PARADOXES

Here's a list of Logical Paradoxes for reference:


Zeno’s Paradox of the Tortoise and Achilles

Zeno of Elea (circa 450 b.c.) is credited with creating several famous paradoxes, but by far the best known is the paradox of the Tortoise and Achilles. (Achilles was the great Greek hero of Homer's The Illiad.) It has inspired many writers and thinkers through the ages, notably Lewis Carroll and Douglas Hofstadter, who also wrote dialogues involving the Tortoise and Achilles.

The original goes something like this:

The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow.
“How big a head start do you need?” he asked the Tortoise with a smile.
“Ten meters,” the latter replied.
Achilles laughed louder than ever. “You will surely lose, my friend, in that case,” he told the Tortoise, “but let us race, if you wish it.”
“On the contrary,” said the Tortoise, “I will win, and I can prove it to you by a simple argument.”
“Go on then,” Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.
“Suppose,” began the Tortoise, “that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?”
“Very quickly,” Achilles affirmed.
“And in that time, how far should I have gone, do you think?”
“Perhaps a meter – no more,” said Achilles after a moment's thought.
“Very well,” replied the Tortoise, “so now there is a meter between us. And you would catch up that distance very quickly?”
“Very quickly indeed!”
“And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?”

“Ye-es,” said Achilles slowly.
“And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance,” the Tortoise continued smoothly.
Achilles said nothing.
“And so you see, in each moment you must be catching up the distance between us, and yet I – at the same time – will be adding a new distance, however small, for you to catch up again.”
“Indeed, it must be so,” said Achilles wearily.
“And so you can never catch up,” the Tortoise concluded sympathetically.
“You are right, as always,” said Achilles sadly – and conceded the race.

Zeno's Paradox may be rephrased as follows. Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room.

What this actually does is to make all motion impossible, for before I can cover half the distance I must cover half of half the distance, and before I can do that I must cover half of half of half of the distance, and so on, so that in reality I can never move any distance at all, because doing so involves moving an infinite number of small intermediate distances first.

Now, since motion obviously is possible, the question arises, what is wrong with Zeno? What is the "flaw in the logic?" If you are giving the matter your full attention, it should begin to make you squirm a bit, for on its face the logic of the situation seems unassailable. You shouldn't be able to cross the room, and the Tortoise should win the race! Yet we know better. Hmm.

Rather than tackle Zeno head-on, let us pause to notice something remarkable.

Suppose we take Zeno's Paradox at face value for the moment, and agree with him that before I can walk a mile I must first walk a half-mile. And before I can walk the remaining half-mile I must first cover half of it, that is, a quarter-mile, and then an eighth-mile, and then a sixteenth-mile, and then a thirty-secondth-mile, and so on. Well, suppose I could cover all these infinite number of small distances, how far should I have walked? One mile! In other words,



At first this may seem impossible: adding up an infinite number of positive distances should give an infinite distance for the sum. But it doesn't – in this case it gives a finite sum; indeed, all these distances add up to 1! A little reflection will reveal that this isn't so strange after all: if I can divide up a finite distance into an infinite number of small distances, then adding all those distances together should just give me back the finite distance I started with. (An infinite sum such as the one above is known in mathematics as an infinite series, and when such a sum adds up to a finite number we say that the series is summable.)
Now the resolution to Zeno's Paradox is easy. Obviously, it will take me some fixed time to cross half the distance to the other side of the room, say 2 seconds. How long will it take to cross half the remaining distance? Half as long – only 1 second. Covering half of the remaining distance (an eighth of the total) will take only half a second. And so one. And once I have covered all the infinitely many sub-distances and added up all the time it took to traverse them? Only 4 seconds, and here I am, on the other side of the room after all.

And poor old Achilles would have won his race.

ADDENDUM

So that you don't get to feeling too complacent about infinities in the small, here's a similar paradox for you to get stirred about...

THOMPSON'S LAMP: Consider a lamp, with a switch. Hit the switch once, it turns it on. Hit it again, it turns it off. Let us imagine there is a being with supernatural powers who likes to play with this lamp as follows. First, he turns it on. At the end of one minute, he turns it off. At the end of half a minute, he turns it on again. At the end of a quarter of a minute, he turns it off. In one eighth of a minute, he turns it on again. And so on, hitting the switch each time after waiting exactly one-half the time he waited before hitting it the last time. Applying the above discussion, it is easy to see that all these infinitely many time intervals add up to exactly two minutes.

QUESTION: At the end of two minutes, is the lamp on, or off?

ANOTHER QUESTION: Here the lamp started out being off. Would it have made any difference if it had started out being on?

