View source for Logical biconditional

{{Short description|Concept in logic and mathematics}}

[[File:Venn1001.svg|220px|thumb|[[Venn diagram|Venn diagram]] of <math>P \leftrightarrow Q</math><br />(true part in red)]]

In [[Logic|logic]] and [[Mathematics|mathematics]], the '''logical biconditional''', also known as '''material biconditional''' or '''equivalence''' or '''biimplication''' or '''bientailment''', is the [[Logical connective|logical connective]] used to conjoin two statements <math>P</math> and <math>Q</math> to form the statement "<math>P</math> [[If and only if|if and only if]] <math>Q</math>" (often abbreviated as "<math>P</math> iff <math>Q</math>"<ref name=":2">{{Cite web|url=http://mathworld.wolfram.com/Iff.html|title=Iff|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-25}}</ref>), where <math>P</math> is known as the ''[[Philosophy:Antecedent (logic)|antecedent]]'', and <math>Q</math> the ''[[Consequent|consequent]]''.<ref name=":1">{{Cite web|url=http://web.mnstate.edu/peil/geometry/Logic/4logic.htm|title=Conditionals and Biconditionals|last=Peil|first=Timothy|website=web.mnstate.edu|access-date=2019-11-25|archive-date=2020-10-24|archive-url=https://web.archive.org/web/20201024171606/http://web.mnstate.edu/peil/geometry/Logic/4logic.htm|url-status=dead}}</ref><ref>{{Cite book|title=Handbook of Logic|last=Brennan|first=Joseph G.|publisher=Harper & Row|year=1961|edition=2nd|pages=81}}</ref> 

Nowadays, notations to represent equivalence include <math>\leftrightarrow,\Leftrightarrow,\equiv</math>.

<math>P\leftrightarrow Q</math> is logically equivalent to both <math>(P \rightarrow Q) \land (Q \rightarrow P)</math> and <math>(P \land Q) \lor (\neg P \land \neg Q) </math>, and the [[Engineering:XNOR gate|XNOR]] (exclusive nor) [[Logical connective|Boolean operator]], which means "both or neither".

Semantically, the only case where a logical biconditional is different from a [[Material conditional|material conditional]] is the case where the hypothesis (antecedent) is false but the conclusion (consequent) is true. In this case, the result is true for the conditional, but false for the biconditional.<ref name=":1" />

In the conceptual interpretation, {{math|1=''P'' = ''Q''}} means "All {{mvar|P}}'s are {{mvar|Q}}'s and all {{mvar|Q}}'s are {{mvar|P}}'s". In other words, the sets {{mvar|P}} and {{mvar|Q}} coincide: they are identical. However, this does not mean that {{mvar|P}} and {{mvar|Q}} need to have the same meaning (e.g., {{mvar|P}} could be "equiangular trilateral" and {{mvar|Q}} could be "equilateral triangle"). When phrased as a sentence, the antecedent is the ''subject'' and the consequent is the ''predicate'' of a universal affirmative proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate).

In the propositional interpretation, <math>P \leftrightarrow Q</math> means that {{mvar|P}} implies {{mvar|Q}} and {{mvar|Q}} implies {{mvar|P}}; in other words, the propositions are logically equivalent, in the sense that both are either jointly true or jointly false. Again, this does not mean that they need to have the same meaning, as {{mvar|P}} could be "the triangle ABC has two equal sides" and {{mvar|Q}} could be "the triangle ABC has two equal angles". In general, the antecedent is the ''premise'', or the ''cause'', and the consequent is the ''consequence''. When an implication is translated by a ''hypothetical'' (or ''conditional'') judgment, the antecedent is called the ''hypothesis'' (or the ''condition'') and the consequent is called the ''thesis''.

A common way of demonstrating a biconditional of the form <math>P \leftrightarrow Q</math> is to demonstrate that <math>P \rightarrow Q</math> and <math>Q \rightarrow P</math> separately (due to its equivalence to the conjunction of the two converse [[Material conditional|conditional]]s<ref name=":1" />). Yet another way of demonstrating the same biconditional is by demonstrating that <math>P \rightarrow Q</math> and <math>\neg P \rightarrow \neg Q</math>.

