A witch, a barber & a paradox

Man likes to think of the human species as the most intelligent on Earth. The way we think, the way we reason, the way we reject alternatives that do not fit with logical thinking, all these assumed properties are used to justify the statement that man is a rational animal.

Of course, reality may well be different! Self-interest, the desire to protect oneself, action taken as part of a plot, political expediency etc. may lead to departure from logical action. It is in this context that paradoxes have a role to play.
What is a paradox? Dictionary defines it as a self-contradictory proposition. That is, if we start with such a proposition assuming it to be true, then after a series of logical steps, we are led to the conclusion that the proposition is false. Since we are accustomed to the belief that a proposition is either true or false, we find this situation very confusing. Nevertheless, paradoxes have added extra charm to how mathematics and logic have developed.
While writing his book on the foundations of mathematics (The Principia Mathematica), Bertrand Russell used a paradox to show how certain kinds of sets of items behave. Without worrying about the sets we can enjoy the paradox, often referred to as the Barber Paradox. It concerns a village barber who shaves all those and only those villagers who do not shave themselves. The question is: Does the barber shave himself? If the answer to this question is “Yes”, then the barber belongs to the class of villagers who do not shave themselves. This contradicts our original assumption that the barber shaves himself. If we opt for the other alternative, namely the answer to our question being “No”, then we have the barber not shaving himself. And so, by his rule, he must shave himself. Thus we face a contradiction both ways. This is what is known as a paradoxical situation. (It will amuse the reader to think of the barber paradox in modern times. In today’s situation, at least in the Western countries, we cannot rule out lady barbers. For a lady barber the above rule does not present any problem.)
Next, consider the story of a prisoner on death row awaiting the final moment at the gallows. A warden walks in and offers him the following choice. He tells the prisoner: “You have to guess whether you will be hanged or shot. We have decided what mode it will be; but if you guess correctly you will be hanged and if you guess wrongly, you will be shot. So make your guess.” After a little thought the prisoner replies: “I will be shot.” This reply put the warden in trouble! For, suppose, the decision is to shoot the prisoner. Since the prisoner has replied correctly, he should be hanged. But if he is to be hanged then the prisoner’s reply was wrong and so he should be shot. Again, either way the action proposed is self-contradictory.
An amusing paradox forms the central theme of the comic opera Rudigore, alternatively named The Witch’s Curse by Gilbert and Sullivan. The witch’s curse was on the family of a nobleman who had persecuted witches. The curse said that the head of the family must perform a criminal act every day, failing which he would die a most painful death on the same day. The result was that, even though by nature they were kind and gentle, these family heads from generation to generation became notorious as evildoers. Not everybody knew that they were performing such criminal acts with great reluctance, as essential action for survival. Rather they were anxiously looking for deliverance from this curse so that they could live a normal life.
A clever man, when he learnt of this scenario came up with a very simple solution. His advice to the current family head was to do nothing. The family head was surprised. Won’t such lack of criminal action invite the death penalty by the end of the day? Our clever hero assured him that provided he deliberately commits no criminal act, he is safe under the terms of the curse. For not doing anything was inviting death, which meant an attempt at suicide. An attempt at suicide was itself a crime and so by following this route the family head was quite safe! Thus the story ends happily with the good Baron free from the evil commitment of performing a crime every day.
As mentioned earlier, paradoxes appear in serous mathematical literature, too. The early work on dynamics, the science of motion has the so-called paradoxes of the Greek philosopher Zeno. The best known of Zeno’s paradoxes is the one relating to a race between the fast Greek runner Achilles and the slow-moving tortoise. The tortoise is given a slight lead, say 10 metres in a race over a 100 metres. All logic tells us that Achilles would win (provided, of course, that he does not sleep like the hare!). But what is wrong with this reasoning? By the time Achilles makes up the gap of 10 metres, the tortoise will have moved forward, say, by one metre. So Achilles has to run to make up that gap. By the time he does so, the tortoise will have moved still forward. And so this goes on. At every stage, Achilles has to make up some gap and so he is always behind the tortoise. In short, the tortoise will win the race.
Obviously there is something wrong with the argument. If it were true the Olympic gold medallist will also lose such a race. Where does the argument fail?
Paradoxes like these serve the purpose of keeping our brain sharp. So while you are relaxing think of the VIP who carried a grenade in his hand baggage while boarding an aircraft. His argument was that this prevented a grenade-carrying terrorist being on the same plane because some statistician had told him that the chance of two random passengers carrying grenades is extremely small.

The writer, a renowned astrophysicist, is professor emeritus at Inter-University Centre for Astronomy
and Astrophysics, Pune University Campus