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The English language functions as a metalanguage in many contexts not limited to linguistics, such as logic, science, and mathematics. In logic, the terms 'syllogism', 'proposition', 'conclusion', 'premise', 'true', 'false', 'sound', 'unsound', 'valid', and 'invalid' are all part of the metalanguage of logic that are also part of English (this is not to be confused with metalogic which is concerned with the boundaries, limits, scope, and foundations of logic itself.) The terms cited above are those used to talk about different kinds of propositions and their relations.
Unsurprisingly, English is also used as a metalanguage in science. Consider the chemical equation H + O2 → H2O. This an example of a balanced equation, one in which energy is conserved. Both the phrase 'balanced equation' and 'energy is conserved' are used to talk about the equation and as such are part of the metalanguage of chemistry. The same ideas can be extended to physics as well as the underlying chemistry and physics in biology, geology, ecology, and so on. The conservation laws, for example, can be said to be part of the metalanguage of all the sciences.
In mathematics there are numerous examples where English is used as a metalanguage. The terms 'one', 'two', 'three' are metalanguage used to talk about the natural numbers 1, 2, and 3. The terms 'axiom', theorem', 'postulate', 'law', and 'proof' are part of the metalanguage of geometryGeometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible to proof, bu. This is not to be confused with metamathematicsMetamathematics is mathematics used to study mathematics. It was originally differentiated from ordinary mathematics in the 19th century to focus on what was then called the foundations problem in mathematics. Important branches include proof theory and m which is concerned with proof theoryLogic Proof theory Proof theory studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures, suc; even metamathematics, however, has its own metalanguage, including such terms as 'consistent', 'inconsistent', 'complete', and 'incomplete'. The terms 'point', 'line','plane', and 'polyhedra' are all part of the metalanguage used to talk about the graphic object language of geometry. In arithmeticArithmetic Arithmetic or arithmetics in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as synonym fo the terms 'plus', 'minus', 'multiply', and 'divide' are part of the metalanguage for the operations of addition, subtraction, multiplication, and division. In algebraAlgebra Algebra (from the Arabic al-jabr meaning reunion connection or completion is a branch of mathematics which may be roughly characterized as a generalization and extension of arithmetic; it also refers to a particular kind of abstract algebra struct the terms 'variable', 'independent variable', 'dependent variable', and 'equation' are part of its metalanguage.
Metalanguages are not limited to English either. The relationship between two foreign languages can also be interpreted as that of a metalanguage to an object language. Spanish can be a metalanguage to English and vice versa. One can talk about Spanish in English and English in Spanish, as is often done when each is taught as a second language.
There are a variety of recognized kinds of metalanguages including embedded, ordered, and nested or hierachical.
Embedded metalanguages, as their name suggests, are metalanguages embedded in an object language. They occur both formally and naturally. This idea is found in Douglas Hofstader's book Godel, Escher, Bach in his discussion of the relationship between formal languages and number theory: "...it is in the nature of any formalization of number theory that it's metalanguage is embedded within it" (pg.270). They occur in informal languages as well, such as in English, where adjectives, adverbs, and possesive pronouns serve as an embedded metalanguage, while nouns, verbs, and in some instances adjectives and adverbs serve as an object language. Thus the term 'red' in the phrase 'red barn' is part of the embedded metalanguage of English and the term 'barn' is part of the object language. A similar example for adverbs is the term 'slowly' in the phrase 'slowly running'.
Ordered metalanguages are analogous to ordered logicOrdered logic, also known as non-commutative logic is a substructural logic that denies the structural rule of exchange. The order of assumption is therefore critical; if is assumed before , then must be used before''. Note: ordered logic is a restrictions. An example of an ordered metalanguage would be the construction of one metalanguage to talk about an object language, then creating another metalanguage to talk about the first metalanguage, and so on for as long as is necessary.
Nested or hierarchical metalanguages are similar to ordered metalanguages in that each level represents a greater degree of abstraction. However, nested metalanguages differ from ordered ones in that each level includes the one below. The paradigmatic example of a nested metalanguage comes from the Linnean taxonomic systemScientific classification or biological classification refers to how biologists group and categorize extinct and living species of organisms. Modern classification has its roots in the system of Carolus Linnaeus, who grouped species according to shared ph in biology. Each level in the system incorporates the one below it. The language used to talk about genus is also used to talk about species, the language that is used to talk about orders is also used to talk about geni, and so on up to kingdoms.
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