# Metalanguage

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In logic and linguistics, a metalanguage is a language used to describe another language, often called the object language. [1] Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quotation marks, or writing on a separate line.[ citation needed ] The structure of sentences and phrases in a metalanguage can be described by a metasyntax. [2]

## Types

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

### Embedded metalanguage

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.

### Ordered metalanguage

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.

### Nested metalanguage

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.

## 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 ${\displaystyle L_{0}}$, 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]

## Types of expressions

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.

### Deductive systems

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]

### Metavariables

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 ${\displaystyle L}$.

### Metatheories and metatheorems

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]

### Interpretations

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

## Role in metaphor

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:
1. Language transfers people's thoughts and feelings (mental content) to others

`ex: Try to get your thoughts across better.`

2. Speakers and writers insert their mental content into words

`ex: You have to put each concept into words more carefully.`

3. Words are containers

`ex: That sentence was filled with emotion.`

4. Listeners and writers extract mental content from words

`ex: Let me know if you find any new sensations in the poem.`

The minor framework views language as an open pipe spilling mental content into the void:
1. Speakers and writers eject mental content into an external space

`ex: Get those ideas out where they can do some good.`

2. Mental content is reified (viewed as concrete) in this space

`ex: That concept has been floating around for decades.`

3. Listeners and writers extract mental content from this space

`ex: Let me know if you find any good concepts in the essay.`

## Metaprogramming

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.

## Dictionaries

• Audi, R. 1996. The Cambridge Dictionary of Philosophy. Cambridge: Cambridge University Press.
• Baldick, C. 1996. Oxford Concise Dictionary of Literary Terms. Oxford: Oxford University Press.
• Cuddon, J. A. 1999. The Penguin Dictionary of Literary Terms and Literary Theory. London: Penguin Books.
• Honderich, T. 1995. The Oxford Companion to Philosophy . Oxford: Oxford University Press.
• Matthews, P. H. 1997. The Concise Oxford Dictionary of Linguistics. Oxford: Oxford University Press. ISBN   978-0-19-280008-4.
• McArthur, T. 1996. The Concise Oxford Companion to the English Language. Oxford: Oxford University Press.

## Related Research Articles

In mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.

Semantics is the linguistic and philosophical study of meaning in language, programming languages, formal logics, and semiotics. It is concerned with the relationship between signifiers—like words, phrases, signs, and symbols—and what they stand for in reality, their denotation.

The Natural semantic metalanguage (NSM) is a linguistic theory that reduces lexicons down to a set of semantic primitives. It is based on the conception of Polish professor Andrzej Bogusławski. The theory was formally developed by Anna Wierzbicka at Warsaw University and later at the Australian National University in the early 1970s, and Cliff Goddard at Australia's Griffith University.

In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence of n elements, where n is a non-negative integer. There is only one 0-tuple, an empty sequence, or empty tuple, as it is referred to. An n-tuple is defined inductively using the construction of an ordered pair.

Zellig Sabbettai Harris was an influential American linguist, mathematical syntactician, and methodologist of science. Originally a Semiticist, he is best known for his work in structural linguistics and discourse analysis and for the discovery of transformational structure in language. These developments from the first 10 years of his career were published within the first 25. His contributions in the subsequent 35 years of his career include transfer grammar, string analysis, elementary sentence-differences, algebraic structures in language, operator grammar, sublanguage grammar, a theory of linguistic information, and a principled account of the nature and origin of language.

Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.

A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined system of abstract thought.

An object language is a language which is the "object" of study in various fields including logic, linguistics, mathematics, and theoretical computer science. The language being used to talk about an object language is called a metalanguage. An object language may be a formal or natural language.

In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning.

Object theory is a theory in philosophy and mathematical logic concerning objects and the statements that can be made about objects.

A metatheory or meta-theory is a theory whose subject matter is some theory. All fields of research share some meta-theory, regardless whether this is explicit or correct. In a more restricted and specific sense, in mathematics and mathematical logic, metatheory means a mathematical theory about another mathematical theory.

In logic, a metatheorem is a statement about a formal system proven 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.

Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.

In analytic philosophy, philosophy of language investigates the nature of language, the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, the constitution of sentences, concepts, learning, and thought.

In mathematical logic, a judgment or assertion is a statement or enunciation in the metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition. In general, a judgment may be any inductively definable assertion in the metatheory.

In computer science, more specifically in automata and formal language theory, nested words are a concept proposed by Alur and Madhusudan as a joint generalization of words, as traditionally used for modelling linearly ordered structures, and of ordered unranked trees, as traditionally used for modelling hierarchical structures. Finite-state acceptors for nested words, so-called nested word automata, then give a more expressive generalization of finite automata on words. The linear encodings of languages accepted by finite nested word automata gives the class of visibly pushdown languages. The latter language class lies properly between the regular languages and the deterministic context-free languages. Since their introduction in 2004, these concepts have triggered much research in that area.

Semantic primes or semantic primitives are a set of semantic concepts that are innately understood but cannot be expressed in simpler terms. They represent words or phrases that are learned through practice but cannot be defined concretely. For example, although the meaning of "touching" is readily understood, a dictionary might define "touch" as "to make contact" and "contact" as "touching", providing no information if neither of these words is understood. The concept of universal semantic primes was largely introduced by Anna Wierzbicka's book, Semantics: Primes and Universals.

This is an index of articles in philosophy of language

In linguistics, the conduit metaphor is a dominant class of figurative expressions used when discussing communication itself (metalanguage). It operates whenever people speak or write as if they "insert" their mental contents into "containers" whose contents are then "extracted" by listeners and readers. Thus, language is viewed as a "conduit" conveying mental content between people.

In logic, a metavariable is a symbol or symbol string which belongs to a metalanguage and stands for elements of some object language. For instance, in the sentence

## References

1. 2010. Cambridge Advanced Learner's Dictionary. Cambridge: Cambridge University Press. Dictionary online. Available from http://dictionary.cambridge.org/dictionary/british/metalanguage Internet. Retrieved 20 November 2010
2. van Wijngaarden, A., et al. "Language and metalanguage." Revised Report on the Algorithmic Language Algol 68. Springer, Berlin, Heidelberg, 1976. 17-35.
3. Hofstadter, Douglas. 1980. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books ISBN   0-14-017997-6
4. Harris, Zellig S. (1991). . Oxford: Clarendon Press. pp.  272–318. ISBN   978-0-19-824224-6.
5. Ibid. p. 277.
6. Borel, Félix Édouard Justin Émile (1928). Leçons sur la theorie des fonctions (in French) (3 ed.). Paris: Gauthier-Villars & Cie. p. 160.
7. Hunter, Geoffrey. 1971. Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. Berkeley:University of California Press ISBN   978-0-520-01822-8
8. Ritzer, George. 1991. Metatheorizing in Sociology. New York: Simon Schuster ISBN   0-669-25008-2
9. Reddy, Michael J. 1979. The conduit metaphor: A case of frame conflict in our language about language. In Andrew Ortony (ed.), Metaphor and Thought. Cambridge: Cambridge University Press