Editing Metalanguage (section)

== In natural language ==

Natural language combines nested and ordered metalanguages. In a natural language there is an infinite regress of metalanguages, each with more specialized vocabulary and simpler syntax.

Designating the language now as <math>L_0</math>, the grammar of the language is a discourse in the metalanguage <math>L_1</math>, which is a sublanguage<ref>{{cite book
 | last =Harris
 | first =Zellig S.
 | author-link =Zellig Harris
 | title =A theory of language and information: A mathematical approach
 | publisher =Clarendon Press
 | date =1991
 | location =Oxford
 | pages =[https://archive.org/details/theoryoflanguage00harr/page/272 272]–318
 | url =https://archive.org/details/theoryoflanguage00harr
 | isbn =978-0-19-824224-6
 | url-access =registration
 }}</ref> nested within <math>L_0</math>.

* The grammar of <math>L_1</math>, which has the form of a factual description, is a discourse in the meta–metalanguage <math>L_2</math>, which is also a sublanguage of <math>L_0</math>. 
* The grammar of <math>L_2</math>, which has the form of a theory describing the syntactic structure of such factual descriptions, is stated in the meta–meta–metalanguage <math>L_3</math>, which likewise is a sublanguage of <math>L_0</math>. 
* The grammar of <math>L_3</math> has the form of a metatheory describing the syntactic structure of theories stated in <math>L_2</math>. 
* <math>L_4</math> and succeeding metalanguages have the same grammar as <math>L_3</math>, differing only in reference.

Since all of these metalanguages are sublanguages of <math>L_0</math>, <math>L_1</math> is a nested metalanguage, but <math>L_2</math> and sequel are ordered metalanguages.<ref>''Ibid''. p. 277.</ref> Since all these metalanguages are sublanguages of <math>L_0</math> they are all embedded languages with respect to the language as a whole.

Metalanguages of formal systems all resolve ultimately to natural language, the 'common parlance' in which mathematicians and logicians converse to define their terms and operations and 'read out' their formulae.<ref>{{cite book
 | last =Borel
 | first =Félix Édouard Justin Émile
 | author-link =Émile Borel
 | title =Leçons sur la theorie des fonctions
 | publisher =Gauthier-Villars & Cie.
 | edition =3
 | date =1928
 | location =Paris
 | pages =160
 | language =fr
 }}</ref>