When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a ''theorem'' and the other its ''reciprocal''.{{Citation needed|date=August 2008}} Thus whenever a theorem and its reciprocal are true, we have a biconditional. A simple theorem gives rise to an implication, whose antecedent is the ''hypothesis'' and whose consequent is the ''thesis'' of the theorem.

It is often said that the hypothesis is the ''sufficient condition'' of the thesis, and that the thesis is the ''necessary condition'' of the hypothesis. That is, it is sufficient that the hypothesis be true for the thesis to be true, while it is necessary that the thesis be true if the hypothesis were true. When a theorem and its reciprocal are true, its hypothesis is said to be the necessary and sufficient condition of the thesis. That is, the hypothesis is both the cause and the consequence of the thesis at the same time.

==Notations==
Notations to represent equivalence used in history include: 
* <math>=</math> in [[Biography:George Boole|George Boole]] in 1847.<ref name="boole1847">{{cite book |last1=Boole |first1=G. |title=The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning |date=1847 |publisher=Macmillan, Barclay, & Macmillan/George Bell |location=Cambridge/London |page=17 |url=https://archive.org/details/mathematicalanal00booluoft}}</ref> Although Boole used <math>=</math> mainly on classes, he also considered the case that <math>x,y</math> are propositions in <math>x=y</math>, and at the time <math>=</math> is equivalence.
* <math>\equiv</math> in [[Biography:Gottlob Frege|Frege]] in 1879;<ref name="frege1879b">{{cite book |last1=Frege |first1=G. |title=Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens |date=1879 |publisher=Verlag von Louis Nebert |location=Halle a/S. |page=15 |language=German}}</ref>
* <math>\sim</math> in [[Biography:Paul Bernays|Bernays]] in 1918;<ref name="bernays1918">{{cite book |last1=Bernays |first1=P. |title=Beiträge zur axiomatischen Behandlung des Logik-Kalküls |date=1918 |publisher=Universität Göttingen |location=Göttingen |page=3}}</ref>
* <math>\rightleftarrows</math> in [[Biography:David Hilbert|Hilbert]] in 1927 (while he used <math>\sim</math> as the main symbol in the article);<ref name="hilbert1927">{{cite journal |last1=Hilbert |first1=D. |title=Die Grundlagen der Mathematik |journal=Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität |orig-date=1927 |date=1928 |volume=6 |pages=65–85 |doi=10.1007/BF02940602 |language=German}}</ref>
* <math>\leftrightarrow</math> in [[Biography:David Hilbert|Hilbert]] and Ackermann in 1928<ref name="hilbert-ackermann1928">{{cite book |last1=Hilbert |first1=D. |last2=Ackermann |first2=W. |title=Grundzügen der theoretischen Logik |edition=1 |date=1928 |publisher=Verlag von Julius Springer |location=Berlin |page=4 |language=German}}</ref> (they also introduced <math>\rightleftarrows,\sim</math> while they use <math>\sim</math> as the main symbol in the whole book; <math>\leftrightarrow</math> is adopted by many followers such as Becker in 1933<ref name="becker1933">{{cite book |last1=Becker |first1=A. |title=Die Aristotelische Theorie der Möglichkeitsschlösse: Eine logisch-philologische Untersuchung der Kapitel 13-22 von Aristoteles' Analytica priora I |date=1933 |publisher=Junker und Dünnhaupt Verlag |location=Berlin |page=4 |language=German}}</ref>);
* <math>E</math> (prefix) in [[Biography:Jan Łukasiewicz|Łukasiewicz]] in 1929<ref name="lukasiewicz1929">{{cite book |last1=Łukasiewicz |first1=J. |editor1-last=Słupecki |editor1-first=J. |title=Elementy logiki matematycznej |orig-date=1929|date=1958 |publisher=Państwowe Wydawnictwo Naukowe |location=Warszawa |edition=2 |language=Polish}}</ref> and <math>Q</math> (prefix) in [[Biography:Jan Łukasiewicz|Łukasiewicz]] in 1951;<ref name="lukasiewicz1951">{{cite book |last1=Łukasiewicz |first1=J. |editor1-last=Słupecki |editor1-first=J. |title=Aristotle's Syllogistic from the Standpoint of Modern Formal Logic |orig-date=1951|date=1957 |publisher=Oxford University Press |location=Glasgow, New York, Toronto, Melbourne, Wellington, Bombay, Calcutta, Madras, Karachi, Lahore, Dacca, Cape Town, Salisbury, Nairobi, Ibadan, Accra, Kuala Lumpur and Hong Kong |edition=2 |language=Polish}}</ref> 
* <math>\supset\subset</math> in Heyting in 1930;<ref name="heyting1929">{{cite journal |last1=Heyting |first1=A. |title=Die formalen Regeln der intuitionistischen Logik |journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse |date=1930 |pages=42–56 |language=German}}</ref>
* <math>\Leftrightarrow</math> in Bourbaki in 1954;<ref name="bourbaki1954b">{{cite book |last1=Bourbaki |first1=N. |title=Théorie des ensembles |date=1954 |publisher=Hermann & Cie, Éditeurs |location=Paris |page=32 |language=French}}</ref>
* <math>\subset\supset</math> in Chazal in 1996;<ref name="chazal1996">{{cite book |last1=Chazal |first1=G. |title=Eléments de logique formelle |date=1996 |publisher=Hermes Science Publications |location=Paris}}</ref> 

