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A set of axioms for logic1

Published online by Cambridge University Press:  12 March 2014

Theodore Hailperin
Theodore Hailperin*
Affiliation:
Cornell University
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Extract

One of the preeminent problems confronting logicians is that of constructing a system of logic which will be adequate for mathematics. By a system's being adequate for mathematics, we mean that all mathematical theorems in general use can be deduced within the system. Several distinct logical systems, all having this end in view, have been proposed. Among these perhaps the best known are the systems referred to as “Principia Mathematica” and “set theory.” In both of these systems (we refer to the revised and simplified versions) there is a nucleus of propositions which can be derived by using only the axioms and rules of the restricted predicate calculus. However, if anything like adequacy for mathematics is to be expected, additional primitives and axioms must be added to the restricted predicate calculus. It is in their treatment of the additional primitive ε, denoting class or set membership, that the above-mentioned systems differ.

In addition to these two, a third and a stronger system has been proposed by W. V. Quine in his paper New foundations for mathematical logic. It is with this system of Quine's that our work is concerned and of which we now give a brief description.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 9 , Issue 1 , 29 March 1944 , pp. 1 - 19
Copyright
Copyright © Association for Symbolic Logic 1944

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Footnotes

1

The material of this paper is contained in the author's doctoral thesis presented to the Graduate Faculty of Cornell University, May 1943. The author wishes to express his thanks to Professor J. B. Rosser for helpful guidance in the development of this work.

References

2 The American mathematical monthly, vol. 44 (1937), pp. 70–80. This is not to be confused with the logical system presented in Quine, , Mathematical logic (New York, 1940)Google Scholar which Rosser and, independently, Lyndon have shown to be inconsistent. (See this Journal, VII 1). On the other hand, despite a critical examination by Rosser, no inconsistencies have been found in New foundations. (See this Journal, IV 15, as well as the cited VII 1).

3 Cf. J. B. Rosser,, in this Journal, IV 17.

4 The ideas for this proof were taken from an unpublished paper of C. D. Firestone on a similar treatment of descriptions.

5 The term is used here merely to render some aid to the intuitive understanding of the axioms.

6 From here on we shall omit references to 4.5.

7 Compare Rosser, this Journal, IV 17.