26 August, 2010

A paradoxical post

I was reading the Wiki list of paradoxes. Some of them are very clever - and really make you think.

Here are a few of note...

Curry's paradox

If this sentence is true, then there is no God.

If the sentence is true, then what it says is true: namely that "if the sentence is true, then there is no God". Therefore, without necessarily believing that there is no God, or that the sentence is true, it seems we should agree that "if the sentence is true, then there is no God". But then this means the sentence is true. So there is no God. [adapted from the link]

Quite brilliant! Of course, anything can be 'proven' by simply inserting it into the latter half of the sentence, so in reality it is meaningless. I can't quite get my head around it but I expect it's simply due to the limitations of language.


Unexpected hanging paradox

A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day. Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on a Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on a Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday. He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.

It's a funny one. It's clearly wrong, and yet it seems foolproof. I like the end part:

The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, will still be an utter surprise to him. Everything the judge said has come true.

So it was a surprise after all!


Barber paradox

Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town 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:

- If the barber does not shave himself, he must abide by the rule and shave himself.

- If he does shave himself, according to the rule he will not shave himself.

This is, of course, an applied version of Russell's paradox. I actually prefer the library book example:

Imagine a library that has catalogs for each section. It has a catalog for the science section, one for the British literature section, one for American literature, one for history and so on. These catalogs are also books in their own right, so they may also be listed in catalogs. Now it also has a master catalog, which lists all books which do not list themselves. Now the question is, does the master catalog list itself? If it does, then on the premise that it lists those books that do not list themselves, it doesn’t list itself. If it doesn’t list itself, then by the same logic, it does.


Liar paradox

An oldie but a goodie. The simplest version of the paradox is this:

This statement is false.

If the statement is true, everything asserted in it must be true. However, because the statement asserts that it is itself false, it must be false. So the hypothesis that it is true leads to the contradiction that it is false. Yet the sentence cannot be false for that hypothesis also leads to contradiction. If the statement is false, then what it says about itself is not true. Hence, it is true. Under either hypothesis, the statement is both true and false.

A criticism of the liar paradox is that it is self-referencing. However, a variation exists that does not self-reference:

Card paradox

Suppose there is a card with statements printed on both sides:

Front: The sentence on the other side of this card is TRUE.

Back: The sentence on the other side of this card is FALSE.

Trying to assign a truth value to either of them leads to a paradox.

I presented a similar argument before using newspapers to show that logical contradictions exist. Neither of the sentences employs self-reference; however, this type of paradox does employ circular referencing. This criticism has also been overcome by a variation that does not self-reference or circular-reference:

Yablo's paradox

The paradox arises from considering the following infinite set of sentences:

(S1): for all k > 1, Sk is false
(S2): for all k > 2, Sk is false
(S3): for all k > 3, Sk is false

The set is paradoxical, because it is unsatisfiable (contradictory), but this unsatisfiability defies immediate intuition. Moreover, none of the sentences refers to itself, but only to the subsequent sentences; this leads Yablo to claim that his paradox does not rely on self-reference. As it continues to infinite, it does not employ circular reference.


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