and so on. Somebody else also use <math>\operatorname{EQ}</math> or <math>\operatorname{EQV}</math> occasionally.{{citation needed|date=September 2023}}{{vague|date=September 2023}}{{clarify|date=September 2023}}

==Definition==

[[Logical equality]] (also known as biconditional) is an operation on two logical values, typically the values of two [[Proposition|proposition]]s, that produces a value of ''true'' if and only if both operands are false or both operands are true.<ref name=":1" />

===Truth table===
The following is a truth table for <math>P \leftrightarrow Q</math>:

{| class="wikitable" style="text-align:center; background-color: #ddffdd;"
|- bgcolor="#ddeeff"
| <math> P </math> || <math> Q </math> || <math> P \leftrightarrow Q </math>
|-
| {{yes2|True}} || {{yes2|True}} || {{yes2|True}}
|-
| {{yes2|True}} || {{no2|False}} || {{no2|False}}
|-
| {{no2|False}} || {{yes2|True}} || {{no2|False}}
|-
| {{no2|False}} || {{no2|False}} || {{yes2|True}}
|}

{{-}}

When more than two statements are involved, combining them with <math>\leftrightarrow</math> might be ambiguous. For example, the statement

:<math>x_1 \leftrightarrow x_2 \leftrightarrow x_3 \leftrightarrow \cdots \leftrightarrow x_n</math>

may be interpreted as

:<math>(((x_1 \leftrightarrow x_2) \leftrightarrow x_3) \leftrightarrow \cdots) \leftrightarrow x_n</math>,

or may be interpreted as saying that all {{math|''x<sub>i</sub>''}} are ''jointly true or jointly false'':
:<math>(x_1 \land \cdots \land x_n) \lor (\neg x_1 \land \cdots \land \neg x_n)</math>

As it turns out, these two statements are only the same when zero or two arguments are involved. In fact, the following truth tables only show the same bit pattern in the line with no argument and in the lines with two arguments:

[[File:Variadic logical XAND.svg|thumb|left|220px|<math>~x_1 \leftrightarrow \cdots \leftrightarrow x_n</math><br />meant as equivalent to<br /><math>\neg~(\neg x_1 \oplus \cdots \oplus \neg x_n)</math><br /><br />The central Venn diagram below,<br />and line ''(ABC&nbsp;&nbsp;)'' in this matrix<br />represent the same operation.]]

