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# Combinatory Logic

A fundamental system of logic based on the concept of a generalized function whose argument is also a function (Schönfinkel 1924). This mathematical discipline was subsequently termed combinatory logic by Curry and "-conversion" or "lambda calculus" by Church. The system of combinatory logic is extremely fundamental, in that there are a relatively small finite numbers of atoms, axioms, and elementary rules. Despite the fact that the system contains no formal variables, it can be used for doing anything that can be done with variables in more usual systems (Curry 1977, p. 119).

Combinator, Lambda Calculus

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## References

Curry, H. B. "Combinatory Logic." §3D5 in Foundations of Mathematical Logic. New York: Dover, pp. 117-119, 1977.Curry, H. and Feys, R. Combinatory Logic, Vol. 1. Amsterdam, Netherlands: North-Holland, 1958.Hindley, J. R.; Lercher, B.; Seldin, J. P. Introduction to Combinatory Logic. London: Cambridge University Press, 1972.Hindley, J. R. and Seldin, J. P. Introduction to Combinators and lambda-Calculus. Cambridge, England: Cambridge University Press, 1986.Holmes, M. R. "Systems of Combinatory Logic Related to Quine's 'New Foundations.' " Annals Pure Appl. Logic 53, 103-133, 1991.Seldin, J. P. and Hindley, J. R. (Eds.). To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. New York: Academic Press, 1980.Schönfinkel, M. "Über die Bausteine der mathematischen Logik." Math. Ann. 92, 305-316, 1924.Schönfinkel, M. "Sur les éléments de construction de la logique mathématique." Math. Inform. Sci. Humaines, No. 112, 5-26 and 59, 1990. [French translation with commentary.]

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Combinatory Logic

## Cite this as:

Weisstein, Eric W. "Combinatory Logic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CombinatoryLogic.html