Re: Mathematical proofs

From: Beth (sneezie_at_unlisted.net)
Date: 02/15/05 

Date: Tue, 15 Feb 2005 05:56:30 GMT

In article <42117639.CA383EED@ANTISPAMbtinternet.com.invalid>, Jim
Spriggs <jim.sprigs@ANTISPAMbtinternet.com.invalid> wrote:

> I note that on the page
> http://mathworld.wolfram.com/IntuitionisticLogic.html there is a link to
> http://mathworld.wolfram.com/ProofbyContradiction.html but that entry
> has yet to be written!

The Proof by Contradiction page does have a link to Reductio ad
Absurdum which says, simply, "A method of proof which proceeds by
stating a proposition and then showing that it results in a
contradiction, thus demonstrating the proposition to be false."

This and the assumption that --A -> A complete my naive understanding
of proof by contradiction. Between your post and Douglas Gwyn's I know
better (although I'm getting the oddest feeling of deja vu, that
somebody corrected me on this once before).

> A substitution instance of Heyting's axiom is
>
> ((-P -> Q) & (-P -> -Q)) -> --P
>
> which gives classically
>
> ((-P -> Q) & (-P -> -Q)) -> P (***)
>
> but _that_ isn't intuitionistically valid. If (***) is what is meant by
> "proof by contradiction" then it is true that "Proofs by contradiction
> are not permissible in intuitionistic logic."

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