| PLEASE IGNORE THIS NOTICE |
Would you be ignoring this notice if you hadn't read it? If you have read it, you have not followed its instructions to ignore it. So therefore this is a paradoxical notice.
Which
is better, eternal happiness or a ham sandwich? It would appear that eternal
happiness is better, but this is really not so! After all, nothing is better
than eternal happiness, and a ham sandwich is certainly better than nothing.
Therefore a ham sandwich is better than eternal happiness.
Proof
that either Tweedledum or Tweedledee exists 
(1) TWEEDLEDUM DOES NOT EXIST
(2) TWEEDLEDEE DOES NOT EXIST
(3) AT LEAST ONE SENTENCE IN THIS BOX IS FALSE
Therefore at least one of them exists.
Proof that Tweedledoo exists
(1) TWEEDLEDOO EXISTS
(2) BOTH SENTENCES IN THIS BOX ARE FALSE
Therefore tweedledoo does not exist.
Proofs that Santa Claus exists
IF THIS SENTENCE IS TRUE THEN SANTA CLAUS EXISTS
THIS SENTENCE IS FALSE AND SANTA CLAUS DOES NOT EXIST
Therefore Santa Claus exists.
Proof that there exists a unicorn
I wish to prove to you that there exists a unicorn. To do this it obviously suffices to prove the (possibly) stronger statement that there exists an existing unicorn. (By an existing unicorn I of course mean one that exists.) Surely if there exists an existing unicorn, then there must exist a unicorn. So all I have to do is prove that an existing unicorn exists. Well, there are exactly two possibilities:
(1) An existing unicorn exists.
(2) An existing unicorn does not exist.
Possibility (2) is clearly contradictory: How could an existing unicorn not exist? Just as it is true that a blue unicorn is necessarily blue, an existing unicorn must necessarily be existing.
Proof that I am Dracula
(1) Everyone is afraid of Dracula.
(2) Dracula is afraid of only me.
Therefore I am Dracula. Doesn't that argument sound
like just a silly joke? Well it isn't; it is valid. Since everyone is afraid
of Dracula, then Dracula is afraid of Dracula. So Dracula is afraid of Dracula,
but also is afraid of no one but me. Therefore I must be Dracula!
Consider the statement in the following box:
| THIS SENTENCE IS FALSE |
Is that sentence true or false? If it is false then it is true, and if it is true then it is false... The following version of the liar paradox was first proposed by the English mathematician P. E. B. Jourdain in 1913. It is sometimes referred to as "Jourdain's Card Paradox".
We have a card on one side of which is written:
| (1) THE SENTENCE ON THE OTHER SIDE OF THIS CARD IS TRUE |
Then you turn the card over, and on the other side is written:
| (2) THE SENTENCE ON THE OTHER SIDE OF THIS CARD IS FALSE ... |
Another popular version of the liar paradox is given by the following three sentences written on a card.
| (1) THIS SENTENCE CONTAINS
FIVE WORDS (2) THIS SENTENCE CONTAINS EIGHT WORDS (3) EXACTLY ONE SENTENCE ON THIS CARD IS TRUE |
Poaching
on the hunting preserves of a powerful prince was punishable by death, but the
prince further decreed that anyone caught poaching was to be given the privilege
of deciding whether he should be hanged or beheaded. The culprit was permitted
to make a statement - if it were false, he was to be hanged; if it were true,
he was to be beheaded. One logical rogue availed himself of this dubious prerogative
- to be hanged 
if
he didn't and to be beheaded if he did - by stating: "I shall be hanged." Here
was a dilemma not anticipated. For, as the poacher put it, "If you now hang
me, you break the laws made by the prince, for my statement is true, and I ought
to be beheaded, but if you behead me, you are also breaking the laws, for then
what I said was false and I should therefore be hanged."
There is a wide variety of puzzles about an island in which certain inhabitants called "knights" always tell the truth, and others called "knaves" always lie. It is assumed that every inhabitant of the island is either a knight or a knave... Suppose A says, "Either I am a knave or else two plus two equals five." What would you conclude.? If A is a knight, then two plus two equals five, which is not true. If A is a knave, then he is speaking the truth, which is not possible. Smullyan comments: The only valid conclusion is that the author of this problem is not a knight. The fact is that neither a knight nor a knave could make such a statement. The visiting logician We are back on the Island of Knights and Knaves, where the following three propositions hold:
(1) knights make only true statements;
(2) knaves make only false ones;
(3) every inhabitant is either a knght or a knave.
These three propositions will be collectively referred to as the "rules of the island." We recall that no inhabitant can claim that he is not a knight, since no knight would make the false statement that he isn't a knight and no knave would make the true statement that he isn't a knight. Now suppose a logician visits the island and meets a native who makes the following statement to him: "You will never know that I am a knight." Do we get a paradox? Let us see. The logician starts reasoning as follows: "Suppose he is a knave. Then his statement is false, which means that at some time I will know that he is a knight, but I can't know that he is a knight unless he really is one. So, if he is a knave, it follows that he must be a knight, which is a contradiction. Therefore he can't be a knave; he must be a knight." So far so good - there is as yet no contradiction. But then he continues reasoning: "Now I know that he is a knight, although he said that I never would. Hence his statement was false, which means that he must be a knave. Paradox!"
