Analytic natural deduction

Abstract

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.