Are paradoxes just clever word play or proof of something else?
Are paradoxes just clever word play or proof of something else?
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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"It is not that I am mad; it is only that my head is different from yours.”
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What the paradox proves is that most things are limited.
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.
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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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.
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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