Bulletin of the Section of Logic 51 (1):91-108 (2022)

In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper, and others proper. Improper inference rules are more complicated and are often harder to understand than the proper ones. In the present paper, we distinguish between proper and improper derivations by using sequent systems. Specifically, we introduce a sequent system \ for classical propositional logic with only structural rules, and prove that \ does not allow improper derivations in general. For instance, the sequent \ cannot be derived from the sequent \ in \. In order to prove the failure of improper derivations, we modify the usual notion of truth valuation, and using the modified valuation, we prove the completeness of \. We also consider whether an improper derivation can be described generally by using \.
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DOI 10.18778/0138-0680.2021.21
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References found in this work BETA

Reasoning with Arbitrary Objects.Kit Fine - 1985 - Oxford and New York: Blackwell.
Arbitrary Reference.Wylie Breckenridge & Ofra Magidor - 2012 - Philosophical Studies 158 (3):377-400.
A Survey of Nonstandard Sequent Calculi.Andrzej Indrzejczak - 2014 - Studia Logica 102 (6):1295-1322.
Formalna Teoria Wartości Logicznych I.Roman Suszko - 1957 - Studia Logica 6 (1):145-237.

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Citations of this work BETA

A Classification of Improper Inference Rules.Katsumi Sasaki - 2022 - Bulletin of the Section of Logic 51 (2):243-266.

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