Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-23T12:01:36.841Z Has data issue: false hasContentIssue false
  • English
  • Français

The system CΔ of combinatory logic1

Published online by Cambridge University Press:  12 March 2014

Frederic B. Fitch
Frederic B. Fitch*
Affiliation:
Yale University
Get access Rights & Permissions [Opens in a new window]

Extract

The system of combinatory logic to be presented in this paper will be called CΔ. It may be compared to various systems of combinatory logic constructed by Schonfinkel [11], Curry [1], Rosser [10], and Curry and Feys [2], and to the author's systems K′ [3] and S [5], and to extensional modifications of K′ [6], [7], [8], [9]. The present system falls within this latter category of extensional or semi-extensional systems, but it is more perspicuous than the others, and the proof of its consistency is more direct. It contains the theory of combinators in full strength. It also contains operators for disjunction, conjunction, negation, existence (that is, non-emptiness), and universality. A considerable part of classical mathematical analysis can be shown to be derivable in CΔ, just as in K′ [4], but with the added advantage of the availability of a limited extensionality principle.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 28 , Issue 1 , March 1963 , pp. 87 - 97
Copyright
Copyright © Association for Symbolic Logic 1964

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

This paper reports one aspect of a more general research program supported by the Office of Naval Research, Group Psychology Branch, Contract No. SAR/Nonr-609(16). Permission is granted for reproduction, translation, publication, and disposal in whole or in part by or for the U.S. Government. The Association for Symbolic Logic retains copyright authority for all other uses. The writer wishes to express his indebtedness to Professor Haskell B. Curry for some helpful criticisms and suggestions.

References

[1]Curry, H. B., Grundlagen der kombinatorischen Logik, American journal of mathematics, vol. 52 (1930), pp. 509–536, 789–834.CrossRefGoogle Scholar
[2]Curry, H. B. and Feys, R., Combinatory logic, vol. 1, Amsterdam, 1958.Google Scholar
[3]Fitch, F. B., An extension of basic logic, this Journal, vol. 13 (1948), pp. 95–106.Google Scholar
[4]Fitch, F. B., A demonstrably consistent mathematics, this Journal, vol. 15 (1950), pp. 17–24, vol. 16 (1951), pp. 121–124.Google Scholar
[5]Fitch, F. B., Symbolic logic, an introduction, New York, 1952.Google Scholar
[6]Fitch, F. B., An extensional variety of extended basic logic, this Journal, vol. 23 (1958), pp. 13–21.Google Scholar
[7]Fitch, F. B., Representations of sequential circuits in combinatory logic, Philosophy of science, vol. 25 (1958), pp. 263–279.CrossRefGoogle Scholar
[8]Fitch, F. B., Quasi-constructive foundations for mathematics, Constructivity in mathematics (ed. by Heyting, A.), Amsterdam, 1959.Google Scholar
[9]Fitch, F. B., A system of combinatory logic, New Haven, 1960. (Technical Report No. 9, Prepared under Contract SAR/Nonr-609(16) for Office of Naval Research, Group Psychology Branch.)Google Scholar
[10]Rosser, J. B., A mathematical logic without variables, Annals of mathematics, vol. 36 (1935), pp. 127–150, Duke mathematical journal, vol. 1 (1935), pp. 328–355.CrossRefGoogle Scholar
[11]Schönfinkel, M., Über die Bausteine der mathematischen Logik, Mathematische Annalen, vol. 92 (1924), pp. 305–316.CrossRefGoogle Scholar