Definition:Biconditional

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Definition

The biconditional is a binary connective:

$p \iff q$

defined as:

$\left({p \implies q}\right) \land \left({q \implies p}\right)$

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:

$v \left({\mathbf A \iff \mathbf B}\right) = \begin{cases} T & : v \left({\mathbf A}\right) = v \left({\mathbf B}\right) \\ F & : \text{otherwise} \end{cases}$


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.


Sources