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Hilbert's Hotel

Hilbert’s Hotel is a (hypothetical) hotel with an infinite number of rooms, each one of which is occupied. The hotel gives rise to a paradox: the hotel is full, and yet it has vacancies.

That the hotel is full is obvious. It has an infinite number of rooms, and an infinite of guests; every room is occupied. That the hotel has vacancies is a little more difficult to demonstrate.

Suppose that a new visitor arrives; can he be accommodated? At first it seems that he cannot, but then the hotel clerk has an idea: He moves the guest in Room 1 to Room 2, and the guest in Room 2 to Room 3, and so on. Every guest is moved to the next room along.

For every guest, in every room, there is another room into which they can be moved. This leaves Room 1 vacant for the new visitor. Although the hotel is full, then, the new guest can be accommodated in Room 1.

It is not only one new guest that can be accommodated; in fact, Hilbert’s Hotel has an infinite number of vacancies. By moving every guest to the room the number of which is double the number of their current room, all of the odd numbered rooms can be vacated for new guests. There are, of course, an infinite number of odd numbered rooms, and so an infinite number of new guests can be accommodated.

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The Arrow Paradox

Zeno’s arrow paradox appears to show that motion is impossible.

It works by taking a snapshot of an arrow at a point (either in space or in time) in its flight. At that point, and at every other, the arrow is motionless. If there is no point, spatially or temporally, at which the arrow is moving, though, then the arrow is motionless. Contrary to appearances, an arrow in flight cannot move.

If we had a film of the arrow in flight, and broke it down to its individual frames, we would see that in each frame the arrow is simply hovering in the air. It is only when you put all the frames together that the arrow appears to move. In each frame, i.e. at each point, the arrow is motionless.

This is true irrespective of whether we think in terms of time or space.

Motion occurs through space, not at a single point in space. To move, something must get from one point to another, and so at each point considered individually, the arrow is still.

Similarly, motion takes time, it doesn’t occur instantaneously. At any specific point in time, therefore, the arrow cannot be moving.

If at every point and at every moment in its flight the arrow is still, though, then how is it possible for it to move from the bow to its target? If the arrow is made of wood at every point in its flight, then it must be wooden; it can’t be plastic. If it is sharp at every point in its flight, then it must be sharp, not blunt. Similarly, if the arrow is motionless at every point in its flight, then it must be still, not moving.

Contrary to appearances, then, arrows cannot move towards targets. In fact, similar reasoning applies to any other alleged case of motion, so it seems that movement in general is impossible.

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The Barber Paradox

The Barber paradox is attributed to the British philosopher Bertrand Russell. It highlights a fundamental problem in mathematics, exposing an inconsistency in the basic principles on which mathematics is founded.

The barber paradox asks us to consider the following situation:

In a village, the barber shaves everyone who does not shave himself, but no one else.

The question that prompts the paradox is this:

Who shaves the barber?

No matter how we try to answer this question, we get into trouble.

If we say that the barber shaves himself, then we get into trouble. The barber shaves only those who do not shave themselves, so if he shaves himself then he doesn’t shave himself, which is self-contradictory.

If we say that the barber does not shave himself, then problems also arise. The barber shaves everyone who does not shave himself, so if he doesn’t shave himself then he shaves himself, which is again absurd.

Even if we try to get clever, saying that the barber is a woman, we do not evade the paradox. If the barber is a woman, then she either shaves herself (and so is one of the people not shaved by the barber), or does not shave herself (and so is one of the people shaved by the barber).

Both cases, then, are impossible; the barber can neither shave himself nor not shave himself. The question ‘Who shaves the barber?’ is unanswerable.

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The Paradox of the Heap

A single grain of sand is not a heap; that's obvious. A heap is a collection of things, you need several things to make it up. Two grains of sand isn't a heap either; a heap is a collection of several things, more than just a couple.
The concept of a heap is fuzzy, though; there’s no precise number that marks the difference between heaps and non-heaps. It’s not as though 37 grains of sand aren’t a heap but 38 are. Defining precisely how many things one needs in order to have a heap is impossible.

This is what gives rise to the paradox of the heap (also called the “Sorites paradox”, sorites being the Greek word for heap).

Suppose that we have a collection of a million grains of sand. That is absolutely, definitely, undeniably a heap.

Because there is no precise number that separates heaps from non-heaps, removing a single grain of sand from a heap will never turn it into a non-heap. If you have a heap of sand, and you take away a single grain, then you still have a heap.

If you have a heap of a million grains of sand, though, and repeatedly take away a single grain of sand, doing so 999,999 times, then what do we have at the end of the process? Is it a heap or not?

