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Metavariable

Metavariable is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Metavariable rather than just read about it. In short: In logic, a metavariable (also metalinguistic variable or syntactical variable) is a symbol or symbol string which belongs to a metalanguage and stands for elements of some object language. For instance, in the sentence Let A and B be two sentences of a language ℒ the symbols A and B are part of the metalanguage in which the statement about the object language ℒ is formulated.

Key takeaways

  • Metavariable belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Metavariable to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Metavariable from memory before moving on to harder problems.

Reference excerpt

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

Let A and B be two sentences of a language ℒ the symbols A and B are part of the metalanguage in which the statement about the object language ℒ is formulated. John Corcoran considered this terminology unfortunate because it obscures the use of schemata and because such "variables" do not actually range over a domain. The convention is that a metavariable is to be uniformly substituted with the same instance in all its appearances in a given schema. This is in contrast with nonterminal symbols in formal grammars where the nonterminals on the right of a production can be substituted by different instances. Attempts to formalize the notion of metavariable result in some kind of type theory.

See also Explicit substitution

Notes

References Corcoran, J. (2006). "Schemata: the Concept of Schema in the History of Logic" (PDF). Bulletin of Symbolic Logic. 12 (2): 219–240. doi:10.2178/bsl/1146620060. S2CID 6909703. Hunter, Geoffrey (1996) [1971]. Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California Press (published 1973). ISBN 9780520023567. OCLC 36312727. (accessible to patrons with print disabilities) Shoenfield, Joseph R. (2001) [1967]. Mathematical Logic (2nd ed.). A K Peters. ISBN 978-1-56881-135-2. Tennent, R. D. (2002). Specifying Software: A Hands-On Introduction. Cambridge University Press. ISBN 978-0-521-00401-5.

Worked examples

Example 1 — a first encounter with Metavariable

Start with the simplest possible case. Write down what Metavariable claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Metavariable before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Metavariable ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Metavariable

In research
Metavariable appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Metavariable in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Metavariable is common in secondary-school and first-year university syllabi. It links to neighbouring topics Logic symbols, Metalogic, Syntax (logic), so understanding it makes those chapters shorter.
In everyday life
Look for Metavariable outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Metavariable in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Metavariable means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Metavariable out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Metavariable in simple terms?

In logic, a metavariable (also metalinguistic variable or syntactical variable) is a symbol or symbol string which belongs to a metalanguage and stands for elements of some object language. For instance, in the sentence Let A and B be two sentences of a language ℒ the symbols A and B are part of th…

Why does Metavariable matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Metavariable?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Metavariable.

Tags

  • Logic symbols
  • Metalogic
  • Syntax (logic)
  • Variables (mathematics)

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