the hypothesis? The nice thing about this sort of hypothesis is that it’s testable and potentially falsifiable. All it takes is to find a flamingo that is not pink, and I can conclude that not all flamingos are pink. Just one observation can change my flamingo world view. It doesn’t matter how many pink flamingos you witness, however, no number can prove the hypothesis short of the number of flamingos that potentially exist. Still, the more you see that are pink, the more probable it becomes that all flamingos are actually pink. How probable you consider that is at any given time is related to how probable you thought it was before you saw the latest one. While considering this, you see someone wearing blue tennis shoes. Does this make it more likely that all flamingos are pink? This is one example of a broader paradox first formally identified by Carl Gustav Hempel, sometimes known as Hempel’s paradox or else the Raven Paradox.

The Raven paradox arises from asking whether observing a green apple makes it more likely that all ravens are black, assuming that you don’t know the answer. It would intuitively seem not. Why should seeing a green apple tell you anything about the colour of ravens? The way to answer this is to re-state ‘All ravens are black’ as ‘Everything that is not black is not a raven.’ In fact, these two statements are logically equivalent. To see this, assume there are just two ravens and two tennis shoes (one right-foot, one left-foot) in the whole world. Now you identify the colour of each of these objects. You observe that both tennis shoes are blue and the other two objects are black. So you announce that everything that is not black (each of the tennis shoes) is not a raven. This is identical to saying that all ravens are black. The logic universalises to any number of objects and colours. Assume now we see just one of the tennis shoes and it turns out to be blue. You can now announce that one possible thing that is not black is not a raven. If you see the other tennis shoe and it is blue, that means that there are now two things that are not black that are not a raven. Each time you see something, it is possible that you would not be able to say this – i.e. you would say instead that you have seen something not black and it is a raven. It is like being dealt a playing card from a deck of four which contains only blue or black cards. You are dealt a black card, and it shows a raven. You know that at least one of the other cards is a raven, and it could be a black card or a blue card. You receive a blue card. Now, before you turn it over, what is the chance it is a raven? You don’t know, but whatever it is, the chance that only black cards show ravens improves if you turn the blue card over and it shows a tennis shoe. Each time you turn a blue card over it could show a raven. Each time that it doesn’t makes it more likely that none of the blue cards shows a raven. Substitute all non-ravens for tennis shoes and all colours other than black for the blue cards, and the result universalises. Every time you see an object that is not black and is not a raven, it makes it just that tiny, tiny bit more likely that everything that is not black is not a raven, i.e. that all ravens are black. How much more likely? This depends on how observable non-black ravens would be if they exist. If there is no chance that they would be seen even if they exist, because non-black ravens never emerge from the nest, say, it is much more difficult to falsify the proposition that all ravens are black. So when you observe a blue tennis shoe it offers less evidence for the ‘all ravens are black’ hypothesis than when it is just possible that the blue thing you saw would have been a raven and not a tennis shoe. More generally, the more likely a non-black raven is to be observed if it exists, the more evidence observation of a non-black object offers for the hypothesis that all ravens are black.

So to summarise, we want to test the hypothesis that all ravens are black. We could go out, find some ravens, and see if they are black. On the other hand, we could simply take the logically equivalent contrapositive of the hypothesis, i.e. that all non-black things are non-ravens. This suggests that we can conduct meaningful research on the colour of ravens from our home or office without observing a single raven, but by simply looking at random objects, noting that they are not black, and checking if they are ravens. As we proceed, we collect data that increasingly less support to the hypothesis that all non-black things are non-ravens, i.e. that all ravens are black. Is there a problem with this approach?

There is no logical flaw in the approach, but the reality is that there are many more non-black things than there are ravens, so if there was a pair (raven, non-black), then we would be much more likely to find it by randomly sampling a raven then by sampling a non-black thing. Therefore, if we sample ravens and fail to find a non-black raven, then we’re much more confident in the truth of our hypothesis that “all ravens are black,” simply because the hypothesis had a much higher chance of being falsified by sampling ravens than by sampling random non-black things.

The same goes for pink flamingos. So we have a paradox traceable to Hempel. I suggest we can do this by appeal to a ‘Possibility Theorem’ which I advance here.

Let’s do this by taking the propositions in the thought experiment in turn. Proposition 1: All flamingos are pink. Proposition 2 (logically equivalent to Proposition 1): Everything that is not pink is not a flamingo. Proposition 3 (advanced here as the Possibility Theorem): If something might or might not exist, but is unobservable, it is more likely to exist than something which can be observed, with any positive probability, but is not observed. If something might or might not exist, it is more likely to exist if it is less likely to be observed than something else which is more likely to be observed, and is not observed. So when I see two blue tennis shoes, I am ever more slightly more confident that all flamingos are pink than before I saw them, and especially so if any non-pink flamingos that might be out there would be easy to spot. And I’d still be wrong, but for all the right reasons, until I saw an orange or white flamingo, and then I’d be right, and sure.

Exercise

Does seeing a blue tennis shoe make it more or less likely that all flamingos are pink, or neither?