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Analytic cut

Published online by Cambridge University Press:  12 March 2014

Raymond M. Smullyan
Raymond M. Smullyan*
Affiliation:
Belfer Graduate School of Science, Yeshtva University
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Extract

The real importance of cut-free proofs is not the elimination of cuts per se, but rather that such proofs obey the subformula principle. In this paper we accomplish this latter objective in a different manner.

In the usual formulations of Gentzen systems, there is only one axiom scheme; all the other postulates are inference rules. By contrast, we consider here some Gentzen type axiom systems for propositional logic and Quantification Theory in which there is only one inference rule; all the other postulates are axiom schemes. This admits of an unusually elegant axiomatization of logic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 33 , Issue 4 , January 1969 , pp. 560 - 564
Copyright
Copyright © Association for Symbolic Logic 1969

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References

[1]Gentzen, G., Investigations into logical deduction, American philosophical quarterly, vol. 1 (1964).Google Scholar
[2]Kleene, S. C., Introduction to metamathematics. Van Nostrana, Princeton, N.J., 1950.Google Scholar
[3]Quine, W. V., Methods of logic, Henry Holt and Company, Inc., New York, 1959.Google Scholar
[4]Smullyan, R. M., Uniform Gentzen systems, this Journal this issue, pp. 549–559.Google Scholar