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{{single source|date=June 2013}}
In [[logic]] and [[mathematics]], the '''logical biconditional''' (sometimes known as the '''material biconditional''') is the [[logical connective]] of two statements asserting "''p'' [[if and only if]] ''q''", where ''q'' is a ''[[hypothesis]]'' (or ''[[antecedent]]'') and ''p'' is a ''[[logical consequence|conclusion]]'' (or ''[[consequent]]'').<ref>Handbook of Logic, page 81</ref> This is often abbreviated ''p iff q''. The operator is denoted using a doubleheaded arrow (↔), a prefixed E (E''pq''), an equality sign (=), an equivalence sign (≡), or ''EQV''. It is logically equivalent to (p → q) ∧ (q → p), or the XNOR (exclusive nor) [[boolean operator]]. It is equivalent to "(not p or q) and (not q or p)". It is also logically equivalent to "(p and q) or (not p and not q)", meaning "both or neither".

The only difference from [[material conditional]] is the case when the hypothesis is false but the conclusion is true. In that case, in the conditional, the result is true, yet in the biconditional the result is false.

In the conceptual interpretation, ''a'' = ''b'' means "All ''a'' 's are ''b'' 's and all ''b'' 's are ''a'' 's"; in other words, the sets ''a'' and ''b'' coincide: they are identical. This does not mean that the concepts have the same meaning. Examples: "triangle" and "trilateral", "equiangular triangle" and "equilateral triangle". The antecedent is the ''subject'' and the consequent is the ''predicate'' of a universal [[affirmative]] [[proposition]].

In the propositional interpretation, ''a'' ⇔ ''b'' means that ''a'' implies ''b'' and ''b'' implies ''a''; in other words, that the propositions are equivalent, that is to say, either true or false at the same time. This does not mean that they have the same meaning. [[Pons asinorum|Example]]: "The triangle ABC has two equal sides", and "The triangle ABC has two equal angles". 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 is to use its equivalence to the conjunction of two converse [[Material conditional|conditional]]s, demonstrating these separately.

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 the
thesis the ''[[necessary condition]]'' of the hypothesis; that is to say, it is sufficient that the hypothesis be true for the thesis to be true; while it is necessary that the thesis be true for the hypothesis to be true also. When a theorem and its reciprocal are true we say that its hypothesis is the [[necessary and sufficient condition]] of the thesis; that is to say, that it is at the same time both cause and consequence.

==Definition==

[[Logical equality]] (also known as biconditional) is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both operands are false or both operands are true.

===Truth table===
The truth table for <math>~A \leftrightarrow B</math> (also written as '''A ≡ B''', '''A = B''', or '''A EQ B''') is as follows:

{| class="wikitable" style="margin: 0 0 1em 1em"
|- bgcolor="#ddeeff" align="center"
|colspan=2|'''INPUT''' || '''OUTPUT'''
|- bgcolor="#ddeeff" align="center"
| A || B || A <math>~ \leftrightarrow ~</math> B
|- bgcolor="#ddffdd" align="center"
|0 || 0 || 1
|- bgcolor="#ddffdd" align="center"
|0 || 1 || 0
|- bgcolor="#ddffdd" align="center"
|1 || 0 || 0
|- bgcolor="#ddffdd" align="center"
|1 || 1 || 1
|}

 

More than two statements combined by <math>~\leftrightarrow~</math> are ambiguous:

<math>~x_1 \leftrightarrow x_2 \leftrightarrow x_3 \leftrightarrow ... \leftrightarrow x_n</math> may be meant as <math>~(((x_1 \leftrightarrow x_2) \leftrightarrow x_3) \leftrightarrow ...) \leftrightarrow x_n</math>,

or may be used to say that all <math>~x_i~</math> are ''together true or together false'': <math>(~x_1 \and ... \and x_n~)~\or~(\neg x_1 \and ... \and \neg x_n)</math>

Only for zero or two arguments this is the same.

The following truth tables show the same bit pattern only in the line with no argument and in the lines with two arguments:

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

[[File:Multigrade operator all or nothing.svg|thumb|right|220px|<math>~x_1 \leftrightarrow ... \leftrightarrow x_n</math> meant as shorthand for <math>(~x_1 \and ... \and x_n~)</math> <math>\or~(\neg x_1 \and ... \and \neg x_n)</math> The Venn diagram directly below, and line ''(ABC  )'' in this matrix represent the same operation.]]
 
The left Venn diagram below, and the lines ''(AB    )'' in these matrices represent the same operation.

