How do you define a paradox? The dictionary defines it as 'a statement that seems to say something opposite to common sense or truth, but which may contain a truth.' I prefer to define it as a headache. Basically, to mathematicians, paradox=contridiction or inconsistency. It could be a statement which you could neither agree nor disagree with. Or a situation which reasonably must happen, but by the same reasoning, could never happen! Like paranormal in the X-Files, can paradoxes be adequetely explained or refuted? Try the following well-known examples below and attempt to understand the reasoning why each is a paradox. They are really interesting to think about and may just make you wonder more about the things happening around you. (excerpts from 'The Riddle of Scheherazade'-Raymond Smullyan)
(1) Paradoxical Statements
'This sentence is false'
Discussion - Not all sentences have to be true or false. This is an example of a sentence which is neither true nor false. Hence there is no way of proving it true or false, nor is there a need to.
Moral of the paradox - Things are not always what they seem.
(2) Barber Paradox - Bertrand Russell
'A barber in a certain town shaved all and only those inhabitants who didn't shave themselves. Did the barber shave himself?'
Discussion - The famous paradox by Bertrand Russell which first appeared circa 1916. Now if the barber shaved himself, he would then be one of the inhabitants who shaved himself, but he would never shave such an inhabitant! If he didn't, he would then be one of the inhabitants who didn't shave himself, and hence has to be shaved by himself, the barber! Thus either case there will be a contradiction. So what gives? Well, the only possible logical explanation(and quite simply really) is that such a barber can never exist! Remember what Sherlock Holmes said? 'Once you have eliminated all the possibilities, whatever remains, however improbable, must be the answer.' By retroanalysis, if we instead assume that the barber indeed exists, whether or not he shaves himself, he could not have claim to 'shave all and only those inhabitants who didn't shave themselves.' Hence, we must conclude that the barber does not exist.
Moral of the paradox - Barbers have trouble shaving themselves.
(3) Refutation of Gotlob Frege's Set Axiom - Bertrand Russell
People familiar with sets will no doubt remember that set theory derives from the basic axiom by the mathematician Gotlob Frege - 'that given any property, there exists the set of all things having the property.'
But this is what Russell proposes. 'Call a set ordinary if it is not an element of itself. Conversely, a set that is an element of itself is extraordinary(eg. set of all sets). Therefore, by Frege's axiom, Z being the set of all ordinary sets, contains as its elements all the ordinary sets, and all ordinary sets are elements of Z. But is Z ordinary?'
Discussion - This was actually the original version of the barber paradox and you can probably guess why Bertrand Russell gave us the more popular version instead. Not really very difficult to understand but knowledge of sets is required. As concluded before, therefore, Z cannot exist. But by Frege's axiom, it does. Hence in place of it, various set theories have arisen with weaken axioms to try to resolve this inconsistency.
Moral of the paradox - Barbers are easier to understand than sets.
(4) Grelling Paradox
'As numbers get larger and larger, more words are usually required to describe them. And for any fixed number of words, there must always be a least number not describable in less than that number of words. In particular, consider the enormous number x which is described as 'the least number not describable in less than eleven words'. But isn't the description of x above less than eleven words?'
Discussion - This variant of the barber paradox is (I believe) far less elegent and difficult to grasp. In particular, the term 'described' is used in two different manners(word play actually). However, in this case, we cannot dismiss the existence of x as easily as before, since intuitively, x must exist! So is there a paradox then?
Moral of the paradox - Arabic numerals are a great discovery.
(5) Quine's Machine
'Quine's machine is a device that only works when it is out of operation. So is it working or when will it be working?'
Discussion - Another variant. This one is simple and succint. Imagine a 'faulty' apparatus continuously switching itself between an 'on' and 'off' state. The moment it starts to work, it will knock itself out again, only to repeat the cycle perpetually. But what state did it first start out as, 'on' or 'off'?
Moral of the paradox - If you must, invent machines which don't work at all, at least it won't create any problems.
(6) Businessman's Paradox - Lisa Collier
'A president of a certain firm offered a reward of $100 to any employee who could provide a suggestion that would save the company money. One employee wrote, "Eliminate the reward!" Did the employee get the reward?'
Discussion - Not a very foolproof variant but delightful in its setting. In this case, the suggestion cannot be accepted or the reward will have to be given, leading to a contridiction.
