View source for If and only if

{{short description|Logical connective}}
{{Redirect-multi|3|⟺|⇔|Iff|other uses|IFF (disambiguation)|and|Arrow (symbol)}}
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{{quote box |quote = {{resize|400%|↔<!-- Unicode 2194 "<->"
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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.

==Definition==

{{multiple image
 | align = right | perrow = 2 | total_width = 200
 | image1 = Venn1000.svg | caption1 = <math>\neg P \and \neg Q</math>
 | image2 = Venn0001.svg | caption2 = <math>P \and Q</math>
 | image3 = Venn1011.svg | caption3 = <math>P \rightarrow Q</math>
 | image4 = Venn1101.svg | caption4 = <math>P \leftarrow Q</math>
 | image5 = Venn1001.svg | caption5 = <math>P \leftrightarrow Q</math>
}}

The [[truth table]] of ''P'' <math>\leftrightarrow</math> ''Q'' is as follows:<ref>[http://www.wolframalpha.com/input/?i=p+%3C%3D%3E+q p <=> q] {{Webarchive|url=https://web.archive.org/web/20161018212628/http://www.wolframalpha.com/input/?i=p+%3C%3D%3E+q |date=18 October 2016 }}. [[WolframAlpha]]</ref><ref>{{citation |title=If and only if |publisher=UHM Department of Mathematics |url=http://www.math.hawaii.edu/~ramsey/Logic/Iff.html |quote=Theorems which have the form "P if and only Q" are much prized in mathematics. They give what are called "necessary and sufficient" conditions, and give completely equivalent and hopefully interesting new ways to say exactly the same thing. |access-date=16 October 2016 |archive-date=5 May 2000 |archive-url=https://web.archive.org/web/20000505112920/http://www.math.hawaii.edu/~ramsey/Logic/Iff.html |url-status=live }}</ref>

{{2-ary truth table|A=P|B=Q
|1|0|0|0|<math>\neg P \and \neg Q</math>
|
|0|0|0|1|<math>P \and Q</math>
|thick
|1|1|0|1|<math>P \rightarrow Q</math>
|
|1|0|1|1|<math>P \leftarrow Q</math>
|thick
|1|0|0|1|<math>P \leftrightarrow Q</math>
}}

It is equivalent to that produced by the [[XNOR gate]], and opposite to that produced by the [[XOR gate]].<ref>{{Cite web|url=http://www.cburch.com/logisim/docs/2.1.0/libs/gates/xor.html|title=XOR/XNOR/Odd Parity/Even Parity Gate|website=www.cburch.com|access-date=2019-10-22|archive-date=7 April 2022|archive-url=https://web.archive.org/web/20220407061623/http://www.cburch.com/logisim/docs/2.1.0/libs/gates/xor.html|url-status=live}}</ref>

==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, ↔, is used as a symbol in logic formulas, while ⇔ 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.

==In terms of Euler diagrams==
<gallery widths="270">
File:Example of A is a proper subset of B.svg|''A'' is a proper subset of ''B''. A number is in ''A'' only if it is in ''B''; a number is in ''B'' if it is in ''A''.
File:Example of C is no proper subset of B.svg|''C'' is a subset but not a proper subset of ''B''. A number is in ''B'' if and only if it is in ''C'', and a number is in ''C'' if and only if it is in ''B''.
</gallery>

[[Euler diagram]]s show logical relationships among events, properties, and so forth. "P only if Q", "if P then Q", and  "P→Q" all mean that P is a [[subset]], either proper or improper, of Q. "P if Q", "if Q then P", and Q→P all mean that Q is a proper or improper subset of P. "P if and only if Q" and "Q if and only if P" both mean that the sets P and Q are identical to each other.

==More general usage==
''Iff'' is used outside the field of logic as well. Wherever logic is applied, especially in [[mathematics|mathematical]] discussions, it has the same meaning as above: it is an abbreviation for ''if and only if'', indicating that one statement is both [[Necessary and sufficient conditions|necessary and sufficient]] for the other. This is an example of [[mathematical jargon]] (although, as noted above, ''if'' is more often used than ''iff'' in statements of definition).

