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.
6. The bicycle

Our examples tend to suggest the following discomfiting fact. A line of reasoning may have a flaw so subtle that we would never suspect its existence but for the fact that the reasoning ends up in a “crazy conclusion.”

In other words, had the conclusion looked banal, or otherwise unremarkable, the flaw in the reasoning would likely have gone unnoticed!

We haven’t seen this yet because our examples have all had “crazy conclusions” which prompted us to inspect their attendant lines of reasoning for subtle errors. But we’re not always so lucky.

Consider this curious case:

A rope is attached to the lower pedal of a normal bicycle and pulled gently backwards as shown. Which way will the bicycle be caused to move? (Assume that the bicycle remains balanced somehow and does not skid along the ground.)

One thinks, if the rope is pulled like that, the pedals will be caused to move in the same way as when the bicycle is normally pedalled. So the bicycle should be caused to move forward.

This reasoning looks natural and the conclusion also seems rather banal. So the reasoning is likely to be accepted. Unfortunately, it is quite wrong. Surprisingly, the bicycle will be caused to move backwards if you pull on the rope like that and you can verify this with any regular bicycle.

Upon seeing this, many people immediately seek the correct explanation for this unexpected behaviour in terms of various physical principles, but we’re not interested in that here. Our question rather is why people tend to endorse the erroneous piece of reasoning above.

It’s partly because the logical error committed is subtle and practically impossible to detect. (We won’t try to diagnose it here.) But, equally important, nothing hints at an error, since the conclusion of the reasoning – that the bicycle will move forward – is entirely innocuous!

So a subtle logical error that culminates in an innocuous conclusion can be quite a hazardous thing. It’s not so bad if it just concerns a bicycle but the hazard could easily be more substantial, as we will see below.

Many people pride themselves on their ability to detect flawed reasoning, but the fact is that some logical errors are so subtle that we actually need the help of a “crazy conclusion” to infer their existence.

Indeed, the situation is a bit worse.

For even when the conclusion is so crazy that it is absolutely certain that the reasoning must contain an error, we often still have difficulty locating the error! In such a case, had there been no indication that anything was amiss, we are practically guaranteed to have swallowed the erroneous reasoning.

Let’s consider a case of this sort to see how “invisible” a logical error can be, even when you know it must be there. This case is 2,000 years old, and no one has yet found the error.