Editing Formal language (section)

=== Formal theories, systems, and proofs ===
{{Main|Theory (mathematical logic)|Formal system}}

[[File:Formal languages.svg|thumb|right|This diagram shows the [[Syntax (logic)|syntactic]] divisions within a [[formal system]]. [[string (computer science)|Strings of symbols]] may be broadly divided into nonsense and [[well-formed formula]]s. The set of well-formed formulas is divided into [[theorem]]s and non-theorems.]]

In [[mathematical logic]], a ''formal theory'' is a set of [[sentence (mathematical logic)|sentences]] expressed in a formal language.

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, which may be interpreted as valid rules of inference, 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. Although a formal language can be identified with its formulas, a formal system cannot be likewise identified by its theorems. Two formal systems <math>\mathcal{FS}</math> and <math>\mathcal{FS'}</math> may have all the same theorems and yet differ in some significant proof-theoretic way (a formula A may be a syntactic consequence of a formula B in one but not another for instance).

A ''formal proof'' or ''derivation'' is a finite sequence of well-formed formulas (which may be interpreted as sentences, or [[proposition]]s) each of which is an axiom or follows from the preceding formulas in the sequence by a [[rule of inference]]. The last sentence in the sequence is a theorem of a formal system. Formal proofs are useful because their theorems can be interpreted as true propositions.

====Interpretations and models====
{{main|Formal semantics (logic)||Interpretation (logic)|Model theory}}

Formal languages are entirely syntactic in nature, but may be given [[semantics]] that give meaning to the elements of the language. For instance, in mathematical [[logic]], the set of possible formulas of a particular logic is a formal language, and an [[interpretation (logic)|interpretation]] assigns a meaning to each of the formulas—usually, a [[truth value]].

The study of interpretations of formal languages is called [[Formal semantics (logic)|formal semantics]]. In mathematical logic, this is often done in terms of [[model theory]]. In model theory, the terms that occur in a formula are interpreted as objects within [[Structure (mathematical logic)|mathematical structures]], and fixed compositional interpretation rules determine how the truth value of the formula can be derived from the interpretation of its terms; a ''model'' for a formula is an interpretation of terms such that the formula becomes true.