Missives from a fly bottle
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Last revised 31 August 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.
1. Beside belief

The term ‘paradox’ is quite old and may be traced as far back as the ancient Greeks, if not further.

Thus, the Greek words para doxa mean something like “beside belief,” and the ancients also used the word paradoxon for the related quality.

So we call something paradoxical if it is out of step with (“beside”) what is natural to believe, e.g. that if you teach students less, they end up learning more, or that you can often do something faster by doing it slower. This is the ordinary sense of the term, preserved through the ages.

If we add ‘logical’ to ‘paradox,’ on the other hand, something more specific is usually meant, at least by modern writers.

Logic is just the study of reasoning – and so a logical paradox, in one good sense, is just a situation in which an unbelievable proposition appears to be generated by a compelling line of reasoning.

In other words, we are shown 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 of course be puzzling, since if the reasoning were truly flawless, the generated conclusion should be perfectly true! This means that the situation in question must involve an illusion – either the “unbelievable conclusion” must really be acceptable after all, or the “compelling reasoning” must contain some subtle flaw.

In most cases like this, a little reflection will show what went wrong and the puzzle is quickly dissolved.

The trouble, however, is that a few cases prove to be rather more stubborn – even after prolonged scrutiny, the reasoning continues to seem completely compelling, while the resulting conclusion continues to seem completely unbelievable!

The unbearable tension produced by this sort of (stubborn) case is what is properly called a “logical paradox,” and we’ll consider some famous ones in what follows.

Before doing that, however, it will be useful to examine some of the simpler (easily resolved) cases first, to drive home the idea of a compelling line of reasoning that terminates in an unbelievable conclusion.

There certainly are such things and it helps to get used to the basic idea!