Home
Hostname: page-component-768dbb666b-l8xdn Total loading time: 0.243 Render date: 2023-02-05T11:24:09.012Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

# New sets of postulates for combinatory logics

Published online by Cambridge University Press:  12 March 2014

## Extract

Sets of postulates for at least three different combinatory logics have been given ([2], [3], [4], [5], [6], [7]). The logics and their sets of postulates are quite similar. In all cases it is the proof of the completeness of the set of postulates which causes the difficulty and makes the main papers so long ([3], [7]). It is the purpose of this paper to present a method of writing down sets of postulates for combinatory logics for which the proof of completeness will be fairly simple. We assume acquaintance with certain descriptive portions of [1], namely sections 1–6, 12, 13, 15. Otherwise the present paper is self-contained.

We first solve our problem for the particular logic studied by Rosser ([7]) and then indicate how the method would apply to other logics.

We start with a set, s, of primitive terms. We define s-combination as in [1], p. 43. The precise contents of s need not concern us. It suffices that there be s-combinations I, J, and Q, with no variables in them, and having properties as described below. I and J are to have the same meaning as in [1], p. 43, namely IA is the same as A and JABCD is the same as AB(ADC). QAB is to have the meaning that A and B, considered as functions, are the same.

Type
Research Article
Information
The Journal of Symbolic Logic , 24 March 1942 , pp. 18 - 27

## 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.)

## References

[1]Church, Alonzo, The calculi of lambda-conversion, Annals of Mathematics studies, no. 6, Princeton University Press, 1941.Google Scholar
[2]Curry, H. B., An analysis of logical substitution, American journal of mathematics, vol. 51 (1929), pp. 363384.CrossRefGoogle Scholar
[3]Curry, H. B., Grundlagen der kombinatorischen Logik, American journal of mathematics, vol. 52 (1930), pp. 509536, 789–834.CrossRefGoogle Scholar
[4]Curry, H. B., Some additions to the theory of combinators, American journal of mathematics, vol. 54 (1932), pp. 551558.CrossRefGoogle Scholar
[5]Curry, H. B., A revision of the fundamental rules of combinatory logic, this JOURNAL, vol. 6 (1941), pp. 4153.Google Scholar
[6]Curry, H. B., Consistency and completeness of the theory of combinators, this JOURNAL, vol. 6 (1941), pp. 5461.Google Scholar
[7]Rosser, J. B., A mathematical logic without variables, Annals of mathematics, ser. 2 vol. 36 (1935), pp. 127150, and Duke mathematical-journal, vol. 1 (1935), pp. 328–355.CrossRefGoogle Scholar
8
Cited by

# Save article to Kindle

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

New sets of postulates for combinatory logics
Available formats
×

# Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

New sets of postulates for combinatory logics
Available formats
×

# Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

New sets of postulates for combinatory logics
Available formats
×
×