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New sets of postulates for combinatory logics

Published online by Cambridge University Press:  12 March 2014

Barkley Rosser*
Affiliation:
Cornell University

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
Copyright
Copyright © Association for Symbolic Logic 1942

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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
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