If and only if

{{multiple image

| align = right | perrow = 2 | total_width = 200

| image1 = Venn1000.svg | caption1 = $\backslash neg\; P\; \backslash and\; \backslash neg\; Q$

| image2 = Venn0001.svg | caption2 = $P\; \backslash and\; Q$

| image3 = Venn1011.svg | caption3 = $P\; \backslash rightarrow\; Q$

| image4 = Venn1101.svg | caption4 = $P\; \backslash leftarrow\; Q$

| image5 = Venn1001.svg | caption5 = $P\; \backslash leftrightarrow\; Q$

}}

The truth table of P $\backslash leftrightarrow$ Q is as follows:[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{{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 }}

{{2-ary truth table|A=P|B=Q

|1|0|0|0|$\backslash neg\; P\; \backslash and\; \backslash neg\; Q$

|

|0|0|0|1|$P\; \backslash and\; Q$

|thick

|1|1|0|1|$P\; \backslash rightarrow\; Q$

|

|1|0|1|1|$P\; \backslash leftarrow\; Q$

|thick

|1|0|0|1|$P\; \backslash leftrightarrow\; Q$

}}

It is equivalent to that produced by the XNOR gate, and opposite to that produced by the XOR gate.{{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}}

The corresponding logical symbols are "$\backslash leftrightarrow$", "$\backslash Leftrightarrow$", and $\backslash equiv$,{{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}} 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, $\backslash leftrightarrow$, is used as a symbol in logic formulas, while $\backslash Leftrightarrow$ or $\backslash equiv$ is used in reasoning about those logic formulas (e.g., in metalogic). In Łukasiewicz's Polish notation, it is the prefix symbol $E$.{{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}}

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: $\backslash iff$ via command \iff or \Longleftrightarrow.{{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}}

In most logical systems, one 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-functional, "P iff Q" follows if P and Q have been shown to be both true, or both false.

Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology.General Topology, reissue {{ISBN|978-0-387-90125-1}} 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."{{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}}

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:{{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}} "Should you need to pronounce iff, really hang on to the 'ff' so that people hear the difference from 'if{{'"}}, implying that "iff" could be pronounced as {{IPA|[ɪfː]}}.

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.For instance, from General Topology, p. 25: "A set is countable iff it is finite or countably infinite." [boldface in original] 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").{{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 }} Moreover, in the case of a 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.

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.

Euler diagrams 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.

Iff is used outside the field of logic as well. Wherever logic is applied, especially in 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 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."

In their Artificial Intelligence: A Modern Approach, Russell and Norvig note (page 282),{{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}} 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 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:

:$\backslash forall$ 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 reason forwards from conditions to conclusions or 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.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.

• Definition
• Equivalence relation
• Logical biconditional
• Logical equality
• Logical equivalence
• If and only if in logic programs
• Polysyllogism

{{Reflist|30em}}

{{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