A paradox is a statement that may be true, but somehow contradicts itself. It is one of the best outcomes of our usage of language, and brings to light situations that usually don't have a proper answer. Although, some paradoxes are known to have come about as a result of actual observations.
Of course, while paradoxes are contradictory situations, they are also dependant upon the way the statement is worded. The real fun in a paradox is thinking about the situation, and how it could possibly manifest itself. I'm listing a bunch of paradoxes I was able to think of and collect. Any additions are always welcome.
Sorites Paradox
One grain of sand does not make a heap. Two grains of sand do not make a heap either. Therefore, three grains do not make a heap either. This can go on, until say, 10000 grains of sand, but that would still not make it a heap of sand. How many grains then, make up a heap of sand?
Sorites Paradox
One grain of sand does not make a heap. Two grains of sand do not make a heap either. Therefore, three grains do not make a heap either. This can go on, until say, 10000 grains of sand, but that would still not make it a heap of sand. How many grains then, make up a heap of sand?
Analogous to this, $1 does not make a man rich, and therefore neither does $2. $3 does not either. This can go on, and thus $1000000 does not make a person rich either. How many dollars then, define "rich"?
This same paradox is often applied to the debate of abortion. A single zygote does not make a person, and neither does one cell more. But at what point does a fetus become a person?
Olber's Paradox
Olber's Paradox has already been discussed here, but deserved a mention, nonetheless. It's a very simple question, but one that holds profound consequences regarding our understanding of the Universe. In the Netwtonian Universe, there are an infinite number of stars, the Universe is infinitely big, and infinitely old. Ergo, everywhere you look in this infinite Universe, there are stars. So this should add up, and the night sky should be as bright as a star. But it is not. Why is the night sky dark?
Fermi's Paradox
The galaxy contains roughly a hundred billion stars. If even a very small fraction of these have planets which develop technological civilizations, there must be a very large number of such civilizations. If any of these civilizations produce cultures which colonize over interstellar distances, even at a small fraction of the speed of light, the galaxy should have been completely colonized in no more than a few million years. Since the galaxy is billions of years old, Earth should have been visited and colonized long ago. Why aren't they here?
This is Fermi's Paradox. There are several arguments for and against it, which I shall not delve into. However, I will add, that all-in-all, it is a generally weak paradox, since it relies upon an individual's opinion of how things should work. Some may be pedantic enough to expect it to have occurred, some optimistic enough to believe that it will occur, and some pessimistic enough to think that a civilization can only reach a certain point before it destroys itself.
Catch-22
Catch-22 is a military term that is confusing and difficult to describe. In short, its basic meaning is that if there was a rule, there is an exception existing to it in such a way that it is in essence, a circular argument. Here is an example from a certain regiment:The rule was simple: If you were sane, you had to go flying. If you were crazy, you were grounded. There was only one catch and that was Catch-22, which specified that a concern for one's safety in the face of dangers that were real and immediate was the process of a rational mind. Now, Orr was crazy and could be grounded. All he had to do was ask to fly, and as soon as he did, he would no longer be crazy and would have to fly more missions. However, Orr would be crazy to fly more missions and sane if he didn't, but if he was sane he had to fly them. If he flew them he was crazy and didn't have to, but if he didn't want to he was sane and had to.
Russell's Paradox
Russell's Paradox is mathematical in nature and is the most famous of the logical or set-theoretical paradoxes. Consider the set of all sets that are not members of themselves. Now, such a set appears to be a member of itself, but only if it is not a member of itself. Hence, the paradox.
You can read more on Russell's Paradox here.
Barber's Paradox
Figaro, a barber in some small Italian village shaves only those people in town who do not shave themselves. The question arises, does Figaro shave himself? If he does, then he cannot, because he is only supposed to shave those men who do not shave themselves. If he does not, then he has to shave himself.
Abian Paradox
There is no set A, in set-theory, such that A is the set of all those sets of which A is not an element. Now, assume that such a set A does in fact, exist. If A is an element of A, then A cannot be an element of A. So A is simultaneously an element of A, and not an element of A.
The Bootstrap Paradox
There are several variations of the Bootstrap Paradox. It's origin lies in trying to lift yourself by your own bootstraps - you can try and try, but no matter how strong you are, you cannot lift yourself up.
