Notes from the underground
barang dot sg
Updated 2 December 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

10. The unicorn

We could easily run through a few more paradoxes but they tell the same broad tale and the essential points have been made.

Some people treat paradoxes 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 one last example, here’s a cheeky “proof” of the existence of a unicorn from the mischievous American logician Raymond Smullyan:
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 that 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 grappling 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!

I wish I could have shown a few more examples above of the patient pursuit and eventual snaring of such elusive prey; the exposing of the precise error in a subtly fallacious line of reasoning. This is an art form in itself and a few examples would have been good to see. (The lone example provided was in the section on logical fatalism, where we took down the Idle Argument.) 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 ...