\begin{definition}[Definition:Logical Equivalence]
If two statements $p$ and $q$ are such that:
:$p \vdash q$, that is: $p$ therefore $q$
:$q \vdash p$, that is: $q$ therefore $p$
then $p$ and $q$ are said to be '''(logically) equivalent'''.
That is:
:$p \dashv \vdash q$
means:
:$p \vdash q$ and $q \vdash p$.
Note that because the conclusion of an argument is a single statement, there can be only one statement on either side of the $\dashv \vdash$ sign.
In symbolic logic, the notion of '''logical equivalence''' occurs in the form of provable equivalence and semantic equivalence.
\end{definition}