In computer terms, before you can load a program into a computer, you must first load the program loader. Seems simple, but if you really think over this, a "program loader" is a program that loads other programs. It is, in itself, a program. How does this program get loaded then? So, the essence of the paradox is: How do you get a program into a computer for the first time if the first program you want to load is the program that loads other programs?
In computer terms, before you can load a program into a computer, you must first load the program loader. Seems simple, but if you really think over this, a "program loader" is a program that loads other programs. It is, in itself, a program. How does this program get loaded then? So, the essence of the paradox is: How do you get a program into a computer for the first time if the first program you want to load is the program that loads other programs?
The Language Paradox
Very similar to the Bootstrap paradox. It is well known that when you write a program, you must compile it first. A compiler itself, though, is a program, which means that the compiler had to be written and compiled. So the question is, which came first, the Programming Language, or the Compiler?
Note: There is an answer to this question.
Grandfather Paradox
Probably the most famous paradox associated with time travel. Supposing that time travel is accomplished some time soon. A time traveler goes into the past and kills his own grandfather. Over the course of time then, his father will never be born, and thus he can never exist.
Visits from the Future
Similar to above, let us say that a time travel machine is invented in the near future, and further, that it is I who invents this machine. I determine right now that as soon as I invent the machine, I shall visit myself at this time. But I have not been visited by myself. So, time travel must not be possible. Or perhaps, it is possible, but I deemed it unnecessary to visit myself at this point in time.
This can also be expanded further: A time travel machine is invented in the future. These people should then visit their ancestors, in the past and teach them how to make time travel machines. Why haven't they visited us yet? (This is a Fermi's-Paradox version of Time Travel)
Zeno's Paradox - The Racetrack
One can never reach the end of a racecourse, for in order to do so one would first have to reach the halfway mark, then the halfway mark of the remaining half, then the halfway mark of the final fourth, then of the final eighth, and so on ad infinitum. Since this series of fractions is infinite, one can never hope to get through the entire length of the track (at least not in a finite time). But things get even worse than this. Just as one cannot reach the end of the racecourse, one cannot even begin to run. For before one could reach the halfway point, one would have to reach the 1/4 mark, and before that the 1/8 mark, etc., etc. As there is no first point in this series, one can never really get started (this is known as the Reverse Dichotomy).
Zeno's Paradox - The Arrow
Suppose you shoot an arrow from a bow. The arrow in flight is really at rest. For at every point in its flight, the arrow must occupy a length of space exactly equal to its own length. After all, it cannot occupy a greater length, nor a lesser one. But the arrow cannot move within this length it occupies. It would need extra space in which to move, and it of course has none. So at every point in its flight, the arrow is at rest. And if it is at rest at every moment in its flight, then it follows that it is at rest during the entire flight. So, the arrow cannot move.
Thomson's Lamp
Suppose you have a lamp with a simple on/off switch. Press the switch when it is off and the lamp will be turned on, press it again and it will be turned off. Now suppose you run the following experiment. You turn the lamp on at the start of a minute.Half a minute later, you turn it off. Half of half of a minute later, you turn it back on, then 7 1/2 seconds later back off again, and so on throughout the midpoints of whatever time remains. Now the question is this. At the end of the minute, will the lamp be on or off? Since the lamp has been turned on and off an infinite number of times, for every time it has been turned on, it has been turned off, and vice versa. At the end of the minute, therefore, it can be neither on nor off. But it must be one or the other.
The Paradox of the Divided Stick
A modern version of a plurality paradox asks what would happen if an infinitely divisible stick were cut in two, then half a minute later each half were again cut in two, then a quarter of a minute later each fourth cut in two, and so on ad infinitum. At the end of one minute what would be left? An infinite number of pieces? Would each piece have any length?
The Matrix Paradox
We're all familiar with the concept of the Matrix, that everything is just a simulation. Let us take this concept further and decide to build a computer that simulates the entire Universe. We'd need a lot of RAM and disk space for this, and we'd need to simulate everything in the universe as it is right now, including the same computer itself. Which would mean we need to take into account everything that this computer is simulating, and we can be quite sure that it needs to simulate the universe as well, and that universe would contain a computer simulating the universe. Thus, we would have to simulate an infinite number of universes in a computer, and so, such a computer can never exist.
One possible answer to this paradox is that we create this computer from outside the Universe. If ever.
