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A simplification of combinatory logic1

Published online by Cambridge University Press:  12 March 2014

Nicolas D. Goodman*
Affiliation:
State University Of New York, Buffalo, New York 14226

Extract

The usual formulations of combinatory logic, as in Church [1] or Curry and Feys [2], have no straightforward semantics. This lack complicates the proofs of basic metasystematic results, prevents the perspicuous formalization of combinatory logic as a deductive theory, and makes the combinators unnecessarily difficult to apply to recursion theory or to the foundations of mathematics. In the present paper we shall give a new formulation of combinatory logic, thought of as an abstract theory of rules. At each stage in the development we shall give a precise semantic interpretation for our axioms. We begin with an abstract treatment of the syntax and semantics of a theory of rules. We then specialize to the theory of the combinators and prove a completeness theorem connecting our axioms with our semantics. On this basis we treat functional abstraction, truth-functions, and the representation of partial recursive functions. In the spirit of Church [1, pp. 62–71], we conclude by extending our methods to the treatment of intensional identity.

The theories developed here are part of a simplification of the theory of constructions described in [3]. Their semantics are motivated by the intended interpretation of the theory of constructions. The present paper is selfcontained and can be read without any familiarity with the theory of constructions, although the motivation of several of the notions considered here is given in [3]. In a future publication we will use the results of the present paper as the basis for a detailed study of a simplified form of the theory in [3].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1972

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Footnotes

1

This work was supported in part by NSF Grant GP-13019.

References

REFERENCES

[1]Church, A., The calculi of lambda-conversion, Annals of Mathematics Studies, no. 6, Princeton University Press, Princeton, N.J., 1941.Google Scholar
[2]Curry, H. B. and Feys, R., Combinatory logic, vol. 1, North-Holland, Amsterdam, 1958.Google Scholar
[3]Goodman, N. D., A theory of constructions equivalent to arithmetic, Intuitionism and proof theory (Kino, A., Myhill, J. and Vesley, R. E., Editors), North-Holland, Amsterdam, 1970, pp. 101120.Google Scholar
[4]Kleene, S. C., Introduction to metamathematics, Van Nostrand, New York, 1952.Google Scholar
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