Missives from a fly bottle
barang dot sg
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.
2. The rope

Imagine that a long rope has been laid snug around the Earth’s equator. (Pretend the Earth is a smooth sphere.)

It’s an engineering project, say, and the men involved have just finished the job.

To their chagrin, the foreman now tells them that a mistake has been made.

The rope should not lie snug on the ground like that, but be raised one foot above the ground, all the way around.

Well, in that case, the rope must obviously be longer and so more rope will be needed. But roughly how much more rope do you think?

A calculation reveals a surprising answer.

The length of the original rope was π × D where D is the Earth’s diameter. So the length of the new rope must be π × (D + 2) since the diameter has simply expanded by two feet, one on each side. But the difference between these values is simply 2π which is about 6.28. And so only about six feet more rope is needed, which is about the height of a man.

Can that be right?

Most people find this very surprising, even unbelievable. Surely you need much more rope than that? Something like a few miles of extra rope would seem more appropriate!

And yet the reasoning above is hard to fault.

This is a simple example of a compelling piece of reasoning that leads to an unbelievable conclusion. Of course, in this case, the conclusion does not stay unbelievable for long. Once the reasoning sinks in, the conclusion comes to be accepted. This is called “swallowing the conclusion,” and is one of the standard ways to respond to a situation of this sort. It’s clearly the right way to go here since the mathematical reasoning is beyond question.

But now let’s consider a case where the opposite happens and we end up “rejecting the reasoning” instead.