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Thread: Paradoxes and proof

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    Paradoxes and proof

    Are paradoxes just clever word play or proof of something else?

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    www.markbastable.co.uk MarkBastable's Avatar
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    Quote Originally Posted by G L Wilson View Post
    Are paradoxes just clever word play or proof of something else?
    Well, the word 'paradox' has a meaning. If you were to look it up, you'd find out what that meaning was. And it's neither of the things you suggest.

  3. #3
    Quote Originally Posted by MarkBastable View Post
    Well, the word 'paradox' has a meaning. If you were to look it up, you'd find out what that meaning was. And it's neither of the things you suggest.
    paradox. a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth.

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    www.markbastable.co.uk MarkBastable's Avatar
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    Quote Originally Posted by G L Wilson View Post
    paradox. a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth.
    Indeed. It's not clever word play. It's a construct in logical thought. And if it expresses a truth, it's not a 'proof of something else' - it's self-referencing, pretty much by definition.

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    A paradoxical statement is a statement that does not contradict the rules of a certain language/rule system, but contradicts the rules of logic.

    For example, the liar's paradox ("this sentence is false") does not violate any rules of grammar, yet assigning to this statement a binary truth value leads to a logical contradiction. Moore's paradox ("It is raining, but I believe it isn't") does/did not violate the rules of predicate logic, yet it is logically incoherent. Russell's paradox does not violate the rules of Georg Cantor's naive set theory, yet leads to a contradiction.

    Paradoxes are useful when assessing the expressive power of a language. The more quickly we run into paradoxes the less expressive a language is. For example, Russell's paradox (again) starts to demonstrate the expressive limits of naive set theory.
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    Quote Originally Posted by Cunninglinguist View Post
    A paradoxical statement is a statement that does not contradict the rules of a certain language/rule system, but contradicts the rules of logic.

    For example, the liar's paradox ("this sentence is false") does not violate any rules of grammar, yet assigning to this statement a binary truth value leads to a logical contradiction. Moore's paradox ("It is raining, but I believe it isn't") does/did not violate the rules of predicate logic, yet it is logically incoherent. Russell's paradox does not violate the rules of Georg Cantor's naive set theory, yet leads to a contradiction.

    Paradoxes are useful when assessing the expressive power of a language. The more quickly we run into paradoxes the less expressive a language is. For example, Russell's paradox (again) starts to demonstrate the expressive limits of naive set theory.
    That is very interesting. I will keep it in mind.

  8. #8
    What the paradox proves is that most things are limited.

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    Existentialist Varenne Rodin's Avatar
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    Paradoxes, if they must prove something, prove that I don't know what the hell is going on regarding several aspects of existence. Besides that, they're a lot of fun.

  10. #10
    Sometimes I wonder if paradoxes are just a manipulation of reality that really serve no purpose. A convoluted lie. Like the "Grandfather paradox." You spend the whole time thinking about this endless loop without ever thinking, "is this even possible?"
    Now timetravel is a whole other conversation, but you know what I'm saying.

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    Quote Originally Posted by R.R.Seeger View Post
    Sometimes I wonder if paradoxes are just a manipulation of reality that really serve no purpose. A convoluted lie. Like the "Grandfather paradox." You spend the whole time thinking about this endless loop without ever thinking, "is this even possible?"
    Now timetravel is a whole other conversation, but you know what I'm saying.
    Timetravellers' paradoxes are almost always groundless and purely hypothetical (edit: viz. they assume without sufficient evidence laws of nature) since modern physics barely knows how time works. If there are multiple, concrete timelines between which information can be exchanged, then the grandfather paradox is not necessarily a paradox, so long as you switch timelines somewhere. If there is only one timeline, the grandfather paradox, if occurred, would indeed be a physical paradox. However, we're having trouble proving that there is even one timeline at all.

    {edit: It might also be said that the grandfather's paradox and other such physical paradoxes aren't actually paradoxes at all. That you can be your own grandfather is either possible or impossible, accords with the laws of physics or does not. In science we must assume physical (real) paradoxes and other such absurdities impossible, else we would not be able to proceed anywhere with our reasoning.}
    Last edited by Cunninglinguist; 09-12-2011 at 06:21 AM.
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    muaz jalil muazjalil's Avatar
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    If I take Paradox as "a statement or proposition that seems self-contradictory" then surely it plays a role in mathematics. Proofs by contradiction are based on this. Granted if you are a follower of Brower or an intuitionist, u may not believe in law of excluded middle ergo proof by contradiction but most mathematician i guess are formalist. Given, u agree with Aristotelian logic, proof by contradiction (based on generating paradoxes) is a legit proof and therefore cannot simply be word play. Additionally paradoxes can show limit of a system, I have in mind Godel's incompleteness theorem or Russells paradox on naive Set theory. Hence I think paradoxes can definitely be used as proofs.

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    Quote Originally Posted by G L Wilson View Post
    What the paradox proves is that most things are limited.

    Paradoxes prove that understanding based in argumental deduction is limited.
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    Quote Originally Posted by Arrowni View Post
    Paradoxes prove that understanding based in argumental deduction is limited.
    If you have no understanding of a thing, is it not limited?

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    Quote Originally Posted by muazjalil View Post
    If I take Paradox as "a statement or proposition that seems self-contradictory" then surely it plays a role in mathematics. Granted if you are a follower of Brower or an intuitionist, u may not believe in law of excluded middle ergo proof by contradiction but most mathematician i guess are formalist. Given, u agree with Aristotelian logic, proof by contradiction (based on generating paradoxes) is a legit proof and therefore cannot simply be word play.
    Though people use paradox and (self-)contradiction synonymously, in all logics self-contradiction is the criterion of impossibility, not paradox. The proof is proof as impossible, not proof as paradox.

    Math cannot be reduced to Aristotelian logic since Aristotelian truth doesn't deal with form, but reality. That apples exist might be called a true proposition in virtue of its correspondence with reality. "Truth" in a mathematical sense is purely formal; that 1+1=2 is true is true in virtue of its form.

    The principle of (non)contradiction is separate from the law of excluded middle, which means that contradiction can exist without the law of excluded middle.
    Last edited by Cunninglinguist; 09-12-2011 at 06:39 AM. Reason: preposition trouble
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