[[File:Variadic logical all or nothing.svg|thumb|right|220px|<math>~x_1 \leftrightarrow \cdots \leftrightarrow x_n</math><br />meant as shorthand for<br /><math>(~x_1 \land \cdots \land x_n~)</math><br /><math>\lor~(\neg x_1 \land \cdots \land \neg x_n)</math><br /><br />The Venn diagram directly below,<br />and line ''(ABC&nbsp;&nbsp;)'' in this matrix<br />represent the same operation.]]
{{-}}
The left Venn diagram below, and the lines ''(AB&nbsp;&nbsp;&nbsp;&nbsp;)'' in these matrices represent the same operation.

===Venn diagrams===
Red areas stand for true (as in [[Image:Venn0001.svg|40px]] for ''[[Logical conjunction|and]]'').

{| border="0" style="width:100%"
| style="vertical-align:top;"|<!--- START LEFT TABLE IN TABLE --->
{|  style="background:#f9f9f9; border:1px solid #ccc; float:left;"
|-
| [[Image:Venn1001.svg|220px]]
|-
| The biconditional of two statements<br />is the [[Negation|negation]] of the [[Exclusive or|exclusive or]]:
|- style="text-align:center;"
|<math>~A \leftrightarrow B~~\Leftrightarrow~~\neg(A \oplus B)</math>

[[File:Venn1001.svg|40px]] <math>\Leftrightarrow \neg</math> [[Image:Venn0110.svg|40px]]
|}<!--- END LEFT TABLE IN TABLE --->
| style="width: 100px"|
| style="vertical-align:top;"|<!--- START CENTRAL TABLE IN TABLE --->
{|  style="background:#f9f9f9; border:1px solid #ccc; margin:auto;"
|-
| [[File:Venn 0110 1001.svg|220px]]
|-
| The biconditional and the<br />exclusive or of three statements<br />give the same result:<br />
<math>~A \leftrightarrow B \leftrightarrow C~~\Leftrightarrow</math><br />
<math>~A \oplus B \oplus C</math>

[[File:Venn 1001 1001.svg|40px]] <math>\leftrightarrow</math> [[File:Venn 0000 1111.svg|40px]]
<math>~~\Leftrightarrow~~</math>

[[File:Venn 0110 0110.svg|40px]] <math>\oplus</math> [[File:Venn 0000 1111.svg|40px]]
<math>~~\Leftrightarrow~~</math> [[File:Venn 0110 1001.svg|40px]]
|}<!--- END CENTRAL TABLE IN TABLE --->
| style="width: 100px"|
| style="vertical-align:top;"|<!--- START RIGHT TABLE IN TABLE --->
{|  style="background:#f9f9f9; border:1px solid #ccc; float:right;"
|-
| [[File:Venn 1000 0001.svg|220px]]
|-
| But <math>~A \leftrightarrow B \leftrightarrow C</math><br />may also be used as an abbreviation<br />for <math>(A \leftrightarrow B) \land (B \leftrightarrow C)</math>

[[File:Venn 1001 1001.svg|40px]] <math>\land</math> [[File:Venn 1100 0011.svg|40px]]
<math>~~\Leftrightarrow~~</math> [[File:Venn 1000 0001.svg|40px]]
|}<!--- END RIGHT TABLE IN TABLE --->
|}



==Properties==
'''Commutativity: Yes'''
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>A \leftrightarrow B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>B \leftrightarrow A</math>
|-
|[[File:Venn1001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn1001.svg|50px]]
|}

'''Associativity: Yes'''
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>~A</math>
|<math>~~~\leftrightarrow~~~</math>
|<math>(B \leftrightarrow C)</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|
|
|<math>(A \leftrightarrow B)</math>
|<math>~~~\leftrightarrow~~~</math>
|<math>~C</math>
|-
|[[File:Venn 0101 0101.svg|50px]]
|<math>~~~\leftrightarrow~~~</math>
|[[File:Venn 1100 0011.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0110 1001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 1001 1001.svg|50px]]
|<math>~~~\leftrightarrow~~~</math>
|[[File:Venn 0000 1111.svg|50px]]
|}

'''Distributivity:'''  Biconditional doesn't distribute over any binary function (not even itself), but [[Logical disjunction#Properties|logical disjunction distributes]] over biconditional.