Theseus' Paradox or The Clone's Paradox
Theseus is famous in Greek mythology as the slayer of the Minotaur, a half-man, half-bull monster who lived in the Labyrinth in the island of Crete. According to Plutarch, the ship in which Theseus sailed back to Athens was preserved for many generations, its old planks being replaced by new ones as they decayed. Now suppose that a few hundred years later, all the original parts of the ship had been replaced, one by one, so that none of the original ship remained. Is the preserved ship still Theseus' ship? Or is it a copy? And if the latter, then at what point did it cease to be Theseus' ship? It seems that if just one plank were replaced, it would still be Theseus' ship. And if it was still his ship, and another plank were replaced, then it should still be Theseus' ship. By this reasoning (which is the same as in the Sorites paradox), it would be Theseus' ship even after all planks are replaced. This problem is not merely another version of the Sorites, however. It involves the notion of identity, of what we mean by something being the "same" object.
Let us suppose that as these planks and nuts and bolts were being replaced, they were being stored in some warehouse. One day, all these are taken and someone puts them back together again. Do we now have two of Theseus' ships?
Similar paradoxes of identity arise in certain science fiction scenarios and in connection with the philosophy of mind. Suppose you are teleported by having your body disintegrated in one place and reassembled in another from new materials. Are you still "you"? Your body is made of different atoms, but it is still you as far as your mind is concerned, right? But what if instead of having your original body disintegrated you merely have a copy made? Then is the copy still you?
Cloning is another topic which can be brought in here. A clone is just a copy of a person, with the same genetic make up, but different materials. Is this clone the same person as the original, is the clone an original, or is this clone simply a non-person?
Hempel's Ravens
Suppose an ornithologist (someone who studies birds) wishes to determine whether or not all ravens are black. The reasonable thing for him to do is to go outside and look for ravens. If he finds even one that isn't black, that proves that not all ravens are black. If, on the other hand, he sees thousands of ravens and every single one of them is black, then that offers support for the proposition that they are all in fact black. Although no amount of observations can ever conclusively prove the hypothesis, each new black raven found provides additional evidence for it. But now suppose that our ornithologist, after seeing thousands of ravens, becomes tired of looking for them, and decides to try a different method. He reasons as follows: The statement "all ravens are black" is logically equivalent to the statement "all nonblack objects are nonravens". When you see a blue sky, a yellow submarine, or any other nonblack nonraven, that supports the proposition that all nonblack objects are nonravens. But in that case, it also supports the proposition that all ravens are black. So all one has to do is look around at ordinary objects to acquire evidence that all ravens are black! No need to go out in the woods in search of ravens, since each nonblack nonraven is also evidence for that hypothesis. This is the paradox known as Hempel's ravens, named after Carl Hempel, who discovered it in 1946. But how can seeing Green Eggs and Ham or a Pink Panther add to the evidence that all ravens are black? Perhaps the answer is that it does add to the evidence, but only by a very tiny amount. Since there are many more nonblack nonravens in the universe than there are black ravens, it is a bad idea to attempt to confirm the hypothesis that all ravens are black by investigating nonblack things. In principle, however, it could be done. The evidence that each nonblack nonraven adds to the proposition may be infinitesimal, but is nevertheless real. Unfortunately, our difficulties do not end here. What our ornithologist didn't notice is that a yellow submarine is not merely an example of a nonblack nonraven. It is also an example of (among other things) a nonwhite nonraven. Thus, a yellow submarine provides evidence not only for the proposition "all ravens are black", but also for the proposition "all ravens are white". But how can it be possible for one fact to support two contradictory claims?
The Self-Amendment Paradox
Constitutions and their implementation are a lot more complex than most people realize. While there are many complexities that would be very interesting, let us examine just one here now to illustrate the point. It is generally agreed that if a constitution is created, there needs to be a way to amend it. Since the Constitution is the highest law of the land, a clause for amending the Constitution must be within the Constitution itself. Article V of our Constitution defines how the Constitution may be amended. Now consider this: what if an amendment was proposed that modified Article 5 itself? It is apparent that having such a clause would allow for self suicide by the Constitution. For an amendment could be proposed that eliminated the contents of Article 5 altogether and replaced it with a statement that the first 10 Amendments were null and void! Not a happy situation, for now we have lost our freedom and the amendment process.