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

{| border="0" style="width:100%"
| style="vertical-align:top;"|
{| style="background: #f9f9f9; border: 1px solid #cccccc;" align="left"
|-
| [[Image:Venn1001.svg|220px]]
|-
| The biconditional of two statements is the [[negation]] of the [[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]]
|}
| style="width: 100px"|
| style="vertical-align:top;"|
{| style="background: #f9f9f9; border: 1px solid #cccccc;" align="center"
|-
| [[File:Venn 0110 1001.svg|220px]]
|-
| The biconditional and the exclusive or of three statements give the same result: 
<math>~A \leftrightarrow B \leftrightarrow C~~\Leftrightarrow</math> 
<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]]
|}
| style="width: 100px"|
| style="vertical-align:top;"|
{| style="background: #f9f9f9; border: 1px solid #cccccc;" align="right"
|-
| [[File:Venn 1000 0001.svg|220px]]
|-
| But <math>~A \leftrightarrow B \leftrightarrow C</math> may also be used as an abbreviation for <math>(A \leftrightarrow B) \and (B \leftrightarrow C)</math>

[[File:Venn 1001 1001.svg|40px]] <math>\and</math> [[File:Venn 1100 0011.svg|40px]]
<math>~~\Leftrightarrow~~</math> [[File:Venn 1000 0001.svg|40px]]
|}
|}

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

'''[[associativity]]: yes'''
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>~A</math>
|<math>~~~\leftrightarrow~~~</math>
|<math>(B \leftrightarrow C)</math>
|    <math>\Leftrightarrow</math>    
|
|
|<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]]
|    <math>\Leftrightarrow</math>    
|[[File:Venn 0110 1001.svg|50px]]
|    <math>\Leftrightarrow</math>    
|[[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 (see [[Logical_disjunction#Properties|there]]) distributes over biconditional.

'''[[idempotency]]: no''' 
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>~A~</math>
|<math>~\leftrightarrow~</math>
|<math>~A~</math>
|    <math>\Leftrightarrow</math>    
|<math>~1~</math>
|    <math>\nLeftrightarrow</math>    
|<math>~A~</math>
|-
|[[File:Venn01.svg|36px]]
|<math>~\leftrightarrow~</math>
|[[File:Venn01.svg|36px]]
|    <math>\Leftrightarrow</math>    
|[[File:Venn11.svg|36px]]
|    <math>\nLeftrightarrow</math>    
|[[File:Venn01.svg|36px]]
|}

'''[[Monotonic function#Boolean_functions|monotonicity]]: no'''
{| style="text-align: center; border: 1px solid darkgray;"
|<math>A \rightarrow B</math>
|    <math>\nRightarrow</math>    
|
|
|<math>(A \leftrightarrow C)</math>
|<math>\rightarrow</math>
|<math>(B \leftrightarrow C)</math>
|-
||[[File:Venn 1011 1011.svg|50px]]
|    <math>\nRightarrow</math>    
||[[File:Venn 1101 1011.svg|50px]]
|    <math>\Leftrightarrow</math>    
||[[File:Venn 1010 0101.svg|50px]]
|<math>\rightarrow</math>
||[[File:Venn 1100 0011.svg|50px]]
|}

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

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

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

'''Non[[Linear#Boolean functions|linearity]]: 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===
[[Biconditional introduction]] allows you to infer that, if B follows from A, and A follows from B, then A [[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, equally inferrable, "I'm alive if and only if I'm breathing."

B → A   
 A → B   
∴ A ↔ B

B → A   
 A → B   
∴ B ↔ A

===Biconditional elimination===
'''Biconditional elimination''' allows one to infer a [[Material conditional|conditional]] from a biconditional: if ( A ↔ B ) is true, then one may infer one direction of the biconditional, '''( A → B ) and ( B → A )'''.

For example, if it's true that I'm breathing [[if and only if]] I'm alive, then it's true that if I'm breathing, I'm alive; likewise, it's true that if I'm alive, I'm breathing.

Formally:

( A ↔ B )  
∴ ( A → B )

also

( A ↔ B )  
∴ ( B → A )

==Colloquial usage==

One unambiguous way of stating a biconditional in plain English is of the form "''b'' if ''a'' and ''a'' if ''b''". Another is "''a'' if and only if ''b''". Slightly more formally, one could say "''b'' implies ''a'' and ''a'' implies ''b''". The plain English "if'" may sometimes be used as a biconditional. One must weigh context heavily.

For example, "I'll buy you a new wallet if you need one" may be meant 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 not meant as a biconditional, since it can be cloudy while not raining.

==See also==
{{Portal|Thinking}}

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

==Notes==
{{Reflist|colwidth=30em}}

==References==
*Brennan, Joseph G. [http://www.archive.org/stream/handbookoflogics012674mbp#page/n90/mode/1up Handbook of Logic, 2nd Edition]. [[Harper & Row]]. 1961

{{Logical connectives}}
{{PlanetMath attribution|id=484|title=Biconditional}}

{{DEFAULTSORT:Logical Biconditional}}
[[Category:Logical connectives]]