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Analytic natural deduction

Published online by Cambridge University Press:  12 March 2014

Raymond M. Smullyan
Raymond M. Smullyan*
Affiliation:
Belfer Graduate School of Science, New York, New York
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Extract

We consider some natural deduction systems for quantification theory whose only quantificational rules involve elimination of quantifiers. By imposing certain restrictions on the rules, we obtain a system which we term Analytic Natural Deduction; it has the property that the only formulas used in the proof of a given formula X are either subformulas of X, or negations of subformulas of X. By imposing further restrictions, we obtain a canonical procedure which is bound to terminate, if the formula being tested is valid. The procedure (ultimately in the spirit of Herbrand [1]) can be thought of as a partial linearization of the method of semantical tableaux [2], [3]. A further linearization leads to a variant of Gentzen's system which we shall study in a sequel.

The completeness theorem for semantical tableaux rests essentially on König's lemma on infinite graphs [4]. Our completeness theorem for natural deduction uses as a counterpart to König's lemma, a lemma on infinite “nest structures”, as they are to be defined. These structures can be looked at as the underlying combinatorial basis of a wide variety of natural deduction systems.

In § 1 we study these nest structures in complete abstraction from quantification theory; the results of this section are of a purely combinatorial nature. The applications to quantification theory are given in § 2.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 30 , Issue 2 , June 1965 , pp. 123 - 139
Copyright
Copyright © Association for Symbolic Logic 1965

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References

REFERENCES

[1]Herbrand, J., Recherches sur la Théorie de la démonstration, Warsaw 1930.Google Scholar
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