Editing Formal proof (section)

== Background ==
=== Formal language ===
{{Main|Formal language}}

A ''formal language'' is a [[Set (mathematics)|set]] of finite [[sequence (mathematics)|sequences]] of [[symbol]]s. Such a language can be defined without [[reference]] to any [[meaning (linguistics)|meaning]]s of any of its expressions; it can exist before any [[Interpretation (logic)|interpretation]] is assigned to it &ndash; that is, before it has any meaning. Formal proofs are expressed in some formal languages.

=== Formal grammar ===
{{Main|Formal grammar|Formation rule}}

A ''formal grammar'' (also called ''formation rules'') is a precise description of the [[well-formed formula]]s of a formal language. It is synonymous with the set of [[String (computer science)|strings]] over the [[alphabet]] of the formal language which constitute well formed formulas. However, it does not describe their [[semantics]] (i.e. what they mean).

=== Formal systems ===
{{Main|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 rule]]s (also called ''inference rules'') or a set of [[axiom]]s, or have both. A formal system is used to [[Proof theory|derive]] one expression from one or more other expressions.

=== Interpretations ===
{{Main|Formal semantics (logic)|Interpretation (logic)}}

An ''interpretation'' of a formal system is the assignment of meanings to the symbols, and truth values to the sentences of a formal system. The study of interpretations is called [[Formal semantics (logic)|formal semantics]]. ''Giving an interpretation'' is synonymous with ''constructing a [[Structure (mathematical logic)|model]].