This was written in 2009. It explains the notion of a logical paradox in a way that comes naturally to me. The conclusion drawn at the end is probably the main thing for me (“We are blinder than we can imagine in matters of straightforward logical reasoning”) but it is not always emphasized by others!
1. Beside belief
The term ‘paradox’ is a fairly old one and its use may be traced as far back as the ancient Greeks, if not further.
Thus, from what I know, the Greek words para doxa mean something like “beside belief,” and the ancients regularly heaped the label paradoxon upon any situation or entity that beggared belief.
We likewise call something paradoxical if it is out of line with what is natural to believe, e.g., that if you teach students less, they end up learning more, or that you can sometimes do something faster by doing it slower. This is the ordinary sense of the term, preserved through the ages.
If we add the word ‘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 perfectly good sense) is just a situation in which a compelling line of reasoning appears to lead directly to an unbelievable proposition. 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 naturally be puzzling, since if the reasoning were truly flawless, the generated conclusion should be perfectly true! What this means is that the situation in question must involve an illusion: either the “compelling reasoning” must contain some subtle flaw, or else the “unbelievable conclusion” must really be acceptable, after all.
In most cases like this, a little reflection will show where the truth lies and the puzzle is quickly dissolved.
The trouble, however, is that a few cases prove to be rather more stubborn, meaning that, 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 many philosophers mean by a “logical paradox,” and I plan to run through 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!