It seems absurd to suppose that a green leaf confirms the hypothesis that all ravens are black! Why should any one suppose that it does?

Well, think how a general hypothesis like ‘All ravens are black’ is confirmed? We check out ravens to see if they are black. We will disregard anything that is not a raven, since whether or not it is black or non-black seems irrelevant. A non-black raven would be a refuting instance for the hypothesis and a black raven will be a confirming instance. The more black ravens we observe, with no observations of ravens of any other colour, then the stronger is the confirmation that indeed ‘All ravens are black’.

That seems sound. However it also seems clearly true that if an observation confirms a hypothesis then it also confirms any statement logically entailed by that hypothesis. For example, if we confirm that all ravens are black, then since ‘All ravens are black’ entails ‘No ravens are red’ then we also confirm that no ravens are red. But this, together with the idea of confirmation outlined in the last paragraph, leads to the paradox that green leaves confirm that all ravens are black.

Here's how it comes about. Think how one would confirm the hypothesis that ‘All non-black things are non-ravens’. Following the above notion of confirmation, we will check out non-black things to see if they are non-ravens. We will disregard things that are black since whether or not they are non-ravens seems irrelevant. A non-black raven will be a refuting instance of the hypothesis, a non-black non-raven – like a green leaf – will be a confirming instance. The more non-black non-ravens we observe, with no observations of non-black ravens, then the stronger will be the confirmation that indeed, ‘All non-black things are non-ravens’. But of course ‘All non-black things are non-ravens’ entails ‘All ravens are black’. Indeed they are logically equivalent; just different ways of stating the same thing. And so, since a green leaf is a confirming instance of the hypothesis that all non-black things are non-ravens, then a green leaf will also be a confirming instance of the hypothesis that all ravens are black. Hence our paradox.

How can we make sense of this? Well, one thing that stands out in the above analysis is that although what counts as a confirming instance seems to vary with the form in which we cast our hypothesis, what counts as a refuting instance remains the same, namely a non-black raven. Of the four possible ‘kinds of thing’ relevant to our hypothesis in either form – i.e. black ravens, non-black ravens, black non-ravens and non-black non-ravens – the existence of black ravens, black non-ravens and non-black non-ravens are all consistent with our hypothesis, in whichever logical form we cast it, and hence could be seen as confirming instances, but only non-black ravens are inconsistent with it and hence potential falsifiers.

So would a shift from a focus on confirmation to a focus on falsification resolve our paradox? If we were to seek out falsifiers, as Popper recommends, there would be no ambiguity about what we would be looking for – non-black ravens.

Popper thinks that hypotheses are not accepted into the body of science because they have been shown to be true, or even probably true, but because they have been subjected to tests, to attempts to falsify them, and have not, so far, been falsified. Negative test results – i.e. those which could have falsified the hypothesis but didn't – ‘corroborate’ the hypothesis. So, for Popper, if we searched diligently for non-black ravens and failed to find them then our hypothesis would be corroborated.

But does focusing on falsification actually remove the paradox? In seeking to falsify ‘All ravens are black’ we seek out ravens to see if we can find any non-black ones. So in seeking to falsify ‘All non-black things are non-ravens’ shouldn't we be seeking out non-black things to see if we can find any that are ravens? So if we check out something green and find that its not a raven, but a leaf, doesn't this corroborate the hypothesis that all non-black things are non-ravens, and hence corroborate the hypothesis that all ravens are black? Wouldn't it be a paradox of corroboration that a green leaf tests and corroborates the hypothesis that all ravens are black?

So why is it that if we are concerned to check that all ravens are black we do not bother ourselves with green leaves? Why is it, if we want to look for the falsifying case of non-black ravens, that we do not check out non-black things to see if they are ravens?

Isn't it just because there are so many fewer ravens than there are non-black things? Checking out non-black things to see if they are ravens would be a very inefficient strategy if we wanted to seek possible falsifiers of our hypothesis – in either of its logically equivalent forms. But suppose we lived in a ‘black’ world, where most things were black; all minerals were black and nearly all organisms. Non-black things in this world, lets suppose, are extremely rare. In this black world there would be far more ravens than non-black things. So in this black world it would be a very inefficient strategy to seek out ravens to see if any were non-black. Much better to seek out the very few non-black things to check if any were ravens. In this black world it would be natural to see a green leaf as a confirming instance of the hypothesis that all non-black things were non-ravens (and hence that all ravens were black).

This suggests that it is falsification and relative frequency that underlie what we naturally regard as confirming (or corroborating) instances. Our hypothesis says of everything that if it is a raven then it is black, and therefore everything apart from a non-black raven could be seen as a confirming instance of the hypothesis. But if the hypothesis is true then the class of non-black ravens is empty. So if we are in doubt about the truth of the hypothesis we will want to check whether that class really is empty. In effect, we will want to look for potential falsifiers. And whether the most efficient way to do this is to seek out ravens to check if they are non-black, or to seek out non-black things to see if they are ravens will be determined by whether ravens or non-black things are least common. Thus in our world, where non-black things are far more frequent than ravens, it is natural to see black ravens, rather than green leaves, as confirming instances of our hypothesis.

If this is right it implies that if scientists, contra Popper, seek to confirm their hypotheses rather to falsify them, then they will nonetheless be adopting the most efficient strategy to find falsifiers, if there are any.