Barber Paradox

Suppose there is a town with just one male barber, and that every man in the town keeps himself clean-shaven. It seems reasonable to make this statement:

The barber shaves all and only those men who do not shave themselves.

Under this scenario, we can ask the following question: Does the barber shave himself? Asking this, however, we discover that the situation presented is in fact impossible.

1) If the barber does not shave himself, he must abide by the statement and shave himself.

2) If the barber shaves himself, according to the statement, he will not shave himself.

So, who shaves the barber?

Source: wikipedia.org
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Unexpected Hanging Paradox

A prisoner is told that he will be hanged on some day between Monday and Friday, but that he will not know on which day the hanging will occur until it is due to happen.

He reasons to himself that he cannot be hanged on Friday, because if he were still alive on Thursday night, he would know that the hanging will occur on Friday; but he has been told he will not know the day of his hanging in advance.

For the same reason, he foresees that he cannot be hanged on Thursday, because if he were still alive on Wednesday night, he would know that the hanging will occur on Thursday.

Using the same argument, he foresees that he cannot be hanged on Wednesday, because if he were still alive on Tuesday night, he would know that the hanging will occur on Wednesday.

And so he reasons that he cannot be hanged on Tuesday, because if he were still alive on Monday night, he would know that the hanging will occur on Tuesday.

Hence, the hanging cannot be on Monday as well, because he now knows that the hanging will occur on Monday; but he has been told he will not know the day of his hanging in advance. Joyfully, he concludes that he will not be hanged.

Nevertheless, the executioner unexpectedly arrives on Thursday, surprising the prisoner. The statement is true after all.

Source: wolfram.com
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Liar Paradox

Consider this paradoxical statement:

"This statement is false."

This statement is paradoxical because there is no way to assign it a consistent truth value. Consider that if "This statement is false" is true, then what it says is the case. But what it says now is that it is false, hence it is false. On the other hand, if it is false, then what it says is not the case; thus, since it says that it is false, it must be true.

A variation of the above statement is this:

"The next sentence is false. The previous sentence is true.”

Now try figuring that out! COOL!

Source: wikipedia.org
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