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

6. The bicycle

Our examples so far point to a certain disconcerting fact about logical reasoning that we should pause briefly to notice.

The disconcerting fact is that 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 actually 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 can’t get into that here. (See this video from the Simons Foundation.) 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. 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 sometimes 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.

All of this is somewhat disconcerting, as mentioned, and we’ll see this a couple of times in the three paradoxes that we have left to discuss. But let’s first consider how “invisible” a logical error can be, even when you know it must be there. Our next paradox is 2,000 years old, and while virtually everyone agrees this time that we must “reject the reasoning,” rather than “swallow the conclusion,” no one has yet satisfactorily explained where and how the reasoning goes wrong.