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!
10. The unicorn
We could possibly run through a few more paradoxes, but they tell the same broad tale and the essential points have basically been made.
Some people treat paradoxes simply as exercises in reasoning. A “seemingly flawless” argument is presented, but it leads to a “seemingly absurd” conclusion, so the challenge is to uncover the flaw in the reasoning, or else explain why the conclusion is not really absurd.
Many of these may be resolved with a little effort and the exercises are often rewarding and entertaining. For a closing example, here’s a cheeky “proof” of the existence of a unicorn from the mischievous American logician Raymond Smullyan:
“To prove that there exists a unicorn, it’s enough to prove the stronger proposition that there exists an existing unicorn.
(An “existing unicorn” is just a unicorn which exists.)
For, obviously, if there exists an existing unicorn, then there must exist a unicorn!
So all we have to do is prove that an existing unicorn exists. But that’s not very hard. How could an existing unicorn not exist? That would be a contradiction!”
It’s not trivial to explain where this “proof” goes wrong, and anyone who spars with problems like these and learns the art of overcoming them will come quickly to appreciate the curious case of the absolutely flawless reasoning with the completely unacceptable conclusion! Indeed, there is such a thing as savouring a paradox before wrestling with it and this experience is available to anyone who is open to it. Sadly, this fact is often lost upon those who regard paradoxes as nothing but affronts to their intelligence.
A really good paradox appears only once in a while, catching everyone by surprise, e.g., once every half century or so. We actually stumble upon them, more than anything else. You cannot really invent a good paradox because they generally betray deep logical or metaphysical prejudices that we cannot detect on our own. As we saw, we can have great difficulty identifying such prejudices even when alerted to their existence – they are very elusive prey!
One thing missing from this survey is an example of the patient pursuit and eventual snaring of such an elusive prey; the exposing of the precise error in a subtly fallacious line of reasoning. This is an art form in itself and an example would have been useful to see. But I have tried to provide some on the rest of this site and so have focused on the more general lesson available to us here, which I’m afraid is this:
Paradoxes reveal that we are often blinder than we can imagine in matters of straightforward logical reasoning ...