Editing If and only if (section)

==Usage==

===Notation===
The corresponding logical symbols are "<math>\leftrightarrow</math>", "<math>\Leftrightarrow</math>",<ref name=":2" /> and <math>\equiv</math>,<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Equivalent|url=https://mathworld.wolfram.com/Equivalent.html|access-date=2020-09-04|website=mathworld.wolfram.com|language=en|archive-date=3 October 2020|archive-url=https://web.archive.org/web/20201003031516/https://mathworld.wolfram.com/Equivalent.html|url-status=live}}</ref> and sometimes "iff". These are usually treated as equivalent. However, some texts of [[mathematical logic]] (particularly those on [[first-order logic]], rather than [[propositional logic]]) make a distinction between these, in which the first, <math>\leftrightarrow</math>, is used as a symbol in logic formulas, while <math>\Leftrightarrow</math> or <math>\equiv</math> is used in reasoning about those logic formulas (e.g., in [[metalogic]]). In [[Jan Łukasiewicz|Łukasiewicz]]'s [[Polish notation]], it is the prefix symbol <math>E</math>.<ref>{{Cite web|url=https://plato.stanford.edu/entries/lukasiewicz/polish-notation.html|title=Jan Łukasiewicz > Łukasiewicz's Parenthesis-Free or Polish Notation (Stanford Encyclopedia of Philosophy)|website=plato.stanford.edu|access-date=2019-10-22|archive-date=9 August 2019|archive-url=https://web.archive.org/web/20190809092951/https://plato.stanford.edu/entries/lukasiewicz/polish-notation.html|url-status=live}}</ref>

Another term for the [[logical connective]], i.e., the symbol in logic formulas, is [[exclusive nor]].

In [[TeX]], "if and only if" is shown as a long double arrow: <math>\iff</math> via command \iff or \Longleftrightarrow.<ref>{{Cite web|url=https://artofproblemsolving.com/wiki/index.php/LaTeX:Symbols|title=LaTeX:Symbol|website=Art of Problem Solving|access-date=2019-10-22|archive-date=22 October 2019|archive-url=https://web.archive.org/web/20191022014053/https://artofproblemsolving.com/wiki/index.php/LaTeX:Symbols|url-status=live}}</ref>

===Proofs===
In most [[logical system]]s, one [[Proof theory|proves]] a statement of the form "P iff Q" by proving either "if P, then Q" and "if Q, then P", or "if P, then Q" and "if not-P, then not-Q". Proving these pairs of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly. An alternative is to prove the [[disjunction]] "(P and Q) or (not-P and not-Q)", which itself can be inferred directly from either of its disjuncts—that is, because "iff" is [[truth-function]]al, "P iff Q" follows if P and Q have been shown to be both true, or both false.

===Origin of iff and pronunciation===
Usage of the abbreviation "iff" first appeared in print in [[John L. Kelley]]'s 1955 book ''General Topology''.<ref>''General Topology,'' reissue {{ISBN|978-0-387-90125-1}}</ref> Its invention is often credited to [[Paul Halmos]], who wrote "I invented 'iff,' for  'if and only if'—but I could never believe I was really its first inventor."<ref name="Higham1998">{{cite book |author=Nicholas J. Higham |title=Handbook of writing for the mathematical sciences |url=https://books.google.com/books?id=9gQd2fJA7Y4C&pg=PA24 |year=1998 |publisher=SIAM |isbn=978-0-89871-420-3 |page=24 |edition=2nd}}</ref>

It is somewhat unclear how "iff" was meant to be pronounced. In current practice, the single 'word' "iff" is almost always read as the four words "if and only if". However, in the preface of ''General Topology'', Kelley suggests that it should be read differently: "In some cases where mathematical content requires 'if and only if' and [[euphony]] demands something less I use Halmos' 'iff{{'"}}. The authors of one discrete mathematics textbook suggest:<ref>{{cite book |title=Discrete Algorithmic Mathematics |last=Maurer |first=Stephen B. |last2=Ralston |first2=Anthony |publisher=CRC Press |year=2005 |isbn=1568811667 |edition=3rd |location=Boca Raton, Fla. |pages=60}}</ref> "Should you need to pronounce iff, really [[Consonant gemination|hang on to the 'ff']] so that people hear the difference from 'if{{'"}}, implying that "iff" could be pronounced as {{IPA|[ɪfː]}}.

=== Usage in definitions ===
Conventionally, [[definitions]] are "if and only if" statements; some texts — such as Kelley's ''General Topology'' — follow this convention, and use "if and only if" or ''iff'' in definitions of new terms.<ref>For instance, from ''General Topology'', p.&nbsp;25: "A set is '''countable''' iff it is finite or countably infinite." [boldface in original]</ref> However, this usage of "if and only if" is relatively uncommon and overlooks the linguistic fact that the "if" of a definition is interpreted as meaning "if and only if". The majority of textbooks, research papers and articles (including English Wikipedia articles) follow the linguistic convention of interpreting "if" as "if and only if" whenever a mathematical definition is involved (as in "a topological space is compact if every open cover has a finite subcover").<ref>{{citation |page=[https://archive.org/details/primerofmathemat0000kran/page/71 71] |first=Steven G. |last=Krantz |title=A Primer of Mathematical Writing |year=1996 |publisher=American Mathematical Society |isbn=978-0-8218-0635-7 |url-access=registration |url=https://archive.org/details/primerofmathemat0000kran/page/71 }}</ref> Moreover, in the case of a [[Definition#recursive definitions|recursive definition]], the ''only if'' half of the definition is interpreted as a sentence in the metalanguage stating that the sentences in the definition of a predicate are the ''only sentences'' determining the extension of the predicate.