What is the difference between implication symbols: $\rightarrow$ and $\Rightarrow$? [duplicate]

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?

Solution 1:

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 $\rho$".

Solution 2:

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.