Metalanguage facts for kids

A metalanguage is like a special language we use to talk about another language. Think of it as a language that describes or explains another language. The language being talked about is called the object language.

For example, when you learn about English grammar, you use words like "noun," "verb," or "sentence." These words are part of a metalanguage because they describe parts of the English language itself. When we write about a word, we often put it in italics or "quotation marks" to show we are talking about the word itself, not using it in a regular sentence.

Different Kinds of Metalanguages

There are a few main types of metalanguages: embedded, ordered, and nested (or hierarchical).

Embedded Metalanguage

An embedded metalanguage is a language that is naturally built into the object language. It's like having the tools to describe a language right inside that language.

For example, in English, words like noun, verb, or word are part of an embedded metalanguage. We use them to talk about how English works.

Ordered Metalanguage

An ordered metalanguage is when you have one metalanguage to talk about an object language, and then another metalanguage to talk about the first metalanguage, and so on. It's like building layers of language on top of each other.

Nested Metalanguage

A nested (or hierarchical) metalanguage is similar to an ordered metalanguage because each level is more abstract. However, in a nested metalanguage, each level includes the one below it.

A good example is the Linnean system used to classify living things in biology. Each level, like "family" or "order," includes the levels below it, like "genus" and "species." The language used to talk about orders also applies to genera, and so on, all the way up to kingdoms.

Metalanguages in Everyday Language

Our everyday languages combine both nested and ordered metalanguages. In natural language, we can always create new ways to talk about language itself.

For example, the rules of a language (its grammar) are described using a metalanguage. Then, the way we describe those rules can be explained by another metalanguage, and so on. Even though these layers exist, they are all part of the main language we speak.

Ultimately, all the special languages used in formal systems, like in math or logic, are explained and understood using our natural, everyday language.

Special Words and Ideas in Metalanguage

When we talk about formal languages, especially in logic, we use specific terms in the metalanguage.

Deductive Systems

A deductive system is a set of rules and starting points (called axioms) that allow us to figure out new true statements (theorems) within a formal system. It's like the instruction manual for how to prove things in that system.

Metavariables

A metavariable is a symbol or group of symbols used in a metalanguage to stand for a symbol or group of symbols in the object language.

For instance, if we say:

Let A and B be any formulas in a formal language.

Here, A and B are not actual symbols from that formal language. They are metavariables in English (our metalanguage) that represent any possible formula in that language.

Metatheories and Metatheorems

A metatheory is a theory that talks about another theory. So, it's a theory about a theory! Statements made in the metatheory about the original theory are called metatheorems. A metatheorem is a true statement about a formal system, but it's proven using the metalanguage, not within the formal system itself. It can discuss ideas that are only in the metalanguage.

Metalanguages in Computer Programming

Computers follow programs, which are sets of instructions written in special formal languages. When people create or work with programming languages, they often use metalanguages. This process is called metaprogramming.

One of the first metalanguages used in computing was Backus–Naur form, created in the 1960s. Today, programming languages like ML, Lisp, m4, and Yacc are often used for metaprogramming.

See also