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Simplified foundations for mathematical logic1

Published online by Cambridge University Press:  12 March 2014

Robert L. Stanley
Robert L. Stanley*
Affiliation:
The University of British Columbia, Vancouver
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Extract

A system SF, closely related to NF, is outlined here. SF has several novel points of simplicity and interest, (a) It uses only one basic notion, from which all the other concepts of logic and mathematics may be built definitionally. Three-notion systems are common, but Quine's two-notion IA has for some time represented the extreme in conceptual economy, (b) The theorems of SF are generated under just three rules of analysis, which unify into a single postulational principle, (c) SF is built solely in terms of what is commonly, known as the “natural deduction” method, under which each theorem is attacked primarily as it stands, by means of a very small body of rules, rather than less directly, through a very large, potentially infinite backlog of theorems. Although natural deduction is by no means new as a method, its exclusive applications have previously been relatively limited, not even reaching principles of identity, much less set theory, relations, or mathematics proper, (d) SF is at least as strong as NF, yielding all of its theorems, which are expressed here in forms analogous to those of the metatheorems in ML. If NF is consistent, so is SF. The main points in the relative consistency proof are set forth below in section seven.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 20 , Issue 2 , June 1955 , pp. 123 - 139
Copyright
Copyright © Association for Symbolic Logic 1955

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Footnotes

1

This system and the three related ones mentioned in footnote 19 below were presented at a meeting of the Association for Symbolic Logic, December 29, 1952. I am deeply indebted to Professor Quine for many helpful suggestions concerning my initial development of this general method, and to the referee for much painstaking constructive criticism of the present writing.

References

BIBLIOGRAPHY

Quine, IA. W. V., Logic based on inclusion and abstraction, this Journal, vol. 2 (1937), pp. 145–152.Google Scholar
Quine, LQ. W. V., On the logic of quantification, this Journal, vol. 10 (1945), pp. 1–12.Google Scholar
Quine, ML. W. V., Mathematical logic, Second printing, Harvard University Press, Cambridge, Mass., 1947.Google Scholar
Quine, NF. W. V., New foundations for mathematical logic, American mathematical monthly, vol. 44 (1937), pp. 70–80.Google Scholar