Definition:Formal Grammar

Definition
Let $\mathcal L$ be a formal language whose alphabet is $\mathcal A$.

The formal grammar of $\mathcal L$ comprises of rules of formation, which determine whether words in $\mathcal A$ belong to $\mathcal L$ or not.

Roughly speaking, there are two types of formal grammar, top-down grammar and bottom-up grammar.

Also known as
The formal grammar may also be called syntax; however, a convenient viewpoint is to think of the formal grammar as explicating the syntax for the associated formal language.

Thus the formal grammar is a means to obtain a syntax for $\mathcal L$, and multiple formal grammars may yield the same syntax.