The elements of ''X'' are ''all and only'' the elements of ''Y'' means: "For any ''z'' in the [[domain of discourse]], ''z'' is in ''X'' if and only if ''z'' is in ''Y''."

==When "if" means "if and only if"==
In their [[Artificial Intelligence: A Modern Approach]], [[Stuart J. Russell|Russell]] and [[Peter Norvig|Norvig]] note (page 282),<ref name="Russell_Norvig_1995_2020">{{cite book | last1=Russell |first1=Stuart J. |author-link= Stuart J. Russell |last2=Norvig |first2=Peter  |title=Artificial Intelligence: A Modern Approach |publisher=[[Prentice Hall]] |orig-date=1995|edition=4 |date=2020 |pages=1136 | isbn = 978-0-13-461099-3 | oclc= 359890490}}</ref> in effect, that it is often more natural to express ''if and only if'' as ''if'' together with a "database (or logic programming) semantics". They give the example of the English sentence "Richard has two brothers, Geoffrey and John".

In a [[database]] or [[logic programming|logic program]], this could be represented simply by two sentences:

:Brother(Richard, Geoffrey).
:Brother(Richard, John).

The database semantics interprets the database (or program) as containing ''all'' and ''only'' the knowledge relevant for problem solving in a given domain. It interprets ''only if'' as expressing in the metalanguage that the sentences in the database represent the ''only'' knowledge that should be considered when drawing conclusions from the database. 

In [[first-order logic]] (FOL) with the standard semantics, the same English sentence would need to be represented, using ''if and only if'', with ''only if'' interpreted in the object language, in some such form as:

:<math>\forall </math> X(Brother(Richard, X) iff X = Geoffrey or X = John).
:Geoffrey ≠ John.

Compared with the standard semantics for FOL, the database semantics has a more efficient implementation. Instead of reasoning with sentences of the form:

:''conclusion iff conditions''

it uses sentences of the form:

:''conclusion if conditions''
 
to [[forward chaining|reason forwards]] from ''conditions'' to ''conclusions'' or [[backward chaining|backwards]] from ''conclusions'' to ''conditions''.

The database semantics is analogous to the legal principle [[expressio unius est exclusio alterius]] (the express mention of one thing excludes all others). Moreover, it underpins the application of logic programming to the representation of legal texts and legal reasoning.<ref>Kowalski, R., Dávila, J., Sartor, G. and Calejo, M., 2023. Logical English for law and education. http://www.doc.ic.ac.uk/~rak/papers/Logical%20English%20for%20Law%20and%20Education%20.pdf In Prolog: The Next 50 Years (pp. 287-299). Cham: Springer Nature Switzerland.</ref>

==See also==
*[[Definition]]
*[[Equivalence relation]]
*[[Logical biconditional]]
* [[Logical equality]]
* [[Logical equivalence]]
* [[Logic programming#negation as failure|If and only if in logic programs]]
* [[Polysyllogism]]

== References ==
{{Reflist|30em}}

==External links==
{{Commons category}}
*{{cite web |url= http://www.math.hawaii.edu/~ramsey/Logic/Iff.html |title= Tables of truth for if and only if |archive-url= https://web.archive.org/web/20000505112920/http://www.math.hawaii.edu/~ramsey/Logic/Iff.html |archive-date= May 5, 2000 }}
*[http://itre.cis.upenn.edu/~myl/languagelog/archives/003470.html Language Log: "Just in Case"]
*[http://hesperusphosphorus.wordpress.com/2009/12/08/just-in-case Southern California Philosophy for philosophy graduate students: "Just in Case"]

{{Common logical symbols}}

[[Category:Logical connectives]]
[[Category:Mathematical terminology]]
[[Category:Necessity and sufficiency]]