Raven paradox

By Martin McBride, 2024-01-18

Categories: recreational maths paradox

The raven paradox was invented by Carl Gustav Hempel in the 1940s. It raises the question of what should be considered evidence for the truth of a hypothesis, in particular when applied to the scientific method.

The scientific method

Scientific enquiry often starts with an observation. For example, we might see a group of ravens, and observe that they are all black.

This may then be followed by conducting some research, that might involve going out and looking for more ravens.

After observing many ravens, and seeing that they too are all black, without exception, we might form a hypothesis that all ravens are black.

We would then go ahead and test that hypothesis by conducting more experiments, which in this case would most likely be further observations of the real world.

What would we be looking for? Well, primarily we would be looking for a non-black raven. If we saw a white raven, for example, that would tell us for certain that the hypothesis was incorrect - that not every raven is black.

If we never saw a non-black raven and only saw more and more black ravens, that wouldn't prove the hypothesis that all ravens are black, but it would certainly give us even more confidence that the hypothesis might be true.

But what about another object we might see? A green leaf or a red apple in the tree where a raven was perched, or a blue car parked nearby? Would they give us additional information?

Well, maybe.

The raven paradox

We can take the statement "all ravens are black", and restate it as a logical statement:

1. If x is a raven, then x is black

Now we can apply contraposition to obtain a logically equivalent statement:

2. If x is not black, then x is not a raven

These two statements are logically equivalent. If either of them is true, then the other must also be true. If either of them is false, then the other must also be false.

This means that if we find evidence that makes one of these statements more likely to be true, then it also makes the other statement more likely to be true. Because if one is true then so is the other.

Now if we see a green leaf, or a red apple, or a blue car, that is certainly evidence to support the statement that a non-black object is not a raven. Those are all things that aren't black and aren't ravens.

But as we have just shown, anything that supports statement 2 also supports statement 1. So seeing a green leaf, or a red apple, or a blue car supports the hypothesis that all ravens are black!

Responses to this paradox

There have been many responses to this paradox. There is no generally accepted answer, but below we will look at some of the main arguments that have been put forward over the years.

The statement is correct, our intuitions are wrong

Many people have a strong intuition that seeing a green apple does not make it more likely that all ravens are black. Seeing a green apple tells us nothing about the colour of a raven.

It is the fact that this counter-intuitive result appears to be confirmed by simple logic that creates the paradox.

So one fairly absolutist argument is that our intuitions are wrong. Seeing a red apple really does support the hypothesis that all ravens are black. Cold logic trumps intuition!

There are a couple of strong counterarguments to this position.

Firstly, if we accept that evidence of non-black, non-ravens supports the hypothesis that all ravens are black, then there is nothing to prevent us from basing the hypothesis entirely on the observation of non-black, non-ravens. In other words, we could convince ourselves that all ravens are black without ever seeing a raven at all. After all, seeing lots of non-black, non-ravens is logically equivalent to seeing lots of black ravens.

Secondly, if we were to accept this as true, then we would have an even bigger problem. We could observe lots of non-green, non-ravens and prove that all ravens are green.

The statement wrong, according to the scientific method

This is another absolutist opinion, in the opposite direction. It is commonly associated with Karl Popper.

In the scientific method, a hypothesis is never fully proven. It can only ever be falsified, for example if a counter-example is found.

If we apply this principle, then of course seeing a green apple doesn't give us any confidence that all ravens are black.

But equally, seeing a black raven doesn't give us any confidence that all ravens are black either!

The only thing the scientific method can do is prove a hypothesis wrong. If we see a different coloured raven that will definitely tell us that the hypothesis isn't true. But unless and until that happens, all we can say is that we haven't seen a non-black raven yet.

So seeing a black raven or a green apple both tell us the same thing - absolutely nothing. There is no paradox.

But in reality, there is a problem with this approach. It isn't how science is done. As we gather more and more information around a hypothesis, we often become more and more confident that it is true. At a certain point the hypothesis becomes a theory and we accept it as being as good as a fact.

An example of this is the theory of gravity. This was hypothesised by Newton in the late 17th century, and his equations provided a very accurate description not only of the behaviour of heavy objects on the Earth but also of the motion of moons and planets in the solar system.