Taking away a single grain of sand cannot turn a heap into a non-heap. We had a heap of sand at that beginning of the process. All we did was take away single grains of sand. Therefore what we have at the end of this process can only be a heap.

What we have at end of the process, though, is a single grain of sand, and, as we said at the beginning, a single grain of sand is obviously not a heap. The single grain, then, both is and is not a heap.

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Hempel's Ravens

If two statements are logically equivalent, if they assert exactly the same thing, then any evidence for one is evidence for the other.

This principle appears to be truism. Consider the two statements “Terry and Judith are my parents” and “I am Terry and Judith’s child”. These statements are logically equivalent, they say the same thing. There is no evidence that would support one of them without supporting the other.

No matter how superficially plausible this principle, however, the Hempel’s Ravens paradox seems to show that it is false. The Hempel’s Ravens paradox uses the principle to prove the absurd conclusion that an observation of a green parrot is evidence that ravens are black. The only way of avoiding this clearly unacceptable conclusion is to reject the principle above.

The paradox goes like this:

Consider the two statements:

(1) “All ravens are black.”
(2) “Everything that isn't black, isn't a raven.”

These two statements say exactly the same thing. The first statement says that everything of a particular kind has a certain property. The second statement says that everything that lacks that property isn’t of that kind.

The two statements are therefore logically equivalent; they are true and false in exactly the same circumstances. If there is anything that is a raven but isn’t black then both (1) and (2) are false; oherwise, they are both true. As the two statements are logically equivalent, any observation that supports one will also support the other.

Suppose, then, that I observe a green parrot. This observation confirms (2), "Everything that isn't black isn't a raven". A green parrot isn’t black and isn’t a raven. The observation is evidence that (2) is true.

Given what has been said so far, my observation of a green parrot must also confirm (1). (1) and (2) are logically equivalent, so any evidence for one is evidence for the other. My observation of a green parrot, then, is evidence for the statement, “All ravens are black”. In fact, any observation of something that isn’t black and isn’t a raven is evidence that ravens are black.

This, though, is absurd; there is no way that we can discover what colour a raven is without looking at a raven.

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The Paradox of Latent Belief

Belief is a cognitive state; believing something is a matter of having a certain kind of positive mental attitude towards it, of thinking that it is true. There are, however, numerous propositions that we believe to be true even though we have never entertained them. Paradoxically, it seems that belief is independent of thought.

Take, for example, the proposition “I have more nostrils than noses.” You know this proposition, and have known it for a long time (and, as the tripartite theory of knowledge explains, belief is necessary for knowledge). However, until you read this page you had never entertained it.

This shows that belief is independent of thought, that you do not need to think a thing in order to know it. You have never engaged in any mental activity that could be described as assenting to the idea that you have more nostrils than noses, and yet you have long known that proposition to be true.

The same can be said of many other propositions: “flamingos have fewer feet than elephants”; “42 has two fewer digits in it than 1966”; etc.

If you are tempted to suggest that before reading this page you did not know these propositions, then consider the following:

You now know that you have more nostrils than noses, that flamingos have fewer feet than elephants, and that 42 has two fewer digits in it than 1966. This page did not teach you any of these things. Therefore, you must already have known them.

It seems that you can know things without ever having entertained them; belief is possible without thought.

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The Liar Paradox

Disney's PinocchioThe Liar Paradox is among the simplest of paradoxes. It can be traced back at least as far as Eubulides of Miletus, a fourth-century B.C. Greek philosopher.

Eubulides’ version of the paradox is this: A man says that he is lying; is what he says true or false?

However we answer this question, difficulties arise.

If we suggest that what the man says is true, then we end in contradiction: if the man’s claim that he is lying is true, then he is lying, in which case what he says is false.

If we suggest that what the man says is false, then we are no better off: if the man’s claim that he is lying is false, then he is not lying, in which case what he says is true.

Both answers give rise to logical contradictions; it cannot be the case either that what the man says is true or that what the man says is false.

The Liar Paradox is sometimes referred to as “Epimenides’ Paradox”, after the sixth-century B.C. Cretan who asserted that all Cretans are liars. The apostle Paul makes reference to Epimenides in Titus 1:12, writing, “It was one of them, their very own prophet who said, ‘Cretans are always liars, vicious brutes, lazy gluttons.’”

Epimenides’ statement alone does not give rise to a paradox. What he says can’t be true, for if Cretans are always liars, and he is a Cretan, then he must be lying, in which case his statement is false. His statement could be false however; it could be that Epimenides is dishonest but that not all Cretans are liars.

