\begin{definition}[Definition:Logical Implication]
In a valid argument, the premises '''logically imply''' the conclusion.
If the truth of one statement $p$ can be shown in an argument directly to cause the meaning of another statement $q$ to be true, then $q$ follows from $p$ by '''logical implication'''.
We may say:
:'''$p$, therefore $q$''' and write $p \vdash q$.
:'''$q$, because $p$''' and write $q \dashv p$.
In symbolic logic, the concept of '''logical consequence''' occurs in the form of semantic consequence and provable consequence.
In the context of proofs of a conventional mathematical nature on {{ProofWiki}}, the notation:
:$p \leadsto q$
is preferred, where $\leadsto$ can be read as '''leads to'''.
\end{definition}