Definition:Biconditional

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Definition

The biconditional is a binary connective:

$p \iff q$

defined as:

$\paren {p \implies q} \land \paren {q \implies p}$

That is:

If $p$ is true, then $q$ is true, and if $q$ is true, then $p$ is true.


$p \iff q$ can be voiced:

$p$ if and only if $q$.


Truth Function

The biconditional connective defines the truth function $f^\leftrightarrow$ as follows:

\(\ds \map {f^\leftrightarrow} {\F, \F}\) \(=\) \(\ds \T\)
\(\ds \map {f^\leftrightarrow} {\F, \T}\) \(=\) \(\ds \F\)
\(\ds \map {f^\leftrightarrow} {\T, \F}\) \(=\) \(\ds \F\)
\(\ds \map {f^\leftrightarrow} {\T, \T}\) \(=\) \(\ds \T\)


Truth Table

The characteristic truth table of the biconditional operator $p \iff q$ is as follows:

$\begin{array}{|cc||c|} \hline p & q & p \iff q \\ \hline \F & \F & \T \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$


Boolean Interpretation

The truth value of $\mathbf A \iff \mathbf B$ under a boolean interpretation $v$ is given by:

$\map v {\mathbf A \iff \mathbf B} = \begin{cases} \T & \text{if } \map v {\mathbf A} = \map v {\mathbf B} \\ \F & \text{otherwise} \end{cases}$


Semantics of the Biconditional

The concept of the biconditional has been defined such that $p \iff q$ means:

If $p$ is true then $q$ is true, and if $q$ is true then $p$ is true.

$p \iff q$ can be considered as a shorthand to replace the use of the longer and more unwieldy expression involving two conditionals and a conjunction.


If we refer to ways of expressing the conditional, we see that:

  • $q \implies p$ can be interpreted as $p$ is true if $q$ is true

and:

  • $p \implies q$ can be interpreted as $p$ is true only if $q$ is true.


Thus we arrive at the usual way of reading $p \iff q$ which is: $p$ is true if and only if $q$ is true.


This can also be said as:

  • The truth value of $p$ is equivalent to the truth value of $q$.
  • $p$ is equivalent to $q$.
  • $p$ and $q$ are equivalent.
  • $p$ and $q$ are coimplicant.
  • $p$ and $q$ are logically equivalent.
  • $p$ and $q$ are materially equivalent.
  • $p$ is true exactly when $q$ is true.
  • $p$ is true iff $q$ is true. This is another convenient and useful (if informal) shorthand which is catching on in the mathematical community.


Also known as

Other names for the biconditional include:


Notational Variants

Various symbols are encountered that denote the concept of biconditionality:

Symbol Origin
$p \iff q$
$p\ \mathsf{EQ} \ q$
$p \equiv q$ 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica
$p = q$
$p \leftrightarrow q$
$\operatorname E p q$ Łukasiewicz's Polish notation


It is usual in mathematics to use $\iff$, as there are other uses for the other symbols.


Examples

Monday iff Tomorrow Tuesday

The following is an example of a biconditional statement:

Either today is Monday if and only if today is the day before Tuesday.


Also see

  • Results about the biconditional can be found here.


Sources

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