The Paradox of the Question
Once upon a time, during a large and international conference of the world's leading philosophers, an angel miraculously appeared and said, "I come to you as a messenger from God. You will be permitted to ask any one question you want - but only one! - and I will answer that question truthfully. What would you like to ask?" The angel said that he would return at the same time the next day. What questions can be asked? The first question proposed was "Is it better to check your oil when the car is hot or when it is cold?", but others said they should not squander this rare opportunity on something so trivial. Next, was proposed this question: "What would be the best question for us to ask, and what is the answer to that question?", but this would count as two questions and could not be asked. Another proposal was to ask the first part of this question in hopes that they'd get another opportunity some time later. Next, this came up: "What is the answer to the question that would be the best question for us to ask?", and this would give them the answer at least. But this was no good, because the answer could be something like "no" or "seven." Finally, this question was approved: "What is the ordered pair whose first member is the question that would be the best one for us to ask you, and whose second member is the answer to that question?" Seems good doesn't it, and safely, this can cover everything. So, the angel returned, and they asked their question. Then he gave this reply: "It is the ordered pair whose first member is the question you just asked me, and whose second member is this answer I am giving you." At the time the philosophers asked the question, it seemed like that question was the ideal one for their peculiar situation. But as it turned out, it was obviously not at all the right thing to ask. The puzzle, then, is this: What went wrong?
Kavka's Toxin Paradox
Suppose you get the following proposal from an eccentric billionaire: "Toxin X is a substance that will make you violently ill for a few hours. However, it has no long term effects of any kind. As an experiment in psychology, I'm offering you a million dollars if tonight at midnight you fully intend to drink toxin X by tomorrow at noon. You don't actually have to drink the toxin; all you have to do is to intend to drink it. Your intention will be tested by a device similar to a polygraph which my people have developed and which has been shown to be 100% accurate. If at midnight you have the intention, a million will be wired to your bank account. The only other conditions are that you are to make no bets, do anything that will cause you to become irrational, or arrange for any way to avoid the effects of the toxin." Suppose you decide that being ill for one day is a reasonable price to pay for a million dollars. Your first thought is to therefore agree to the proposal. It then occurs to you that you won't even have to become sick in order to win the money. All you have to do is to intend to drink the toxin. You don't actually have to carry out your intention. But now if you know ahead of time that you don't actually have to drink the toxin, then you can't really intend to drink it. So you tell yourself you really do have to drink it. But then if at midnight you really did intend to drink the toxin, and you got the million, then come the next day you would no longer have any reason to drink it: you've already been paid and drinking the toxin would make you unnecessarily sick. Is there any way for you to win the money?
The Prisoner's Dillemma
The Prisoner's Dilemma is a short parable about two prisoners who are individually offered a chance to rat on each other for which the "ratter" would receive a lighter sentence and the "rattee" would receive a harsher sentence.
The problem results from the fact that both can play this game, that is, defect, and if both do, then both do worse than they would had they both kept silent. This peculiar parable serves as a model of cooperation between two or more individuals (or corporations or countries) in ordinary life in that in many cases each individual would be personally better off not cooperating (defecting) on the other.
Galileo's Paradox
In his final scientific work, the Two New Sciences, Galileo made two apparently contradictory statements about the positive whole numbers. First, some numbers are perfect squares, while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares. And yet, for every square there is exactly one number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other. (This is an early use, though not the first, of a proof by one-to-one correspondence of infinite sets.)
Abilene Paradox
The Abilene Paradox is a paradox in which the limits of a particular situation force a group of people to act in a way that is directly the opposite of their actual preferences, illustrated in this story: A sunny afternoon, a family playing cards on a terrace. One of them thinks they should move -- not that he cares, on the contrary, but he thinks the others want to -- so he proposes a trip to Abilene. No fun, hot, bad food. Back home one of them admits that he had preferred to stay home. Everybody would have liked that, only they did not admit to it when it was still time to enjoy the afternoon.In other words, a bad decision was made, but was carried out against the will of the participants.
The Liar's Paradox
If a compulsive liar tells you, "I am lying." Is he lying, or is he telling the truth?
Similarly, "This sentence is false."
The Gambler's Paradox
A gambler keeps gambling, and by probability he does win a couple times. He then spends all his wins on gambling further. Thus the gambler gambles forever.