'''Idempotency: No'''<br />
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>~A~</math>  
|<math>~\leftrightarrow~</math> 
|<math>~A~</math> 
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>~1~</math> 
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nLeftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>~A~</math> 
|-
|[[File:Venn01.svg|36px]] 
|<math>~\leftrightarrow~</math> 
|[[File:Venn01.svg|36px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn11.svg|36px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nLeftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn01.svg|36px]]
|}

'''Monotonicity: No'''
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>A \rightarrow B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|
|
|<math>(A \leftrightarrow C)</math>
|<math>\rightarrow</math>
|<math>(B \leftrightarrow C)</math>
|-
||[[File:Venn 1011 1011.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
||[[File:Venn 1101 1011.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
||[[File:Venn 1010 0101.svg|50px]]
|<math>\rightarrow</math>
||[[File:Venn 1100 0011.svg|50px]]
|}

'''Truth-preserving: Yes'''<br />
When all inputs are true, the output is true.
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>A \land B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>A \leftrightarrow B</math>
|-
|[[File:Venn0001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn1001.svg|60px]]
|}

'''Falsehood-preserving: No'''<br />
When all inputs are false, the output is not false.
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>A \leftrightarrow B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>A \lor B</math>
|-
|[[File:Venn1001.svg|60px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn0111.svg|50px]]
|}

'''[[Hadamard transform|Walsh spectrum]]: (2,0,0,2)'''

'''Nonlinearity: 0''' (the function is linear)

==Rules of inference==
Like all connectives in first-order logic, the biconditional has rules of inference that govern its use in formal proofs.

===Biconditional introduction===
{{Main|Biconditional introduction}}
Biconditional introduction allows one to infer that if B follows from A and A follows from B, then A [[If and only if|if and only if]] B.

For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive" or equivalently, "I'm alive if and only if I'm breathing." Or more schematically:

  B → A &nbsp;&nbsp;
 <u> A → B &nbsp;&nbsp;</u>
  ∴ A ↔ B

  B → A &nbsp;&nbsp;
 <u> A → B &nbsp;&nbsp;</u>
  ∴ B ↔ A

===Biconditional elimination===
Biconditional elimination allows one to infer a [[Material conditional|conditional]] from a biconditional: if A <small>↔</small> B is true, then one may infer either A <small>→</small> B, or B <small>→</small> A.

For example, if it is true that I'm breathing [[If and only if|if and only if]] I'm alive, then it's true that ''if'' I'm breathing, then I'm alive; likewise, it's true that ''if'' I'm alive, then I'm breathing. Or more schematically:

  <u>A ↔ B&nbsp;&nbsp;</u>
  ∴ A → B

  <u>A ↔ B&nbsp;&nbsp;</u>
  ∴ B → A

==Colloquial usage==

One unambiguous way of stating a biconditional in plain English is to adopt the form "''b'' if ''a'' and ''a'' if ''b''"—if the standard form "''a'' if and only if ''b''" is not used. Slightly more formally, one could also say that "''b'' implies ''a'' and ''a'' implies ''b''", or "''a'' is necessary and sufficient for ''b''". The plain English "if'" may sometimes be used as a biconditional (especially in the context of a mathematical definition<ref>In fact, such is the style adopted by Wikipedia's manual of style in mathematics.</ref>). In which case, one must take into consideration the surrounding context when interpreting these words.

For example, the statement "I'll buy you a new wallet if you need one" may be interpreted as a biconditional, since the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional). However, "it is cloudy if it is raining" is generally not meant as a biconditional, since it can still be cloudy even if it is not raining.

==See also==

* [[If and only if]]
* [[Logical equivalence]]
* [[Logical equality]]
* [[Engineering:XNOR gate|XNOR gate]]
* [[Biconditional elimination]]
* [[Biconditional introduction]]

==References==
{{reflist}}

==External links==

{{Logical connectives}}
{{Mathematical logic}}

{{DEFAULTSORT:Logical Biconditional}}
[[Category:Logical connectives|Biconditional]]
[[Category:Equivalence (mathematics)]]

{{Sourceattribution|Logical biconditional}}