Editing If and only if (section)

{{short description|Logical connective}}
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<br/>Logical symbols representing ''iff''&nbsp;&nbsp;
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In [[logic]] and related fields such as [[mathematics]] and [[philosophy]], "'''if and only if'''" (often shortened as "'''iff'''") is paraphrased by the [[biconditional]], a [[logical connective]]<ref>{{Cite web |title=Logical Connectives |url=https://sites.millersville.edu/bikenaga/math-proof/logical-connectives/logical-connectives.html |access-date=2023-09-10 |website=sites.millersville.edu}}</ref> between statements.  The biconditional is true in two cases, where either both statements are true or both are false. The connective is [[biconditional]] (a statement of '''material equivalence'''<!--boldface per WP:R#PLA-->),<ref>{{cite book |last=Copi |first=I. M. |last2=Cohen |first2=C. |last3=Flage |first3=D. E. |year=2006 |title=Essentials of Logic |edition=Second |location=Upper Saddle River, NJ |publisher=Pearson Education |page=197 |isbn=978-0-13-238034-8 }}</ref> and can be likened to the standard [[material conditional]] ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is false.

In writing, phrases commonly used as alternatives to P "if and only if" Q include: ''Q is [[Necessary and sufficient conditions#Simultaneous necessity and sufficiency|necessary and sufficient]] for P'', ''for P it is necessary and sufficient that Q'', ''P is equivalent (or materially equivalent) to Q'' (compare with [[material conditional|material implication]]), ''P precisely if Q'', ''P precisely (or exactly) when Q'', ''P exactly in case Q'', and ''P just in case Q''.<ref>Weisstein, Eric W. "Iff." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Iff.html {{Webarchive|url=https://web.archive.org/web/20181113005302/http://mathworld.wolfram.com/Iff.html |date=13 November 2018 }}</ref> Some authors regard "iff" as unsuitable in formal writing;<ref>E.g. {{citation |title=Reading, Writing, and Proving: A Closer Look at Mathematics |series=[[Undergraduate Texts in Mathematics]] |first1=Ulrich |last1=Daepp |first2=Pamela |last2=Gorkin|author2-link=Pamela Gorkin |publisher=Springer |year=2011 |isbn=9781441994790 |url=https://books.google.com/books?id=4QKcaXrVZb0C&pg=PA52 |page=52 |quote=While it can be a real time-saver, we don't recommend it in formal writing.}}</ref> others consider it a "borderline case" and tolerate its use.<ref>{{citation |title=Engineering Writing by Design: Creating Formal Documents of Lasting Value |first1=Edward J. |last1=Rothwell |first2=Michael J. |last2=Cloud |publisher=CRC Press |year=2014 |isbn=9781482234312 |page=98 |url=https://books.google.com/books?id=muXMAwAAQBAJ&pg=PA98 |quote=It is common in mathematical writing}}</ref> In [[Formula (mathematical logic)|logical formulae]], logical symbols, such as <math>\leftrightarrow</math> and <math>\Leftrightarrow</math>,<ref name=":2">{{Cite web|last=Peil|first=Timothy|title=Conditionals and Biconditionals|url=http://web.mnstate.edu/peil/geometry/Logic/4logic.htm|access-date=2020-09-04|website=web.mnstate.edu|archive-date=24 October 2020|archive-url=https://web.archive.org/web/20201024171606/http://web.mnstate.edu/peil/geometry/Logic/4logic.htm|url-status=dead}}</ref> are used instead of these phrases; see {{Section link||Notation}} below.