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.
3. The helicopter

A certain wealthy man, who happens to own a helicopter, needs to travel from Madrid, where he lives, to New York.

He notices that Madrid and New York lie at the same latitude on the globe, six hours apart, and hits upon this bright idea.

Why not get in the helicopter, hover steadily in the air, and simply let the Earth rotate beneath. In six hours, New York would rotate into view, and the helicopter may then be lowered back to the ground.

The man’s logic sounds straightforward, but it also sounds a bit too good to be true. Can it really be that easy to cross the Atlantic? After all, even a jet plane needs about eight hours for the trip, pushing its engines hard. Why would airlines be expending all that effort if this man’s method were available?

And yet what was wrong with his reasoning? Is it that no helicopter could hover in the air that long? Would the Atlantic winds blow it away? Would Boeing and Airbus raise a ruckus?

No, the trouble is that the man has simply overlooked the Law of Inertia. If he tried his own suggestion, he would be dismayed to find that his helicopter would simply inherit the inertia of the rotating Earth, and would thus remain fixed above Madrid, rotating in space together with the Earth.

So his reasoning does not compel for long. Of course, it’s easy for us to say this now, because the Law of Inertia has become so familiar. But there was a time when this sort of logic would have been readily accepted. (The fact that a stone thrown straight up comes back straight down was once deemed evidence that the Earth was not moving!)

At any rate, the larger point is that this is the opposite case from the previous one. As before, we have a “compelling” line of reasoning that leads to an unbelievable conclusion, but, this time, rather than swallowing the conclusion, it proves correct to reject the reasoning.

These are our two basic options in such a situation, and we should now consider a case where it is unclear whether we should swallow the conclusion or reject the reasoning, even after prolonged reflection.

This is when we start to approach a genuine logical paradox.