The Paradox of the Stone
God is all-powerful, or as theologians put it, ďomnipotentĒ; there is nothing that he cannot do. This is part of the definition of ďGodĒ.
Can God create a stone that is so heavy that he cannot lift it? Either he can or he canít.
If God canít, then he isnít all-powerful. If God canít create a stone that he canít lift, then there is something that he canít do. Since God is all-powerful, though, he can do everything; thereís nothing that he canít do. That includes creating a stone that he canít lift. Since God is omnipotent, then, he must be able to create a stone that is so heavy that he canít lift it.
If God can create a stone that is so heavy that he canít lift it, though, then he also isnít all-powerful. If God can create a stone that is so heavy that he canít lift it, then thereís something that he canít do: lift that stone. Since God is omnipotent, then, he must be unable to create a stone that is so heavy that he canít lift it.
Godís omnipotence, then, means that he is both able and unable to create a stone that is so heavy that he canít lift it. He canít be both able and unable to do this, though; either he can or he canít. So which is it to be?
Theseus is remembered in Greek mythology as the slayer of the Minotaur. For years, the Athenians had been sending sacrifices to be given to the Minotaur, a half-man, half-bull beast who inhabited the labyrinth of Knossos. One year, Theseus braved the labyrinth, and killed the Minotaur.
The ship in which he returned was long preserved. As parts of the ship needed repair, it was rebuilt plank by plank. Suppose that, eventually, every plank was replaced; would it still have been the same ship?
A strong case can be made for saying that it would have been: When the first plank was replaced, the ship would still have been Theseus' ship. When the second was replaced, the ship would still have been Theseus' ship. Changing a single plank can never turn one ship into another. Even when every plank had been replaced, then, and no part of the original ship remained, it would still have been Theseus' ship.
Suppose, though, that each of the planks removed from Theseus' ship was restored, and that these planks were then recombined to once again form a ship. Would this have been Theseus' ship? Again, a strong case can be made for saying that it would have been: this ship would have had precisely the same parts as Theseus' ship, arranged in precisely the same way.
If this happened, then, then it would seem that Theseus had returned from Knossos in two ships. First, there would have been Theseus' ship that has had each of its parts replaced one by one. Second, there would have been Theseus' ship that had been dismantled, restored, and then reassembled. Each of them would have been Theseus' ship.
Theseus, though, sailed in only one ship. Which one?
The Tristram Shandy Paradox
Tristram Shandy is a novelist writing an auto-biography. Unfortunately, he writes very slowly; each day of his life takes him a year to write about.
The Tristram Shandy paradox asks: If Shandy continues at this rate for eternity then will his book ever be finished?
Russell, who invented this paradox, suggested that the book would be finished. Given an infinite amount of time, for every day in Shandyís life there is a year to spend writing about it; there are, after all, an infinite number of years in which to write the autobiography. The autobiography therefore can be completed.
This doesnít seem right though. With each passing year, Shandy completes his writing about one day, but leaves another three hundred and sixty-four days undocumented. Every year, then, there are three hundred and sixty-four days more for Shandy to write about; the more time passess, the further behind he falls.
How can it be that Shandy ever falls further behind, and yet that given an eternity he will complete his work?
The Two Envelope Paradox
Youíre on a game show. Youíre given a choice between two envelopes containing money, knowing that one of the envelopes contains twice as much as the other. You get to keep the contents of whichever envelope you choose.
Having chosen the envelope, you open it, and find that it contains $1000. Before the game ends, though, you get one chance to change your mind, to exchange your envelope for the other one. The two envelope paradox arises because no matter which envelope you chose in the first place, it always seems that swapping is the rational thing to do.
Suppose that you chose the envelope containing the least money. If you swap for the other envelope, then youíll double your money to $2000. You could gain $1000 by swapping.
Suppose that you chose the envelope containing the most money. If you swap for the other envelope, then youíll halve your money to $500. You could lose $500 by swapping.
If you decide to exchange envelopes, then, then you could gain twice as much as you could lose. Your chances of gaining are equal to your chances of losing. Exchanging envelopes is therefore the rational thing to do.
Had you chosen the other envelope, though, then you could have reasoned in precisely the same way. Whatever amount of money you had taken from the other envelope, you could have reasoned that by exchanging you had twice as much to gain as to lose, and that your chances of gaining and losing were equal, and so that you should choose to swap.
Whichever envelope you choose in the first place, then, youíre better off swapping it for the other one when you get the chance.
The Unexpected Hanging
A murderer had been found guilty of a particularly heinous crime. The judge sentencing the murderer decides that death is too good for him; he wants to make him suffer. He passes his sentence, "You will be taken from this place, and hanged from the neck until you are dead. Before that, though, you will suffer anguish, waiting, never knowing whether this will be the day that you will die. One morning, sometime in the next week, it will happen, but until it does you will live in fear."
The murderer leaves the courtroom with a light heart, knowing that the sentence handed down to him cannot be carried out.
He reasons like this:
Suppose that on the seventh morning I am alive. I will know that that is the day that I am to die. But the judge said that I would not know the day that I am to die. Therefore I will not be hanged on the seventh day. The sixth day is the last day that it could be.
But in that case, if I am alive on the sixth morning then I will know that it is the sixth day on which I am to be hanged. But the judge said that I would not know the day that I am to die. Therefore I will not be hanged on the sixth day.
He continues, applying the same reasoning to the fifth day, and then to the fourth, and so on, concluding that he cannot be hanged on any day according to the judgeís instructions. The sentence handed down to him cannot be carried out.