God's Stone
Can god create a stone so heavy that even he cannot lift it? This paradox basically goes to question the possibility of an all-powerful deity, as well as his capabilities.
Or, as Homer Simpson put it, "Can Jesus microwave a burrito so hot that he himself cannot eat it?"
True or False?
The statement below is true.The above statement is false.
Preceding Quotations
"is preceded by it's quotation." is preceded by it's quotation.1
Control Paradox
The Control paradox states that a live or conscious man will also be controlled either by others, or by themselves, so the idea of control will be operating on them. Certainly, medically speaking, man need a certain amount of control systems to keep working, but this is more a philosophy about whether if we have free will we are completely free. It could easily be changed around: No man is free from freedom, because even when they are free from others' control, they are under their own control. Can free will exist?
Always, Never
"Always" and "never" always render the statements in which they are contained untrue, and should therefore never be used.
The Force and the Object
What happens when an irresistible force meets an immovable object?
The Cat and the Sandwich Paradox
If you glue a sandwich on a cat's back, (butter side up), then throw the cat out the window - what happens? Cats always land on their feet, while we all know that if you drop a sandwich, it will always land with the buttered side down. Perhaps it will hover in the air indefinitely in a state of quantum indecision.
Star Trek vs. Star Wars Paradox
If you have one red-suited security person from Star Trek (the original series), and an Imperial Storm Trooper from Star Wars at your command, and you make them shoot at each other with their respective zap-guns - what happens? Red suits from Star Trek always die, and Imperial Storm Troopers always miss.
The Wife Beater's Paradox
Ask someone this question, "When did you stop beating your wife?" or "Have you stopped beating your wife?" This is not a simple question, and needs a carefully phrased reply, to avoid the inevitable come-back to 'I have not'. How is one to understand this denial, as saying you continue to beat your wife, or that you once did but do so no longer, or that you never have, and never will? It is a question of what the 'not', or negation means, in this case. If 'stopped beating' means 'beat before, but no longer', then 'not stopped beating' covers both 'did not beat before' and 'continues to beat'. And in that case 'I haven't' is an entirely correct answer to the question, if you in fact did not beat your wife.
The Golf Paradox
In Britain, you cannot join a golf club unless you have had your handicap measured properly. You cannot have your handicap measured properly unless you are a member of a golf club.1
The Lucky Fishes Paradox
Fishes, real or symbolic, in odd numbers are considered lucky. Also, the number 13 is considered unlucky. The paradox is, are 13 fishes lucky or unlucky?
Infinite Circle Paradox
It is observed easily, that the larger a circle is, the less the curvature of its circumference. Does that mean that a straight line is actually a circle of infinite radius?
Shampoo User's Paradox
Ever notice those shampoo bottles that come with these instructions:
Wet hair
Massage shampoo into scalp
Leave for two minutes
Rinse
Repeat
This would finish the shampoo off in a single shower!
Wet hair
Massage shampoo into scalp
Leave for two minutes
Rinse
Repeat
This would finish the shampoo off in a single shower!
The Quit Smoking Paradox
This one used to happen to me back when I was trying to quit a while ago. I went through several of those quit-smoking techniques, and none of them worked. One of them was, "Every time you successfully resist the urge to have a cigarette, place some money into a jar. At the end of the month, take out all the money, and reward yourself with a nice treat."
I tried, this, and at the end of the month, I spent all the money on more cigarettes.
The JavaScript Paradox #1
A user goes to a certain page, say index1.html. Index1.html, in turn redirects him to Index2.html, which redirects him back to Index1.html. The user is stuck in this loop now.
The JavaScript Paradox #2
A user goes to a page say, index.html, which launches a new window upon exit of that page. This new window in turn, contains the same page. The user can never exit from this loop.
Mailer Daemon Paradox
Although this does not happen, it is a theoretical possibility provided that technology had not taken care of it.
Two friends set their respective email accounts to "vacation mail," which would respond with an automated message to every message sent to them. Then, one friend mails the other one to say that he's going on vacation. The email is responded to with a vacation mail, which in turn is responded to with a vacation mail. This continued, until either or both accounts fill up, after which Mailer Daemon's are exchanged, and responded to.
Exceptions
There is an exception to every rule. Except this one.2
1 Thanks to wossname!
2 Murphy's Law is most likely an actual exception
