In logic, do the $\Longrightarrow$ and $\rightarrow$ signify different things?

Question

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.

Answer 1

Long story short, $\rightarrow$ is a logic operator, whereas $\implies$ is a statement (where you know the outcome must be true).

$\rightarrow$ : can have result "false". $\\$ $\implies$ : Always true, by definition cannot be false. Not used as an operator.

Answer 2

The logical definition for "$A\to B$" is equivalent to $\neg A\vee B$:

"$A\Rightarrow B$" is hereby defined as an $A\to B$ which is always true (tautology).

E.g. "$n>4\to n>2$" is always true, hence it holds "$n>4\Rightarrow n>2$".

The converse does not hold: "$n>2\to n>4$" is an eligible statement, but "$n>2\Rightarrow n>4$" does not hold.

Answer 3

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.