Talk:Proof by contradiction

message: This is a decent article, though it is lacks any source.. Luckily, there seem to be several good web sites which can be cited for the majority of this article's content. Here's what a quick google search yielded: , ,  Jellocube27 20:25, 11 March 2007 (UTC)
replies: text: Hello User:Jellocube; I have just added the IEP link, thanks.  — Gamall Wednesday Ida (t · c) 16:52, 19 December 2016 (UTC)
thread_title: Please help add sources

message: Is the tilda symbol ~ in common use to mean logical NOT? I have never seen its usage before. I am more familiar with ¬, if others agree then I think it should be changed. 
The ¬ symbol is used later on in the article, so either ~ or ¬ should be used throughout (not both) for consistency.Bah23 12:51, 5 March 2007 (UTC)
replies: text: It is extremely common.  It was introduced by Giuseppe Peano, and is widely used, for example in Principia Mathematica.  I agree with you that the usage in the article should be consistent. -- Dominus 16:48, 5 March 2007 (UTC)
replies: text: I have replaced all instances of ~ by \lnot, which seems to me to be the standard symbol. cf Negation.  I work in CS fields related to logic (though I'm not a logician), and I cannot recall having seen ~ in a paper. (Though it would still be clear enough in context that I may have seen it and not been shocked enough to commit such instances to memory.)  — Gamall Wednesday Ida (t · c) 17:05, 19 December 2016 (UTC)
thread_title:  Use of tilda (~) for logical NOT 

message: I notice that there is a detailed discussion of contraposition in one of the comments above, but none of it seems to have made it into the article Reductio ad absurdum itself.  It may be helpful to have a simple-minded comparison of the two as part of the article.  Also, if anybody cares to explain the difference between "proof by contradiction" and law of excluded middle", I would be much obliged. Katzmik (talk) 12:43, 26 October 2008 (UTC)
replies: text: Hello Katzmik; I have just added a paragraph Proof by contradiction#rel. Is it helpful?  — Gamall Wednesday Ida (t · c) 19:46, 19 December 2016 (UTC)
thread_title:  contraposition 

message: I just made a minor edit to make statement names consistent. In the main part of the article, P was the name of the statement to be proven, and Q was the name of the statement that was contradicted to prove P. The section "Relationship with other proof techniques" in one place used both the name Q, and the compound statement "If P, then Q" as the statement to be proven. I changed the former to P, and the latter to "If A, then B" to show that A and B were components of P, not the P and Q in the main part.
And that distinction is related to what I think is a common misapplication; that is, confusing elements of the proof method, with elements of the statement to be proven. And it points to a problem with many claimed proofs-by-contradiction in general. The section describes how you can prove P="If A, then B" by proving the contrapositive "If ~B, then ~A". But that is not the only form of a proof that uses contraposition.
The section "Principle" lists two forms of proof by contradiction, but many claimed examples I've seen fit neither. They (1) assume that P is false, (2) show that a statement ~Q follows from ~P where (3) Q is a known truth that is independent of P. In a proof by contradiction, both Q and ~Q have to follow from ~Q.
In the Pythagorean example of proof by contradiction, the claimed contradiction is "a^2+b^2<c^2" and "a^2+b^2=c^2". But the latter does not follow from "a+b≤c," it is a truth that is known independently. What the method proves, is "if a+b≤c, then a^2+b^2<c^2". The contrapositive of this is "if a^2+b^2>=c^2, then a+b>c. Since we know that a^2+b^2=c^2, we can conclude that a+b>c.
And not only is Cantor's proof not a proof by contradiction, he never claimed it was one. JeffJor (talk) 22:16, 9 January 2018 (UTC)
thread_title:  Is it really contradiction? 

message: For the section "No least positive rational number", the proof is technically not a contradiction. In fact the statement itself is negative, so assuming its affirmation is just regular constructive practice. — Preceding unsigned comment added by Matteo.Ferrari95 (talk • contribs) 08:28, 6 May 2019 (UTC)
replies: text: You're right in raising this issue.  It's important!   This issue - the difference between proof by contradiction and refutation by contradiction - has been dealt with now.   John Baez (talk) 10:51, 3 October 2022 (UTC)
thread_title:  Is it really contradiction? PART II 

message: In the section 'law of excluded middle' https://en.wikipedia.org/wiki/Proof_by_contradiction#Law_of_the_excluded_middle, the claim is made that a proof by contradiction requires the law of excluded middle. This isn't shown nor referenced, and I couldn't find a statement resulting to this claim on the page Law of excluded middle either ShearedLizard (talk) 17:19, 20 February 2018 (UTC)
thread_title:  Necessity of law of excluded middle is not shown

message: Gauss seems to have written Q.E.A. sometimes.--SilverMatsu (talk) 15:43, 5 January 2022 (UTC)
thread_title:  about Q.E.A. 

message: Jochen Burghardt asks for clarification regarding the passage about the BHK interpretation of proof by contradiction. I think this is a fair point, and he remarks that such clarification is not available in on the BHK page (the passage about absurdity comes close). I am not sure how to proceed. The BHK page has a fairly narrow explanation of what the BHK interpretation is, namely it claims to be about proofs, whereas the present page takes the BHK interpretation in its wider (and more acurate) sense explaining what the intuitionistic conditions for validity are. So until the BHK page is fixed, we cannot link to a reasonable explanation there. Any ideas what to do, apart from going after a major edit of the BHK page? AndrejBauer (talk) 18:56, 27 May 2024 (UTC)
thread_title:  Clarification of BHK interpretation of proof by contradiction.