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This is a place to ask questions about math terminology, and to post links to other resources out on the web.
Billy Bob
2004-04-03 15:05:28
Paradoxes

What is a paradox? OR What does it mean?

TomM
2004-04-03 19:09:43
Re: Paradoxes

A paradox is a statement in logic that can be neither true nor false, even though it apparently must be one or the other. The simplest is the classic "liar's paradox": "I am lying."

If you consider the possibility that it is true, them I am lying. But if I am lying, what I'm saying (the statement) is false, so we get a contradiction.

If you consider the possibility that it is false, then I am not lying, and the statement is true.

In a Knights and Liars puzzle, neither a knight nor a Liar may say "I'm a Liar." The statement is equivalent to "Everything I say is a lie." It would be a false statement if the speaker were a knight, but a knight cannot make false statements. It would be a true statement if made by a Liar, but Liars cannot make true statements.

Aspiring Novice
2004-04-03 20:04:32
Re: Paradoxes

To be a little more precise, a paradox occurs when seemingly justified premises lead to a contratiction.

So, for instance, with the Liar's Paradox that TomM quoted, if we start from the assumption that all statements are either True or False, the statement "this statement is false" leads to an apparent contradiction, since it can be neither true nor false, thus countering the original premise.

Usually what this means, however, is simply that your premise is wrong. If we use the contradiction above to show that not all statements are either true or false, or that the statement quoted above isn't a "true" statement, then we have no paradox. If, however, we also prove that the premises are correct, then it is probably the case that our system of logical reasoning is flawed.

In a sense we could go from this to say that there are no "real" paradoxes, only problems that haven't been solved in logic. Until we solve all the problems in logic, however (not that I'm suggesting it could even theoretically be done), it's perfectly fair to call contradictions that arise from premises that you want to hold on to and the conclusions that you draw from them paradoxes.

Sam
2004-04-03 20:05:43
Re: Paradoxes

Oops, my name is not Aspiring Novice (though that I may always be), but is, in fact, Sam. As in "Sam I am."

TomM
2004-04-03 21:25:56
Re: Paradoxes

>>In a sense we could go from this to say that there are no "real" paradoxes, only problems that haven't been solved in logic.

It sounds like you believe that paradox can be eliminated by defining your logical system more precisely. I suggest you Google "Gödel's Incompleteness Theorem." It is inherent in logical systems that they cannot avoid paradox.

Sam
2004-04-04 15:16:28
Re: Paradoxes

Well, not quite I think. Godel proves that in any logical system there are statements that cannot be answered, which is not quite the same as saying that there always exist paradoxes. But this is minor quibbling, and I certainly was not forgetting his proof (A while ago on this site I said that Godel's proof was considered to be one of the most important proofs in the history of math, and was told "Lol! Ever heard of Pythagoras?" Well, I know who I'd pick...)

Anyway, I pointedly qualified my statement with "not that I'm suggesting it could even theoretically be done," and was simply making a more general point of paradoxes in general. Most things that people call paradoxes (Liar's, Achilles, Pop Quiz, the Barber) have all been said to have been "solved" by various philosophers or logicians. Whether or not you agree with the solutions, the point still remains that a paradox is generally seen as something that needs to be solved, either by denying the premises or by adjusting the system of logic.

Gamer
2004-04-04 17:03:59
Re: Paradoxes

I think there are a few paradoxes, and everything comes back to them :) For example, Achilles could be blamed on division by zero, where Barber could be blamed on liar/this statement is false type logic. (of course, assuming that the barber is male. If there is a female barber, it's not a problem!)

The point
2004-05-01 12:27:46
Re: Paradoxes

Female barbers could shave...

Federico Kereki
2004-05-27 19:53:32
The barber paradox

The barber paradox isn't actually a paradox -- you're just asking for an impossibility. Smullyan compares it to saying that there's a man who's taller than 6', and also shorter than 6' -- it's just impossible, no paradox there.

The "This sentence is false" case is a different one, because everyone understands what the sentence says, the sentence does exist... but you cannot assign either a "true" or a "false" value to it; if we assume everything must be one or the other, therein lies the paradox.

Cheyenne
2005-09-24 21:13:23
But isn't reality a paradox..

since on an atomic level it exists in quantum states.

Like the idea of schrodinger's cat.

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