It has been suggested that this article be merged into Formal grammar and Talk:Formal grammar#Formation rules. (Discuss) Proposed since March 2009.

In mathematical logic, **formation rules** are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. A grammar only addresses the location and manipulation of the strings of the language. It does not describe anything else about a language, such as its semantics (i.e. what the strings mean).

A *formal language* is an organized set of symbols the essential feature of which is that it can be precisely defined in terms of just the shapes and locations of those symbols. Such a language can be defined, then, without any reference to any meanings of any of its expressions; it can exist before any formal interpretation is assigned to it -- that is, before it has any meaning. First order logic is expressed in some formal language. A formal grammar determines which symbols and sets of symbols are formulas in a formal language.

Main article: Formal system |

A *formal system* (also called a *logical calculus*, or a *logical system*) consists of a formal language together with a deductive apparatus (also called a *deductive system*). The deductive apparatus may consist of a set of transformation rules (also called *inference rules*) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions.

The formation rules of a propositional calculus may, for instance, take a form such that;

- if we take Φ to be a propositional formula we can also take Φ to be a formula;
- if we take Φ and Ψ to be a propositional formulas we can also take (Φ Ψ), (Φ Ψ), (Φ Ψ) and (Φ Ψ) to also be formulas.

A predicate calculus will usually include all the same rules as a propositional calculus, with the addition of quantifiers such that if we take Φ to be a formula of propositional logic and α as a variable then we can take (α)Φ and (α)Φ each to be formulas of our predicate calculus.