Notes from the underground
barang dot sg
Updated 28 January 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

1. Beyond belief

The term ‘paradox’ is fairly old and may be traced all the way back to the ancient Greeks, if not further.

Thus, the Greek words para doxa mean something like “beyond belief” or “contrary to belief,” and the ancients also used the term paradoxon to describe any proposition that they found difficult to believe, or otherwise absurd to contemplate.

Paradox of choice
We likewise call something paradoxical if it goes against what is natural to believe, e.g., that you can often do something faster by doing it slower, or that if you teach students less, they sometimes learn more.

This is the ordinary sense of the term ‘paradox,’ preserved through the ages.

Once you add the word ‘logical’ to ‘paradox,’ on the other hand, something more specific is usually meant, at least by modern writers, and this is what this essay is really about.

Logic is just the study of reasoning, and so a ‘logical paradox’ is said to arise when a “compelling” line of reasoning leads inexorably to an “unbelievable” conclusion. In other words, you are faced with a piece of reasoning that seems to be flawless, but which culminates in a conclusion that is very hard to believe; perhaps even downright absurd.

Such a situation would naturally be intolerable to the rational mind, since if the reasoning were truly flawless, the generated conclusion should be perfectly true! And, sure enough, in the large majority of such cases, an element of deception proves to be present: either the “unbelievable” conclusion turns out to be acceptable, after all, or else the “compelling” reasoning proves to contain a subtle flaw. A little reflection indicates where the deception lies and the intolerable state of affairs is quickly resolved. I will give some examples of this below.

The trouble, however, is that a few such cases prove to be rather more stubborn. Even after careful and prolonged scrutiny, the reasoning at hand continues to seem completely compelling, while the resulting conclusion continues to seem completely unbelievable! Something must give but it seems that nothing will!

Paradox cartoon

The unbearable tension produced by this sort of (stubborn) case is what logicians and philosophers properly mean by a “logical paradox,” and I’ll run through some famous ones—some would say notorious ones—in what follows.

Before doing that, however, let’s examine some of the simpler, “quickly resolved,” cases first, to drive home the idea of a compelling line of reasoning that appears—at first sight—to terminate in an unbelievable conclusion. There certainly are such things and it helps to get used to the basic idea!