Abstract

The paper is inspired by and explicitly presupposes the readers’ knowledge of the Kürbis normalization procedure for the Milne tree-like natural deduction system _C_ for classical propositional logic. The novelty of _C_ is that for each conventional connective, it has only _general_ introduction and elimination rules, whose paradigm is the rule of proof by cases. The present paper deals with the Milne–Kürbis troublemaker—adding universal quantifier—caused by extending the normalization procedure to \(\mathbf {C^{\exists }_{\forall }} \), the first-order variant of _C_. For both quantifiers, \( \mathbf {C^{\exists }_{\forall }}\) has only general quantifier rules, whose paradigm is the indirect rule of existential elimination. We propose the classical first-order tree-like natural deduction system \( \mathbf {NC^{\exists }_{\forall }}\) such that it contains all the propositional and half the quantifier rules of \(\mathbf {C^{\exists }_{\forall }}\) and differs from it with respect to the other half of the quantifier rules and the notion of a deduction. We show that in the case of \(\mathbf {NC^{\exists }_{\forall }} \), whose specifics is based on the Quine treatment of _flagged_ variables in the linear-style natural deduction system as well as on the Aguilera-Baaz treatment of _characteristic_ and _side_ variables for sequent-style calculi, the troublemaker doesn’t occur. In order to handle another specifics of \(\mathbf {NC^{\exists }_{\forall }}\) —an auxiliary notion of a _finished_ deduction that makes deductions _nonhereditary_—we show soundness and completeness of \( \mathbf {NC^{\exists }_{\forall }} \). While emphasizing the former with the help of the Smirnov technique, we additionally indicate a fixable gap in the Quine soundness argument. At last, the philosophical significance of the main result of this paper is developed in more detail.