The Paradox Salad
The Paradox Salad
What came first? The chicken or the egg?
By M. Awais Aftab
Which came first, the chicken or the egg? Was it the egg that got laid first or was it the chicken [pun intended]? The question, however, was not hatched for mere amusement. It is not just a question about chickens and eggs; it is also a question about how life and universe began. Such philosophical problems are often phrased in comical paradoxes: Can Achilles overtake the tortoise in a race? Are all Cretans liars? Can the barber shave himself? As baffling as it may seem, these are the questions which the great minds of humanity have been trying to solve for more than two millennia...
This paradox consists of a single sentence (in various versions) and yet becomes frustratingly cyclic: "This sentence is false". If this sentence is true, then this sentence has to be false (by its own accord), and if this sentence is false, then it has to be true (by negation). And so on. It is frustrating because this sentence does not seem to conform to the binary idea of true value i.e. a sentence is either true or false, but this sentence seems to be neither.
Another famous version of the Liar's Paradox is Epimenides paradox. A Cretan says: "All Cretans are liars". If all Cretans lie, then this statement is a lie too, and hence Cretans speak the truth, but if Cretans speak the truth, this means the sentence is true and all Cretans lie...
Possible Approach: Many philosophers have offered their solution to the Liar's paradox. One of these suggested answers says that there is nothing paradoxical about this statement because all statements imply their truth in implicit assertion. For example, to say "It is true that one is less than two" is no different than to say "One is less than two" because the latter already contains the assertion of its truth without mentioning it. Seen from this point of view, "This statement is false" is logically equivalent to "This statement is true and this statement is false". Since this is a simple contradiction, this statement is therefore simply inaccurate, not paradoxical.
The Hangman Paradox
Imagine a judge who tells a prisoner that he is to be hanged but that he will be hanged any weekday in the coming week and that the day of hanging would be a surprise to him. The prisoner however, reasons that he cannot be hanged since the condition of hanging being a surprise can never be logically met. This is how he reasons: Since hanging is supposed to be a surprise, it cannot be on Friday, because if he has not been hanged by Thursday, there is just one day left and hence there would be no surprise. It cannot also be Thursday, because if he has not been hanged by Wednesday, there would be two days left, Thursday and Friday, and since Friday is eliminated, Thursday is the only day left and it would not be a surprise. Extending this reasoning, he concludes that he cannot be hanged on Wednesday, Tuesday or Monday. However, the prisoner was hanged on Tuesday, and as expected, it came as a total surprise to him. But what was wrong with his reasoning?
Possible Approach: The argument starts by assuming that the prisoner would be alive on Thursday and would not have been killed in the days before. Hence, on Thursday, he would come to know that Friday is the last day left. However, this information cannot be possibly known before Thursday, and by extending this certainty backwards in time, the prisoner is making a logical mistake.
This problem presents us with this hypothetical situation: there is a town in which there is only one male barber. Some people in the town shave themselves, while others get it done by the barber. The barber obeys a simple logical rule: he shaves only those men who do not shave themselves.
Now, the problem emerges when the barber needs a shave. Does the barber shave himself? If he doesn't, then he has to abide by the rule and shave himself. And if he does shave himself, then he has to abide by the rule and does not shave himself.
Barber's paradox is actually a popular version of Russell's Paradox, which was discovered by Bertrand Russell and is based on the set theory of mathematics. There are some sets which are members of themselves, and there are some sets which are not members of themselves [such as a null set]. Russell asks to consider the set of all sets which are not members of themselves. The questions arises, is this set a member of itself?
First consider the possibility that it is a member of itself. But how can it be a member of this set, because the set contains only those sets which are not members of themselves.
So, let us consider the second possibility that it is not a member of itself, but if it is not a member of itself, it is a set which is not a member of itself, and therefore should be included in the set of all sets which are not members of themselves!
Possible Approach: While it may be easy to say the barber of the paradox does not exist in reality, to say that such a set does not exist is to threaten the very foundation of set theory in mathematics as it did in the early 19th century. Russell himself and other mathematicians responded to this paradox by modifying the set theory in a way that would avoid this paradox from emerging in the first place.
Ship of Theseus
Imagine a ship. It gets damaged as time passes. Its damaged parts are replaced by new parts. Eventually with time, all the parts of the ship have been replaced by new parts. Is it still the same ship? Furthermore, if all the parts that had been removed were reassembled to create a ship, which of these two ships would be the original ship?
Possible Approach: The answer to this paradox depends on what is exactly meant by the words "the same". If the same means consisting of those exact particles of matter of which it was composed, then obviously the ship is not the same. But if same means that the ship possess the same design and structure and spatio-temporal location, then yes, it is the same ship. [By the way, this paradox can also be applied to human body. In a matter of 10-15 years, all the particles that form our body are eventually replaced by new particles. Do we still remain the same persons as we were before?]
It is a proof of obvious contradiction. For example, proving that 1 = 2
(1) X = Y
(2) X^2 = XY [Multiply both sides by X]
(3) X^2 – Y^2 = XY – Y^2 [Subtract Y^2 from both sides]
(4) (X+Y)(X-Y) = Y(X-Y) [Factor both sides]
(5) (X+Y) = Y [Divide both sides by (X-Y)]
(6) Y+Y = Y [Since X=Y]
(7) 2Y = Y [Add the Y's]
(8) 2 = 1 [Divide both sides by Y]
Possible Approach: The problem exists in line (5). The term (X-Y) is equal to zero because X=Y, and X-X= 0. In line (5) we divide both sides by (X-Y) i.e. zero, and anything divided by zero is an undefined entity in mathematics.
Can an omnipotent being (i.e. God) create a rock so heavy that He cannot lift it? If He cannot lift it, then He is not omnipotent, and if He cannot create such a rock, then He is still not omnipotent.
Possible Approach: Most philosophers prefer to either modify the notion of omnipotence or do away with the idea of omnipotence altogether. One possible answer to this paradox can be that an omnipotent being can do the logically impossible tasks i.e. an omnipotent being can create a rock that it cannot lift and then still lift it up. Such an omnipotent being would also be possible to make two plus two equal to five. The paradox is resolved only at the expense of logical consistency.
Zeno's Paradox: Achilles and the Tortoise
Zeno of Elea was a Greek philosopher who invented many paradoxes, one of which deals with a race between Achilles and the tortoise. Since Achilles is over-confident of his victory, he gives the tortoise a head start of 100m in the race. Both Achilles and Tortoise start at the same instant and at constant speeds. By the time Achilles has covered the 100m, the Tortoise has moved ahead by a distance of one meter. When Achilles covers that 1m, by that time the Tortoise has moved ahead by a distance of 0.01 m. When Achilles covers that 0.01 m, the Tortoise has moved ahead by 0.0001 m. Whenever Achilles reaches where the Tortoise had been, he still has some distance to cover, no matter how small it is. Achilles would have to take an infinite number of such steps to overtake the Tortoise, which means that he will always be behind the Tortoise!
Possible Approach: The paradox assumes that an infinite number of steps cannot be taken, but this is a false assumption. Sum of infinite terms can actually be a finite number, as students of calculus would be well aware.
The total of infinite number of steps can be a finite distance covered in a finite time. This should not sound surprising because if we take a finite amount of distance, say one meter, we can divide it into an infinite number of pieces, and if we add up those infinite number of pieces, we would again get a finite distance of one meter. Hence, Achilles would overtake the Tortoise and win the race.
[Cover Story published in Us Magazine 11 December 2009.]