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.

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

Some people treat paradoxes simply 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 a closing example, here’s a cheeky “proof” of the existence of a unicorn from the mischievous American logician 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 which 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!”

For, obviously, if there exists an

So all we have to do is prove that an

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 wrestling 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

One thing missing from this survey is an example of the patient pursuit and eventual snaring of such an elusive prey; the exposing of the precise error in a subtly fallacious line of reasoning. This is an art form in itself and an example would have been useful to see. 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 ...

Achilles & the tortoise

The surprise exam

Newcomb’s problem

Newcomb’s problem (sassy version)

Seeing and being

Logic test!

Philosophers say the strangest things

Favourite puzzles

Books on consciousness

Philosophy videos

Phinteresting

Philosopher biographies

Philosopher birthdays

Draft

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