Editing Proof by contradiction (section)

==Notation==

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Proofs by contradiction sometimes end with the word "Contradiction!".  [[Isaac Barrow]] and Baermann used the notation Q.E.A., for "''quod est absurdum''" ("which is absurd"), along the lines of [[Q.E.D.]], but this notation is rarely used today.<ref>{{cite web|url=http://mathforum.org/kb/message.jspa?messageID=1175481|title=Math Forum Discussions}}</ref> A graphical symbol sometimes used for contradictions is a downwards zigzag arrow "lightning" symbol (U+21AF: ↯), for example in Davey and Priestley.<ref>B. Davey and H.A. Priestley, ''[[Introduction to Lattices and Order]]'', Cambridge University Press, 2002; see "Notation Index", p. 286.</ref>  Others sometimes used include a pair of [[Hand of Eris|opposing arrows]] (as <math>\rightarrow\!\leftarrow</math>{{citation needed|reason=Give a reference for each notational variant.|date=October 2021}} or <math>\Rightarrow\!\Leftarrow</math>),{{citation needed|date=October 2021}} struck-out arrows (<math>\nleftrightarrow</math>),{{citation needed|date=October 2021}} a stylized form of hash (such as U+2A33: ⨳),{{citation needed|date=October 2021}} or the "reference mark" (U+203B: ※),{{citation needed|date=October 2021}} or <math>\times\!\!\!\!\times</math>.<ref>Gary Hardegree, ''Introduction to Modal Logic'', Chapter 2, pg. II–2.  https://web.archive.org/web/20110607061046/http://people.umass.edu/gmhwww/511/pdf/c02.pdf</ref><ref>The Comprehensive LaTeX Symbol List, pg. 20.  http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf</ref>