\begin{definition}[Definition:Biconditional/Semantics of 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.
\end{definition}