Metavariable

For the term as used in computer science and programming, see Metasyntactic variable.

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]: 220

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

Notes

1. Hunter 1973, p. 13.
2. Shoenfield 2001, p. 7.
3. Corcoran 2006, p. 220.
4. Tennent 2002, pp. 36–37, 210.
5. Masahiko Sato, Takafumi Sakurai, Yukiyoshi Kameyama, and Atsushi Igarashi. "Calculi of Meta-variables^{[permanent dead link]}" 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. ISBN 3-540-40801-0. pp. 484–497

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 (26 June 1973). Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California Press. ISBN 9780520023567.
• 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.