Metalanguage

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

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.

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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• Metalanguage, Principia Cybernetica.
• Willard McCarty (submitted 2006) Problematic Metaphors, Humanist Discussion Group, Vol. 20, No. 92.