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Combinatory logic with discriminators

Published online by Cambridge University Press:  12 March 2014

John T. Kearns*
Affiliation:
State University of New York at Buffalo

Extract

In this paper, I present a modified and extended version of combinatory logic. Schönfinkel originated the study of combinatory logic (in [2]), but its development is primarily due to H. B. Curry. In the present paper, I will make use of both the symbolism (with some modification) and the results of Curry, as found in [1].

What is novel about my version of combinatory logic is a kind of combinators which I call discriminators. These combinators discriminate between different symbols, and yield values which are determined by the symbols which are their arguments. If the normal combinators investigated by Curry are considered as functions taking (combinations of) symbols as arguments and yielding symbols as values, these combinators can rearrange and cancel their arguments or introduce new symbols.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1970

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References

[1] Curry, Haskell B. and Feys, Robert, Combinatory logic, Vol. 1, North-Holland, Amsterdam, 1958.Google Scholar
[2] Schönfinkel, M., Über die Bausteine der mathematischen Logik, Mathematische Annalen, vol. 92 (1924), pp. 305316; English transl., From Frege to Gödel, ed. by Jean van Heijenoort. Google Scholar
[3] Turing, A. M., Computability and λ-definability, this Journal , vol. 2 (1937), pp. 153163.Google Scholar
[4] Turing, A. M., On computable numbers, with an application to the Entscheidungs problem, Proceedings of the London Mathematical Society, vol. 42 (1936), Series 2.Google Scholar