Logic systems
Logic and Metalogic

Definition (formal system). A formal system consists of a formal language with a formal grammar, which can be used to generate a set of well-formed expressions using formation rules known as syntax.

Formal systems first appeared in mathematical logic in order to mathematically represent mathematical language and reasoning, but they are also used in other contexts: computer science, artificial intelligence, chemistry...

Definition (formula). For a given formal language, a (well-formed) formula is the generic name given to a (well-formed) expression in that language.