Moral of the paradox - Never offer rewards for employee suggestions.
(7) The Surprise Examination(aka Unexpected Hanging) - Raymond Smullyan
'On Monday morning, the professor informed the class that they will get a surprise examination this week. Surprise meaning that they will not know beforehand when they will get it. The class then reasons that they will not get the examination on Friday, since that is the last day of the week, and if they haven't had the examination by Thursday, they will know it is on Friday, which they are not supposed to. Hence Thursday is out too, since come Wednesday and they still haven't had the examination, they will know that it is on Thursday, it being the last possible day of the week now. Therefore, Wednesday, Tuesday and Monday can similiarly be ruled out one by one. Hence they conclude it is not possible to have a surprise examination at all. Unfortunately for the class, on Friday morning, the professor started to hand out the examination scripts, which naturally surprised everyone!'
Discussion - Other version is that of a prisoner sentenced with an unexpected hanging. Like the class, he reasons that the hanging can never occur, yet it is precisely because of this reasoning which allows the event to materialize(and hence result in a paradox). Or is it? Actually, the crux of the paradox is in the procedure of the reasoning. Reasoning 'backwards' in such a manner can often lead to fallacious results. Specifically, Friday can only be ruled out if and only if Thursday has past. So assuming that the professor had intended to set the examination on Thursday, but had forgotten to bring the scripts on that day, then he would be unable to do it the next day, because the class would now be expecting the examination. That is to say, Thursday cannot be ruled out at all, since after Wednesday, the professor could still set the examination on either Thursday or Friday(remember that the class cannot rule out Friday yet since Thursday hasn't past). While intuitively, there is no way the professor can set the examination on Friday since Thursday must pass by then, there is no way we can reach this conclusion until Thursday has actually past!(ie. the event must actually occur, who knows whether Thursday will really pass!) Hence similiarly, the other days cannot be ruled out either simply because the flow of time is forward! This solves the paradox since the professor can now set the examination on any day of the week(even Friday) just as he should.
However, consider the case of another paradox. This time round, the class is a studious but not so logical one(or so it seems). They just studied hard and came to class each day expecting the surprise examination. Every morning, the professor will be greeted with the question, 'We will be having the examination today, right?' Finally, on Friday morning, the professor had no choice but to scrap the examination!!!
Moral of the paradox - Don't announce surprise examinations to avoid paradoxes!
(8) Knights and Knaves - Raymond Smullyan
'On the Island of Knights and Knaves, each inhabitant is either a knight or a knave, and knights make only true statements and knaves make only false ones. One day, I met an inhabitant who said to me, "You will never know that I am a knight." Is he a knight or a knave?'
Discussion - Now if you ever know that he is a knight, that will falsify his statement, making him a knave, which is a contridiction. Hence you will never know that he is a knight. This now makes the statement true, so you know he's a knight, but this again falsifies the statement and contridicts the initial assumption! But think about it. Since you can never know whether he is a knight or a knave due to the paradox, he is indeed telling the truth and he must therefore be a knight. Although this in turn falsifies the statement, it only means that he has somehow managed to utter a statement which is both true and false at the same time! This isn't strange considering the existence of sentences which are neither true nor false.
Moral of the paradox - Don't try to lie if you can't.
(9) Godel's Incompleteness Theorem
'Let us define an a logician to be accurate if everything he can prove is true; he never proves anything false. One day, another inhabitant from the Island of Knights and Knaves told an accurate logician, "You cannot prove that I am a knight." Is he a knight or a knave?'
Discussion - If he is a knave, the statement is false, meaning that the logician could prove that he is a knight, which is something he cannot do. Hence he must be a knight, and the statement is true. So the logician knows that he is a knight but cannot prove that he is! This comes from the Incompleteness Theorem by Kurt Godel which states that for every system there must always be true sentences which are not provable in the system itself. The proof : For a specific system, we assign to each sentence a number, called the Godel number of the sentence. Then we construct a sentence G which states that its Godel number belongs to the set S, which consists of numbers that are not Godel numbers of sentences provable in the system. Hence G is true if and only if it is not provable in the system.
Moral of the paradox - It is sometimes difficult to prove what you know.
