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Normality, Non-contamination and Logical Depth in Classical Natural Deduction
Studia Logica ( IF 0.7 ) Pub Date : 2019-02-18 , DOI: 10.1007/s11225-019-09847-4
Marcello D’Agostino , Dov Gabbay , Sanjay Modgil

In this paper we provide a detailed proof-theoretical analysis of a natural deduction system for classical propositional logic that (i) represents classical proofs in a more natural way than standard Gentzen-style natural deduction, (ii) admits of a simple normalization procedure such that normal proofs enjoy the Weak Subformula Property, (iii) provides the means to prove a Non-contamination Property of normal proofs that is not satisfied by normal proofs in the Gentzen tradition and is useful for applications, especially in formal argumentation, (iv) naturally leads to defining a notion of depth of a proof, to the effect that, for every fixed natural k , normal k -depth deducibility is a tractable problem and converges to classical deducibility as k tends to infinity.

中文翻译:

经典自然演绎中的正态性、非污染性和逻辑深度

在本文中,我们对经典命题逻辑的自然演绎系统进行了详细的证明理论分析,其中 (i) 以比标准 Gentzen 式自然演绎更自然的方式表示经典证明,(ii) 承认简单的归一化程序,例如正则证明具有弱子公式性质,(iii) 提供了证明正则证明的非污染性质的方法,该性质在 Gentzen 传统中被正则证明所不满足,并且对应用非常有用,尤其是在形式论证中,(iv)自然导致定义证明深度的概念,其结果是,对于每个固定的自然 k ,正常的 k 深度推导性是一个易于处理的问题,并且随着 k 趋于无穷大而收敛到经典推导性。
更新日期:2019-02-18
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