What is the difference between implication symbols: $\rightarrow$ and $\Rightarrow$? I do not understand the difference between $\rightarrow$ and $\Rightarrow$. Sometimes I see implication truth tables labeled with the former, sometimes with the latter.
Aren't they synonyms of logical implication or is there any difference?
 A: There's a subtle difference.  "$P\rightarrow Q$" is the statement that $P$ implies $Q$, which may be a true or false statement.  "$P \Rightarrow Q$" is the assertion that $P\rightarrow Q$ is true.
A: As far as I'm aware, these two symbols mean the same thing. In my limited experience, Logicians seem to use "$\rightarrow$", while people not studying logic seem to use $"\implies"$ more often.
A: As others have said, in practice both are used for the material conditional, and I certainly wouldn't want to say that someone is 'wrong' in using the one symbol rather than the other, but personally I have my reasons for separating between the two:
I use $\rightarrow$ for the material implication, so that I can use the $\Rightarrow$ for logical implication. For example, I would use $P \land Q \Rightarrow P$ to make the meta-logical statement that the logic statement $P$ is logically implied by the logic statement $P \land Q$. Likewise, I use $\leftrightarrow$ for the material biconditional, and $\Leftrightarrow$ to express logical equivalence.  For example: $P \leftrightarrow Q \Leftrightarrow Q \leftrightarrow P$ expresses the meta-logical statement that the logic statement $P \leftrightarrow Q$ is logically equivalent to the logic statement $Q \leftrightarrow P$ (of course, some use $\equiv$ to express logical equivalence, but I have also seen $\equiv$ used to express the material conditional ...)
In short: people use different symbols, and that's just fine as long as you make clear what they mean and how you use them, but to me the single horizontal line signals something about the syntax of logic, while the double line to me signals something semantical. Indeed, in my eyes this distinction mirrors the distinction between $\vdash$ and $\vDash$ where $\vdash$ is about purely syntactical derivations, and the $\vDash$ is about semantical implication.
A: Usually, $\Rightarrow$ denotes implication in the metalanguage, whereas $\rightarrow$ denotes implication in the formal language that you want to talk about. For example, $$M \models \sigma \rightarrow \tau \ \Rightarrow \ M \models \rho$$ is translated as "if $M$ is a model of $\sigma \rightarrow \tau$, then $M$ is a model of $\rho$".
A: There is no universally observed difference between the two symbols.
$\Rightarrow$ tends to be used more often in undergraduate instruction, where the logical symbols are used to explain and elucidate ordinary mathematical arguments -- for example, in real analysis.
$\to$ tends to be favored in formal mathematical logic, where the focus is modeling ordinary mathematical arguments as formal mathematical objects that follow precise rules and can be studied as a subject in themselves.
But this split is not observed by all authors, and you cannot expect that a random text you encounter will be following it.