(10) Newcombe Paradox
'A chest has two drawers. There is either a hundred dollar bill in each drawer or a thousand dollar bill in each drawer. You can choose either to take the two bills in both drawers or just the bill in the bottom drawer. Obviously you would want to get more money and choose both drawers, when you can obtain $100 or $1000 more. However, before you decide, I also inform you that if you choose both drawers, there will be only be $100 in each, whereas if you choose just the bottom drawer, there will be $1000 in each. So you should choose just the bottom drawer when you can get $1000 compared to $200. But why don't you open the top drawer too and get another $1000 instead?'
Discussion - This paradox is actually quite difficult to phrase to eliminate the loopholes, but you should be able to see the inconsistency quite easily. In fact, the original version uses an imaginery perfect predictor to determine the outcome. This paradox works on the concept of altering the outcome based on the choice made. In this case, you must assume that the choice available is linear, ie. that one choice will only lead to one outcome. For example, choosing both drawers will yield $200 and choosing the bottom drawer will yield $1000. So you can no longer choose both drawers once you have decided on the bottom one(as suggested in the question), since the outcome has already been fixed by your initial choice.
Moral of the paradox - Don't think too much when someone offers you money.
(11) Prisoner's Dilemma
'You and I are accomplices in crime who have been arrested. We have the choice of keeping quiet or accusing the other of the crime. If we both keep quiet, we both get 1 year in jail. If we both accuse each other, we both get 3 years in jail. If only one of us accuses the other, the accuser goes scott free while the accused gets 5 years in jail. So what is your best strategy if you do not know of my decision?'
Discussion - If I keep quiet, you will serve less time by accusing me(none versus 1 year). If I accuse you, you will also serve less time by accusing me(3 versus 5). Hence you should accuse me in any case. Similiarly, I would reason that I should accuse you too. Thus we both accuse each other and get the worst penalty, while had we both kept quiet instead, we would only serve 1 year each!
The trick is in realising two things. That the outcome would depend on the choices of both parties and that both parties would reason similiarly and hence make the same choice. Therefore, it now becomes obvious that we should both keep quiet(instead of both accusing each other) to get the lesser charge, even though individually, we are better off accusing the other!
Moral of the paradox - Squeallers will not have a good end.
(12) Klepper's Experiment 6
'There are 200 people and each is given $5. As one of the 200, you have the choice of keeping the $5 or giving it up to the common pool. Each person decides independently and in the end, the money in the pool is doubled and shared amongst all 200 people. What is your choice?'
Discussion - If you keep the $5, you will get the $5 and in addition, a share of the money from the pool. If you contribute to the pool, you will only stand to gain if more than half the people contributes. Clearly the 'dominant strategy' is to keep the $5, since your choice will not influence the others nor the pool. Thus all 200 people ended up with $5. But then again, had everyone contributed his $5, then everyone will end up with $10 instead...
As above, the reasoning for the best startegy may not alway yield the best results. This is especially so when the result depends on the choice of more than one person. Individually, everyone would be better off keeping the $5, but in the end, each stood to gain more had he contirbuted the same $5 to the common pool.
Moral of the paradox - No one would pay even $51 for an ABM in downtown Pittsburgh.
(13) Envelope Paradox
'There are two envelopes, one of which contains twice as much money as the other. You first pick one of the envelopes and open it. Then you have the choice of keeping the contents or trading it for the money in the other envelope. Should you trade?'
Discussion - Say you got $x in the first envelope. So the other envelope contains with equal probability $2x or $x/2. So by trading, you could gain $x or lose $x/2. Since you stand to gain more than you lose, you conclude that you should trade, regardless of the amount you find(which is ridiculous). Hence chances are that you will be better off with the envelope you did not pick, which is inconsistent, since both envelope have equal chance of having more money!
In this case, the fallacy is in the reasoning. Whether you picked the envelope with more money or the one with less money are two independent occurences. Hence you either found less money($x) and can trade it to get($2x), or you found more money($y) and can trade it to get($y/2). Since the total amount of money in both envelopes are fixed, x+2x=y+y/2, hence 2x=y. So you can see that the amount of money you stand to gain($x) and the amount of money you stand to lose($y/2), is actually the same! So there is no advantage whether or not to trade(as it should logically be).
Moral of the paradox - When in doubt, do not trade.
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