Metavariable

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]

Notes

^ Hunter 1973, p. 13.
^ Shoenfield 2001, p. 7.
^ Corcoran 2006, p. 220.
^ Tennent 2002, pp. 36–37, 210.
^ 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.

Mathematical logic

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