Although, technically, the scientific method can only falsify a hypothesis, nobody expects to wake up one morning and discover that gravity no longer works.

Of course, Newton's theory was eventually refined by Einstein's theory of general relativity. But, as often happens with well-established theories, gravity is still used to describe everyday situations.

Bayesian approach

Imagine if nobody had ever seen ravens before. Then someone notices one and observes that it is black. As described earlier, they might try to observe more ravens and gradually become increasingly confident that all ravens are black. More scientists might get involved and supply more observations.

But each new sighting becomes less and less significant. By the time the 1000th raven is seen, the fact that it is black just like the others adds a tiny bit of confidence to the hypotheses but not as much as the first few sightings. There have been so many sightings already that one extra (unless it is non-black) doesn't change things very much.

Then one day news comes in of a raven in a different country, on the other side of the world. This is important, of course, because it might have evolved slightly different features, and so might be non-black. If the raven turns out to be black, that adds more weight to the hypothesis. But as more and more black ravens are seen in the second country, they too become less significant.

The important point here is that, while every sighting of a black raven helps to confirm the hypothesis, not every sighting is equally significant. After observing a very large number of black ravens, observing one more tells us very little that is new.

Now we can also apply the same reasoning to non-black, non-ravens. We might say, yes, it is quite true that observing a green apple increases our confidence that all ravens are black. But only by a tiny, tiny amount. The evidence provided by seeing a green apple is nowhere near as strong as the evidence of seeing a black raven. This is because:

There are far, far more objects that aren't ravens.
Most people have observed far, far more non-black, non-ravens than they have black ravens.

So, in reality, seeing a non-black, non-raven might technically increase our confidence in the hypothesis, but in practical terms it is a negligible effect.

Non-equivalence of the cases

We might consider that there are 3 possible hypotheses:

All ravens are black. That some, but not all, ravens are black. And that no ravens are black.
Some, but not all, ravens are black.
No ravens are black.

Seeing a black raven does not prove that all raven are black. Seeing a black raven is consistent with either hypothesis 1 or 2. But it does disprove hypothesis 3, that no ravens are black.

On the other hand, seeing a non-black, non-raven doesn't prove any of the hypotheses. It is consistent with any one of them being true.

This means that seeing a black raven favours the hypothesis that all ravens are black, simply in the sense that it disproves one of the 3 hypotheses. But a non-black, non-raven is completely consistent with any of the 3 hypotheses. This is not surprising, of course, because seeing such an object tells us nothing about ravens.

Context of the experiment

A further consideration might be the wider context of the experiment. In the case of the raven paradox, the experiment is the search for various types of objects.

However, our interpretation of the result depends on the type of experiment we perform.

Imagine we conducted an experiment by searching a large photography database, such as Flickr or Google Images, and examining a random selection of the images of the many objects we found there.

In Experiment 1, we might search exclusively for images of birds. The results could be very useful. We would find lots of black ravens (which support the hypothesis), and we might find a non-black raven (which would instantly disprove the hypothesis).

As for the non-black, non-ravens, well since there are a finite number of images in the database, at a minimum we might argue that a non-black, non-crow means there is one less possibility of finding a non-black crow, so it might add a tiny amount of support to the theory.

In Experiment 2 we might search only for black objects. This search would be more or less useless. Every item would either be a black raven (and already know that these exist) or a black non-raven (and again we all know that such items exist). With no possibility of finding a non-black raven (or a non-black anything else), the search tells us nothing.

In Experiment 3 we might search only for undersea photos that were taken by deep-sea divers. In this case, seeing a non-black, non-raven tells us nothing at all, because there is no possibility of finding a raven, of any colour, under the ocean

Summary

These have been a selection of the many views put forward over the years since the problem was proposed. It is still not a settled issue, with many potential resolutions, each with its supporters.

Finally

Are there, in fact, any non-black ravens? Yes, there are! White ravens exist. They can either be albino (lacking pigment) or leucistic (having pigment that is expressed in a reduced form, making them slightly yellow rather than pure white).

That doesn't really affect the arguments here, of course, because the idea of a non-black raven could be substituted with any other unusual item.