Notes from the underground
barang dot sg
Updated 2 December 2023
What’s a logical paradox?

1. Beyond belief
2. Swallowing the conclusion
3. Rejecting the reasoning
4. Hempel’s ravens
5. Descartes’s dream
6. The bicycle
7. The heap
8. Logical fatalism
9. Newcomb’s paradox
10. The unicorn
11. References

4. Hempel’s ravens

A certain man’s backyard is frequented by the occasional black raven, and he soon starts to wonder if all the ravens in his village are black.

So he tramps about his village over the next few days, trying to spot as many ravens as he can. Unfortunately, through a lack of luck and knowledge of where to look, he fails to find a single one. Nor does he know how many ravens nest in his village, really.

On the third day of this forlorn venture, it dawns upon him that these two propositions are logically equivalent:
All ravens are black.
All non-black things are non-ravens.
As he says, neither proposition can be true without the other being true as well. Accordingly, he reasons, any evidence for the second proposition will automatically be evidence for the first!

So he sets about trying to confirm the second proposition instead. He begins to look for non-black things in order to check if they are non-ravens and quickly finds that they invariably seem to be! Thus, in a yard nearby, he spots a pink pig; on a branch above, a green leaf. Before long, he finds numerous non-black, non-ravens, including a blue van, some oranges, and a white shoe. Indeed, every non-black thing that he comes across turns out to be a non-raven. The evidence thus accumulating, he becomes increasingly confident that all the non-black things in his village are non-ravens.

Accordingly, he reminds himself, he should now be just as confident that all the ravens in his village are black!

At this point, however, the absurdity of it all began to sink in. Could a white shoe, for example, constitute any evidence whatsoever for the hypothesis that all ravens are black? It seemed a little hard to believe, almost preposterous, really.

And yet what was wrong with the man’s reasoning? If a black raven can contribute towards raising your confidence that all ravens are black, then a non-black, non-raven (such as a white shoe) ought likewise to raise your confidence that all non-black things are non-ravens. So it should also raise your confidence that all ravens are black, since this is a logically equivalent proposition, just worded in a different way.

What went wrong?

Carl G. Hempel
This curious problem was raised by the German philosopher Carl G. Hempel in the 1940s and its correct resolution is still being debated.

Some say that the man’s reasoning is perfectly correct and that, strange as it may first seem, a white shoe really does help to confirm the hypothesis that all ravens are black. Hempel himself certainly thought so. (Swallowing the conclusion.)

This conclusion can be made more “palatable,” they add, by allowing that a white shoe may provide less confirmation of the hypothesis in question than a black raven does. After all, there are considerably more non-black things than ravens in this world. So it is much easier to find a non-black, non-raven than it is to find a black raven, thus “cheapening” the former’s contribution towards confirming the hypothesis that all ravens are black. Even so, its contribution is not zero.

Others find even this watered-down conclusion unpalatable and insist that something must be wrong with the man’s reasoning. (Rejecting the reasoning.) For example, the American philosopher W. V. Quine contended that, while observing a black raven can legitimately raise the man’s confidence that all ravens are black, the inverse is simply not true. Observing a non-black, non-raven can give him no reason whatsoever to think that all non-black things might be non-ravens.

W. V. Quine
The reason, Quine said, is that artificial categories like ‘non-black’ and ‘non-raven’ have no tendency to reflect regularities found in our environment, unlike natural categories like ‘raven’ and ‘black,’ which have evolved through natural selection to serve our predictive, generalizing needs. If a girl eats some avocadoes and finds them to be delicious, she has every right to suspect that avocadoes might in general be delicious. But if she eats some non-avocadoes, then, whatever she discovers, she cannot likewise project her findings onto other non-avocadoes, because non-avocadoes don’t form a “natural kind.” According to Quine, the man above reasoned in error from failing to see this.

Given that Hempel and Quine, both towering intellects, disagreed on how to handle this situation, we can see that we are in challenging territory. There is still no consensus on the correct way to resolve this paradox and I cannot see my way through it either. This is a case where a “compelling” line of reasoning yields an “unbelievable” conclusion, but it’s unclear if we should swallow the conclusion or reject the reasoning. Even those who reject the reasoning differ considerably on where the reasoning goes wrong.

This is entirely characteristic of a logical paradox and this one is known as Hempel’s paradox of the ravens, if you’d like to look it up further. (Among philosophers of science, it’s more sombrely called the paradox of confirmation.) You can also find a catalogue of various other suggestions for resolving this paradox on Wikipedia.

Of course, some people say that a good paradox should never be resolved, but this is usually meant tongue in cheek. On a traditional view, there will always be a “correct” way to resolve a paradox, even if it takes us a while to see which way to go. For the moment, let’s leave Hempel’s paradox behind and consider another difficult paradox, where it’s also not easy to see which way to go.