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An extensional variety of extended basic logic

Published online by Cambridge University Press:  12 March 2014

Frederic B. Fitch
Frederic B. Fitch*
Affiliation:
Yale University
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Extract

The system K′ of “extended basic logic” lacks a principle of extensionality. In this paper a system KE′ will be presented which is like K′ in many respects but which does possess a fairly strong principle of extensionality by way of rule 6.37 below. It will be shown that KE′ is free from contradiction. KE′ is especially well suited for formalizing the theory of numbers presented in my paper, On natural numbers, integers, and rationals. The methods used there can be applied even more directly here because of the freedom of KE′ from type restrictions, but the details of such a derivation of a portion of mathematics will not be presented in this paper. It is evident, moreover, that KE′ contains at least as much of mathematical analysis as does K′, and perhaps considerably more. The method of carrying out proofs in KE′ is closely similar to that used in my book Symbolic logic, and could be expressed in similar notation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 23 , Issue 1 , March 1958 , pp. 13 - 21
Copyright
Copyright © Association for Symbolic Logic 1958

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References

1 See the writer's papers, An extension of basic logic, this Journal, vol. 13 (1948), pp. 95–106Google Scholar, and A definition of negation in extended basic logic, this Journal, vol. 19 (1954), pp. 29–36Google Scholar. The former paper will be referred to below as EBL, and the latter as DN.

2 Ibid., vol. 14 (1949), pp. 81–84.Google Scholar

3 For the amount of mathematical analysis contained in extended basic logic, see the writer's paper, A demonstrably consistent mathematics, this Journal, vol. 15 (1950), pp. 17–24, vol. 16 (1951), pp. 121–124.Google Scholar

4 Ronald Press, New York, 1952.