# Definition:Biconditional

## 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:

 $\displaystyle f^\leftrightarrow \left({F, F}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\leftrightarrow \left({F, T}\right)$ $=$ $\displaystyle F$ $\displaystyle f^\leftrightarrow \left({T, F}\right)$ $=$ $\displaystyle F$ $\displaystyle f^\leftrightarrow \left({T, T}\right)$ $=$ $\displaystyle T$

### 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:

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

## 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 this operator include:

• equivalence
• material equivalence
• logical equivalence
• logical equality

## 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.