====== (Material) Biconditional ======

A compound logical //statement// which is //true// exactly when both sides have the same truth value.

Example of a biconditional statement:

> A number is //even// exactly when it is //divisible by two// without remainder.

===== Other names =====

  * Logical biconditional

===== Description =====

A //biconditional// is //true// if both sub-statements have identical truth values.

<aside float-right>
<figure>
{{ :logic:glossary:biconditional.svg|Biconditional Venn-diagram}}
<figcaption>Venn-diagram: biconditional A ≡ B</figcaption>
</figure>
</aside>

^ A ^ B ^ <span "A if and only if B">A ≡ B</span> ^
| true | true | true |
| true | false | false |
| false | true | false |
| false | false | true |

This is a special case ofthe  [[glossary:conditional|conditional]]: For every true //biconditional// statement <span "A if and only if B">A ≡ B</span>, it holds that both <span "if A, then B">A → B</span> and <span "if B, then A">B → A</span> must be true.

It follows that:

> <span "if A, then B">A → B</span>   –   <i>if</i> A, <i>then</i> B.
> <span "and">∧</span> <span "if B, then A">B → A</span>   –   <i>and, if</i> B, <i>then</i> A.
> <span "Therefore: A if and only if B"><span>∴</span> <span>A ≡ B</span></span>   –   Therefore: B <i>if and only if</i> A.

===== Logical symbol =====

In the context of this website, the identity sign (<span>≡</span>, pronounced: "if and only if" or also: "is identical to") is used for //biconditionals//. In other publications, symbols like <span>↔</span>, <span>⇔</span>, or occasionally the tilde symbol (<span>~</span>) are used. Especially in computer sciences and related fields, the neologism  <span :en "if and only if">iff</span>, (short for: "if and only if") is common.

===== Application =====

The //biconditional// is a very expressive logical operator, which also means that outside of formal systems (such as mathematics and logic) there are actually very few practical uses where such statements would hold.

In particular, this includes the following situations:

==== 1. Tautological statements ====

Obviously, //biconditional// statements are always valid if [[glossary:antecedent|antecedent]] and [[glossary:consequent|consequent]] (A and B) are //identical// (e.g. "if it rains, it rains").

This also applies to situations where the tautology only exists //indirectly//, for example through the definitions of the terms. An example from mathematics was already stated above: "if a number is divisible by 2, it is even", which also entails: "if a number is //not// even, it is //not// divisible by 2", since "even" and "divisible by 2" are synonymous by definition.

Such tautologies can be constructed via several intermediate steps, as the following example shows:

> If today is Monday, then the day after tomorrow is Wednesday.

This is tautological, because "Wednesday" can be defined as "the day after Tuesday" and Tuesday in turn as "the day after Monday". It follows that Wednesday is by definition "two days after Monday". Thus the statement is tautological; Consequently the converse is also true:

> If the day after tomorrow is Wednesday, then today is Monday.

==== 2. Empty complementary set ====

Even independently of a true tautology, A and B can describe identical sets: In set theory this can be described as follows: <span "the difference of A with B is the empty set">𝔸 ∖ 𝔹 = ∅</span> (the difference set of A with B is empty).

For example, the following sets:

> 𝔸 = All astronauts who flew to the moon.
> 𝔹 = Everybody who has taken rock samples from the moon.

Even though 𝔸 and 𝔹 do not //necessarily// describe the same set (it is possible to fly to the moon without bringing rock samples), the difference set between these groups is empty*. Therefore, the //biconditional// <span "A if and only if B">A ≡ B</span> is valid. 

<aside box>
* Note: I //suspect// that these groups are identical, but have not found a source for this. This is therefore only meant to be a (purely theoretical) example. If you have more information about this, please contact me.
</aside>

===== Relevance =====

The formal logical fallacy "[[logic:formal_fallacies:denying_the_antecedent|Denying the antecedent]]" is based on a confusion of [[glossary:conditional|conditional]] and //biconditional//.

===== See also =====

  * [[glossary:conditional|Conditional]]
  * [[logic:formal_fallacies:denying_the_antecedent|Denying the antecedent]]

===== More information =====

  * [[wp>Logical biconditional]] on //Wikipedia//

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