Metalanguage

There are a variety of recognized types of metalanguage, including embedded, ordered, and nested (or hierarchical) metalanguages.

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."^[3]

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 an 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 ${\displaystyle L_{0}}$ , the grammar of the language is a discourse in the metalanguage ${\displaystyle L_{1}}$ , which is a sublanguage^[4] nested within ${\displaystyle L_{0}}$ .

• The grammar of ${\displaystyle L_{1}}$ , which has the form of a factual description, is a discourse in the metametalanguage ${\displaystyle L_{2}}$ , which is also a sublanguage of ${\displaystyle L_{0}}$ .
• The grammar of ${\displaystyle L_{2}}$ , which has the form of a theory describing the syntactic structure of such factual descriptions, is stated in the metametametalanguage ${\displaystyle L_{3}}$ , which likewise is a sublanguage of ${\displaystyle L_{0}}$ .
• The grammar of ${\displaystyle L_{3}}$  has the form of a metatheory describing the syntactic structure of theories stated in ${\displaystyle L_{2}}$ .
• ${\displaystyle L_{4}}$  and succeeding metalanguages have the same grammar as ${\displaystyle L_{3}}$ , differing only in reference.

Since all of these metalanguages are sublanguages of ${\displaystyle L_{0}}$ , ${\displaystyle L_{1}}$  is a nested metalanguage, but ${\displaystyle L_{2}}$  and sequel are ordered metalanguages.^[5] Since all these metalanguages are sublanguages of ${\displaystyle L_{0}}$  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.^[6]

There are several entities commonly expressed in a metalanguage. In logic usually the object language that the metalanguage is discussing is a formal language, and very often the metalanguage as well.

A deductive system (or, deductive apparatus of a formal system) consists of the axioms (or axiom schemata) and rules of inference that can be used to derive the theorems of the system.^[7]

A metavariable (or metalinguistic or metasyntactic variable) is a symbol or set of symbols in a metalanguage which stands for a symbol or set of symbols in some object language. For instance, in the sentence:

Let A and B be arbitrary formulas of a formal language ${\displaystyle L}$ .

The symbols A and B are not symbols of the object language ${\displaystyle L}$ , they are metavariables in the metalanguage (in this case, English) that is discussing the object language ${\displaystyle L}$ .

A metatheory is a theory whose subject matter is some other theory (a theory about a theory). Statements made in the metatheory about the theory are called metatheorems. A metatheorem is a true statement about a formal system expressed in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory.^[8]

An interpretation is an assignment of meanings to the symbols and words of a language.

Michael J. Reddy (1979) argues that much of the language we use to talk about language is conceptualized and structured by what he refers to as the conduit metaphor.^[9] This paradigm operates through two distinct, related frameworks.

The major framework views language as a sealed pipeline between people:

The minor framework views language as an open pipe spilling mental content into the void:

Computers follow programs, sets of instructions in a formal language. The development of a programming language involves the use of a metalanguage. The act of working with metalanguages in programming is known as metaprogramming.

Backus–Naur form, developed in the 1960s by John Backus and Peter Naur, is one of the earliest metalanguages used in computing. Examples of modern-day programming languages which commonly find use in metaprogramming include ML, Lisp, m4, and Yacc.

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