Meta language
Broadly, any metalanguage is language or symbols used when language itself is being discussed or examined. In logic and linguistics, a metalanguage is a language used to make statements about statements in another language (the object language). Expressions in a metalanguage are often distinguished from those in an object language by the use of italics, quotation marks, or writing on a separate line. The structure of sentences and phrases in a metalanguage can be described by a metasyntax.
There are a variety of recognized metalanguages, including embedded, ordered, and nested (or, hierarchical).
An embedded metalanguage is a language formally, naturally and firmly fixed in an object language. This idea is found in Douglas Hofstadter's book, Gödel, Escher, Bach, in a discussion of the relationship between formal languages and number theory: "... it is in the nature of any formalization of number theory that its metalanguage is embedded within it". It occurs in natural, or informal, languages, as well—such as in English, where words such as noun, verb, or even word describe features and concepts pertaining to the English language itself.
An ordered metalanguage is analogous to ordered logic. An example of an ordered metalanguage is the construction of one metalanguage to discuss an object language, followed by the creation of another metalanguage to discuss the first, etc.
A nested (or, hierarchical) metalanguage is similar to an ordered metalanguage in that each level represents a greater degree of abstraction. However, a nested metalanguage differs from an ordered one in that each level includes the one below. The paradigmatic example of a nested metalanguage comes from the Linnean taxonomic system in biology. Each level in the system incorporates the one below it. The language used to discuss genus is also used to discuss species; the one used to discuss orders is also used to discuss genera, etc., up to kingdoms.
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 L0, the grammar of the language is a discourse in the metalanguage L1, which is a sublanguage nested within L0. The grammar of L1, which has the form of a factual description, is a discourse in the metametalanguage L2, which is also a sublanguage of L0. The grammar of L2, which has the form of a theory describing the syntactic structure of such factual descriptions, is stated in the metametametalanguage L3, which likewise is a sublanguage of L0. The grammar of L3 has the form of a metatheory describing the syntactic structure of theories stated in L2. L4 and succeeding metalanguages have the same grammar as L3, differing only in reference. Since all of these metalanguages are sublanguages of L0, L1 is a nested metalanguage, but L2 and sequel are ordered metalanguages. Since all these metalanguages are sublanguages of L0 they are all embedded languages with respect to the language as a whole.
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