In logic, do the $\Longrightarrow$ and $\rightarrow$ signify different things? Are there contexts where one is more appropriate than the other?
I had believed that the $\Longrightarrow$ was for metalogic, and the $\rightarrow$ was for logic. However, recently, I've noticed $\Longrightarrow$ used more often than $\rightarrow$ in non-metalogical logical contexts.
From what I've seen, $\longrightarrow$ and $\implies$ mean material implication, "if then". $\vdash$ and $\therefore$ are used for logical implication. $\implies$ seems to be more common in modern books and $\longrightarrow$ seems to be more common in older books.
Thus $p \implies q$ means that p is a sufficient condition for q and is false precisely when $\neg p \land q$. $p \vdash q$ means q follows from p by axioms, definitions, and theorems.