Paradoxes

In logic, a paradox is a contradictory or implausible conclusion which seems to follow by valid argument from true premises. Aside from Zeno’s paradox, and Maxwell’s demon, proposed by James Clerk Maxwell, Erwin Schrödinger’s cat and Olber’s paradox, all dealt with in other places, the most interesting paradoxes are all logical challenges.

The paradox of the Spanish barber concerns a barber who is asked how business is doing. “Not badly,” says the barber. “I shave everybody in the village who does not shave himself.” The problem: who shaves the barber?

Logicians with a smattering of Greek will
share my delight in this Athenian street sign.

Epimenides of Crete is credited with one of the simplest paradoxes: “All Cretans are liars”, meaning by implication that this statement (made by a Cretan) was necessarily untrue. But if it is untrue, then not all statements made by Cretans are liars, and so on.

There is a more complex form of this paradox, consisting of two sentences. “The next sentence is false” and “The previous sentence is true”. Or "there are two erors in this sentence".

The dilemma of the crocodile: a crocodile seizes a child, but promises to let the child go, if the father guesses correctly whether he will do so or not. If the father offers as his guess the opinion that the crocodile will not return the child, what should an honest crocodile do?

A lawyer is trained by a teacher who says “you must pay me for your tuition after you win your first case”. Several years go by, during which time the teacher gets annoyed because the lawyer has yet to win a case, and sues the lawyer, saying “If I win my case, you must pay me, but if I lose, you have won your first case and must pay me.”

“Not so fast”, says the young lawyer. “If I lose the case, I have yet to win a case and need not pay you. But if I win, then by the court’s judgement, I do not have to pay you.” Does the lawyer have to pay?

Some words describe themselves, so “short” is a short word, but “long” is not a long word, “English” is an English word, but “German” is not a German word, and so on. We call words which describe themselves as autological, while words which do not describe themselves are called heterological.

But what about the word “heterological” — does it describe itself or not? If “heterological” is heterological, then it describes itself, and so it is autological. But if the word is autological, then that means it is a word that does not describe itself … or something.

Paradoxes can be useful ways of extending our knowledge, or at least ways of finding the right questions to ask. Fermi’s conjecture, also known as Fermi’s paradox, was offered by Enrico Fermi. In simple terms, it asks why, if the Galaxy is filled with intelligent and technological civilizations, haven’t they come to us yet?

There are several possible answers to this question (good taste on the ETs’ part, distance, or a recognition that contact with a superior civilisation is damaging to the more primitive one), but as we only have the vaguest idea what the right conditions for life and intelligence in our Galaxy, this paradox probably has no ready answer.

Paradoxes are also useful as a form of the mathematical proof called reductio ad absurdum, an argument which comes to an absurd or contradictory conclusion, hence showing that an initial assumption must be wrong. This shows up well in the so-called grandfather paradox of relativity. Imagine that your grandfather has just built a time machine, which you then use to go back in time, to give your grandfather the plans, so he can later build the time machine.

You reach him at a time where he has yet to meet your grandmother, he refuses to believe you, and in an argument, he steps out into a road, and is run over and killed by a passing car. He has now died before he met your grandmother, so you do not exist, since one of your parents does not exist, and the time machine does not exist, so you cannot be there in any case.

This is one of the arguments physicists use to support their belief in the causality principle. Others say the fact that we have never met time travellers proves that there will never be any, but this is probably one of the most useless of paradox types.

Let's get back to more practical stuff!