Paul, though, excludes this dissolution of the paradox, proceeding to say in the following verse, “This testimony is true.” This leaves Paul asserting that Epimenides truly said that he (and all other Cretans) are liars, which takes us back to Eubulides’ paradox above; Paul cannot be right.

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The Preface Paradox

Many authors introduce their books with a caution: it is inevitable that somewhere in this book there is an error. This is a common claim in prefaces. But do the authors that write these claims believe them or not?

If the author is asked of each specific claim in the book Is this an error? then he will say No. For each individual claim that the author makes, he believes that it is true.

If the author believes that each claim is true, though, then mustn’t he believe that every claim is true? A collection of claims, none of which is an error, contains no errors. The author believes that his book is a collection of such claims; he believes that it contains no errors.

Yet the author also believes that somewhere in the book he will have made a mistake. Aware of his fallibility, he believes that not every claim in the book is true, that somewhere in the book there is an error.

What is really odd about this is not that authors have inconsistent beliefs, it is that the author is being perfectly rational in believing both that his book does and does not contain errors.

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The Problem of the Specious Present

Nihilism is the view that nothing exists. There are different kinds of nihilism; one can be a moral nihilist, for instance, holding that morality does not exist, or a religious nihilist, holding that God does not exist. The problem of the specious present supports a universal nihilism, the view that nothing whatsoever exists.

In order for something to exist it must have duration, it must exist for a certain amount of time. To say that something exists for no time at all, that at the very moment that it comes into existence it also passes out of it, is to say that it doesn’t exist at all. Unicorns exist for no time at all; so do square circles. Things that exist for no time at all don’t exist. In order for something to exist it must have duration.

The past and the future do not exist; they are not there, in the world. Perhaps the past once existed, and perhaps its effects can still be seen in the world today, but the past doesn’t exist now; if it exists now, then where is it? And perhaps the future will exist one day, but it doesn’t exist yet; again, if it exists now, then where is it? The past and the future clearly do not exist; the universe consists only of the gap between them, the present.

How large is the gap between the past and the future? What is the duration of the present? A minute? A second? A nano-second?

Clearly the present does not last as long as a minute. A minute consists of different temporal parts. First comes its beginning, then its middle, and then its end. Each of its parts occurs at a different time. If its beginning is present then its middle and end are future. If its middle is present, then its beginning is past and its end is future. If its end is present then its beginning and middle are past. If the present lasted as long as a minute then it would consist of past, present, and future elements, but that would be absurd; the present must be wholly present.

The same, though, could be said if the present were of shorter duration, lasting only a second, or even only a nano-second. In either case, the present would have temporal parts: a beginning, a middle, and an end. If its beginning were present then its middle and end would be future. If its middle were present, then its beginning would be past and its end would be future. If its end were present then its beginning and middle would be past. If the present has any duration at all then it consists of past, present, and future elements, but that, as I said before, would be absurd.

The present, then, has no duration; there is no gap between the past and the future. It has already been seen, though, that to say that something has no duration is to say that it does not exist. The present, then, like the past and the future, does not exist.

If there is neither past, nor present, nor future, though, then what is there? Nothing. Nothing exists at all. Universal nihilism is true.

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The Paradox of the Stone

God is all-powerful, or as theologians put it, “omnipotent”; there is nothing that he cannot do. This is part of the definition of “God”.

So can God create a stone that is so heavy that he cannot lift it? Either he can or he can’t.

If God can’t, then he isn’t all-powerful. If God can’t create a stone that he can’t lift, then there is something that he can’t do: create the stone.

If God can create a stone that is so heavy that he can’t lift it, though, then he also isn’t all-powerful. If God can create a stone that is so heavy that he can’t lift it, then there’s something that he can’t do: lift that stone.

There is, therefore, no way of answering the question above that preserves God’s omnipotence. If there is an omnipotent God, then he neither can nor can’t create a stone so heavy that he cannot lift it. This, though, is absurd; he must be either able or unable to perform this feat.

This is the paradox of omnipotence. Many critics of theism have used it to argue that the concept of omnipotence is self-contradictory, that there can be no omnipotent being, and so that God cannot exist.

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Theseus' Ship

CLICK to read essay on PERSONALITYTheseus 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?

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The Tristram Shandy Paradox

Tristram ShandyTristram Shandy is an author 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 Shandy’s 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 doesn’t 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?

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The Two Envelope Paradox

Which envelope?You’re on a game show. You’re 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 you’ll 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 you’ll halve your money to $500. You could lose $500 by swapping.

If you decide to exchange envelopes, 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, you’re better off swapping it for the other one when you get the chance.

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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 judge’s instructions. The sentence handed down to him cannot be carried out.

He is hanged on the morning of the third day, much to his surprise.

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