Missives from a fly bottle
barang dot sg
Last revised 18 October 2017
What’s a logical paradox?


This was written in 2009. It explains the notion of a logical paradox in a way that comes very naturally to me. The conclusion drawn at the end is probably the most important thing for me (“We are blinder than we can imagine in matters of straightforward logical reasoning”) but I find that it is rarely emphasized by others.
4. Hempel’s ravens

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

So he tramps about his village over the next couple of days, trying to spot as many ravens as he possibly can. Unfortunately, through a combination of bad luck and being unsure where to look, he fails to find a single raven. 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 the proposition, “All ravens are black,” is logically equivalent to the proposition, “All non-black things are non-ravens.” As he correctly observes, if either of these propositions is true, then so must be the other.

Accordingly, he gleefully reasons, any evidence for the second proposition will equally be evidence for the first!

So he starts simply to look for non-black things, and finds this to be woefully easy. Moreover, they prove invariably to be non-ravens! Thus, in a yard nearby is a whole herd of pink pigs; on a branch above, a hundred green leaves. Before long, he finds thousands of non-black non-ravens, including countless brown sparrows, red berries, blue flowers and even a couple of yellow postal delivery vans, and starts becoming increasingly confident that all the non-black things in his village are non-ravens.

Accordingly, he reminds himself, he should now be increasingly 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 neighbouring herd of pink pigs, for example, really constitute any evidence (whatever) for the proposition that all the ravens in his village were black? It seemed a little hard to believe.

And yet what was wrong with his reasoning? A herd of pink pigs surely helps confirm the proposition that all non-black things are non-ravens. So it must also help confirm the proposition that all ravens are black, since this is a logically equivalent proposition, just worded in a different way.

What went wrong?

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 seem, a herd of pink pigs really does help confirm the hypothesis that all ravens are black. Hempel himself certainly thought so. (Swallowing the conclusion.)

Others find this too much to bear and insist that something must be wrong with the man’s reasoning. For example, some claim that raven and black are natural categories, unlike non-black and non-raven, which are quite artificial. So the man’s two propositions cannot really be equated in the way that he wants. (Rejecting the reasoning.)

Thankfully, we don’t need to resolve this issue here. Our point is just that this is a case where a “compelling” line of reasoning yields an “unbelievable” conclusion, but it’s terribly unclear if we should swallow the conclusion or reject the reasoning. People tend to disagree even after considerable reflection!

This is entirely characteristic of a logical paradox and this famous one is lovingly known as Hempel’s paradox of the ravens. (Among philosophers of science, it’s more sombrely called the paradox of confirmation.)

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 one, where it’s also not easy to see which way to go.