## Paradox Lost

In mathematical logic, a paradox is a self-contradictory statement. The best known is “This sentence is a lie.” Another is Bertrand Russell’s “barber paradox”. In a village, there is a barber who shaves everyone who  does not shave themselves. So, who shaves the barber? Neither ‘the barber’ nor ‘someone else’ is logically acceptable. If it is the barber, then he shaves himself — but we are told that he doesn’t. But, if it is someone else, then the barber does not shave himself…but we are told that he shaves all such people, so he does shave himself !

In the real world. there are plenty of get-outs (are we talking about shaving beards here, or legs, or what? Is the barber a woman? Can such a barber actually exist anyway?) But in mathematics, a more carefully stated version of Russell’s paradox ruined the life’s work of Gottlob Frege, who attempted to base the whole of mathematics on set theory — the study of collections of objects, and how these can be combined to form other collections.

Here’s another famous (alleged) paradox:

Protagoras was a Greek lawyer who lived and taught in the fifth century BC. He had a student, and it was agreed that the student would pay for his teaching after he had won his first case. But, the student did not get any clients, and eventually Protagoras threatened to sue him. Protagoras reckoned that he would win either way: if the court upheld his case, the student would be required to pay up, but if Protagoras lost, then by their agreement the student would have to pay anyway. The student argued exactly the other way round: if Protagoras won, then by their agreement the student would not have to pay, but if Protagoras lost, the court would have ruled that the student did not have to pay.

Is this a genuine logical paradox or not?

Your comments, answers are welcome 🙂

Ref; Professor Ian Stewart’s Cabinet of Mathematical Curiosities

More later,

Nalin Pithwa

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