Metavariable Explained

In logic, a metavariable (also metalinguistic variable^[1] or syntactical variable)^[2] 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 considers this terminology unfortunate because it obscures the use of schemata and because such "variables" do not actually range over a domain.^[3]

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.^[4]

Attempts to formalize the notion of metavariable result in some kind of type theory.^[5]

See also

• Explicit substitution

References

• Corcoran . J. . John Corcoran (logician) . 2006 . Schemata: the Concept of Schema in the History of Logic . Bulletin of Symbolic Logic . 12 . 2 . 219–240 . 10.2178/bsl/1146620060 . 6909703 .
• Book: Hunter. Geoffrey. Geoffrey Hunter (logician). Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. 26 June 1973. University of California Press . 9780520023567.
• Book: Shoenfield . Joseph R. . Joseph R. Shoenfield . Mathematical Logic . 1967 . . 2nd . 978-1-56881-135-2 . 2001.
• Book: Tennent. R. D.. Specifying Software: A Hands-On Introduction. 2002. Cambridge University Press. 978-0-521-00401-5.

Notes and References

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5. Masahiko Sato, Takafumi Sakurai, Yukiyoshi Kameyama, and Atsushi Igarashi. "Calculi of Meta-variables" in Computer Science Logic. 17th International Workshop CSL 2003. 12th Annual Conference of the EACSL. 8th Kurt Gödel Colloquium, KGC 2003, Vienna, Austria, August 25-30, 2003. Proceedings, Springer Lecture Notes in Computer Science 2803. . pp. 484–497