Cellini and Bellini ... whenever Cellini made a sign, he inscribed a false statement on it, and whenever Bellini made a sign, he inscribed a true statement on it. Also, we shall assume that Cellini and Bellini were the only sign-makers of their time... You come across the following sign:
| THIS SIGN WAS MADE BY CELLINI |
Who made the sign? If Cellini made it, then he wrote a true sentence on it - which is impossible. If Bellini made it, then the sentence on it is false - which is again impossible. So who made it? Now, you can't get out of this one by saying that the sentence on the sign is not well-grounded! It certainly is well-grounded; it states the historical fact that the sign was made by Cellini; if it was made by Cellini then the sign is true, and if it wasn't, the sign is false. So what is the solution? The solution, of course, is that I gave you contradictory information. If you actually came across the above sign, then it would mean either that Cellini sometimes wrote true inscriptions on signs (contrary to what I told you) or that at least one other sign-maker sometimes wrote false statements on signs (again, contrary to what I told you). So this is not really a paradox, but a swindle.
On a Monday morning, a professor says to his class, "I will give you a surprise examination someday this week. It may be today, tomorrow, Wednesday, Thursday, or Friday at the latest. On the morning of the examination, when you come to class, you will not know that this is the day of the examination."
Well, a logic student reasoned as follows: "Obviously I can't get the exam on the last day, Friday, because if I haven't gotten the exam by the end of Thursday's class, then on Friday morning I'll know that this is the day, and the exam won't be a surprise. This rules out Friday, so I now know that Thursday is the last possible day. And, if I don't get the exam by the end of Wednesday, then I'll know on Thursday morning that this must be the day (because I have already ruled out Friday), hence it won't be a surprise. So Thursday is also ruled out." The student then ruled out Wednesday by the same argument, then Tuesday, and finally Monday, the day on which the professor was speaking.
He concluded: "Therefore I cannot get the exam at all; the professor cannot possibly fulfil his statement." Just then, the professor said: "Now I will give you your exam." The student was most surprised! ...
Let me put myself in the student's place. I claim that I could get a surprise examination on any day, even on Friday! Here is my reasoning: Suppose Friday morning comes and I haven't got the exam yet. What would I then believe? Assuming I believed the professor in the first place (and this assumption is necessary for the problem), could I consistently continue to believe the professor on Friday morning if I hadn't gotten the exam yet? I don't see how I could. I could certainly believe that I would get the exam today (Friday), but I couldn't believe that I'd get a surprise exam today. Therefore, how could I trust the professor's accuracy? Having doubts about the professor, I wouldn't know what to believe. Anything could happen as far as I'm concerned, and so it might well be that I could be surprised by getting the exam on Friday.
Actually, the professor said two things: (1) You will get an exam someday this week; (2) You won't know on the morning of the exam that this is the day. I believe it is important that these two statements should be separated. It could be that the professor was right in the first statement and wrong in the second.
On Friday morning, I couldn't consistently believe that the professor was right about both statements, but I could consistently believe his first statement. However, if I do, then his second statement is wrong (since I will then believe that I will get the exam today.) On the other hand, if I doubt the professor's first statement, then I won't know whether or not I'll get the exam today, which means that the professor's second statement is fulfilled (assuming he keeps his word and gives me the exam). So the surprising thing is that the professor's second statement is true or false depending respectively on whether I do not or do believe his first statement. Thus the one and only way the professor can be right is if I have doubts about him; if I doubt him, that makes him right, whereas if I fully trust him, that makes him wrong!
In a certain village there
is a man, so the paradox runs, who is a barber; this barber shaves all and only
those men in the village who do not shave themselves. Query: Does the barber
shave himself? Any man in this village is shaved by the barber if and only if
he is not shaved by himself. Therefore in particular the barber shaves himself
if and only if he does not. We are in trouble if we say the barber shaves himself
and we are in trouble if we say he does not. Quine disarms the paradox thus:
What are we to say to the argument that goes to prove this unacceptable conclusion?
Happily it rests on assumptions. We 
are
asked to swallow a story about a village and a man in it who shaves all and
only those men in the village who do not shave themselves. This is the source
of our trouble; grant this and we end up saying, absurdly, that the barber shaves
himself only if he does not. The proper conclusion to draw is just that there
is no such barber. We are confronted with nothing more than what logicians have
been referring to for a couple of thousand years as a reductio ad absurdum.
We disprove the barber by assuming him and deducing the absurdity that he shaves
himself if and only if he does not. The paradox is simply a proof that no village
can contain a man who shaves all and only those men in it who do not shave themselves.
Another "swindle", like Cellini and Bellini above.
"Interesting"
and "uninteresting" numbers
The question arises: Are there any uninteresting numbers? We can prove that there are none by the following simple steps. If there are dull numbers, then we can divide all numbers into two sets - interesting and dull. In the set of dull numbers there will be only one number that is the smallest. Since it is the smallest uninteresting number it becomes, ipso facto, an interesting number. We must therefore remove it from the dull set and place it in the other. But now there will be another smallest uninteresting number. Repeating this process will make any dull number interesting."
Here are some paradoxical pictures. Can you spot the paradoxes?
1.
Notice how strangely the pillars of the second floor are constructed? And the top floor appears to be built in a NW-SE manner while the second floor is built in a SW-NE manner!
2. 
Which hand is drawing which hand?
3.
Is the waterfall on the same level as its source? Or is it flowing uphill?
4.
Wonder how you're going to construct something like this?
So these are just a FEW examples of the MANY paradoxes in Mathematics, everyday life and in art.
Sources taken from www.wordsmith.demon.co.uk/someparadoxes.
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