A completeness-proof method for extensions of the implicational fragment of the propositional calculus
A completeness-proof method for extensions of the implicational fragment of the propositional calculus
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命题演算蕴涵片段扩展的完备性证明方法
DOI：
10.1305/ndjfl/1093883174
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发表时间：
1980
期刊：
影响因子：
--
通讯作者：
D. Batens
中科院分区：
文献类型：
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作者：
D. Batens
经典命题演算（PC）是强完备的传统证明（即，如果. t= A，则a h A）是基于公式的最大相容集的概念，并且因此基于强的某些性质（即，PC-）否定。本文提出了一种完备性证明方法，它不涉及极大相容集，而只涉及下列集合：（i）非平凡的（不是所有公式都是成员），（ii）演绎闭的（所有句法结果都是成员），（iii）蕴涵饱和的（对于所有B，如果A不是成员，则A D B是成员）。如果将这种证明方法应用于包含强否定的逻辑，则集合对于强否定是一致的。我将首先把证明方法应用于PC的蕴涵片段的一个特定扩展，然后证明它也适用于蕴涵片段本身和作为蕴涵片段扩展的大量逻辑。如果这样一个逻辑是由语义学来表征的，那么公理系统的表达就很简单（从证明方法的角度来看），反之亦然。完备性证明方法特别适用于基于实质蕴涵的次协调逻辑（见[1 M6]）。[1]次协调逻辑是这样一种逻辑，根据这种逻辑，至少有一些不一致的理论是非平凡的（语言中的一些句子不能从 * 的公理推导出来）。由于他们的意见，本文件的编排方式有了实质性的改进。
The traditional proof that the classical propositional calculus (PC) is strongly complete (i.e., if a. t= A, then a h A) is based on the notion of a maximal consistent set of formulas, and hence on certain properties of strong (i.e., PC-)negation. In this paper* I present a completeness-proof method which does not refer to maximal consistent sets, but only to sets which are: (i) nontrivial (not all formulas are members), (ii) deductively closed (all syntactical consequences are members), and (iii) implication saturated (for all B, A D B is a member if A is not a member). If this proof method is applied to logics that contain strong negation, the sets turn out to be consistent with respect to strong negation. I shall first apply the proof method to a specific extension of the implicational fragment of PC, and next show that it also applies to the implicational fragment itself and to a large number of logics that are extensions of the implicational fragment. If such a logic is characterized by a semantics, the articulation of an axiomatic system is straightforward (in view of the proof method) and vice versa. The completeness-proof method is especially fit for paraconsistent logics that are based on material implication (see [1M6]). 1 Paraconsistent logics are logics according to which at least some inconsistent theories are nontrivial (some sentences of the language are not derivable from the axioms of the *I am indebted to the referee and especially to the editor. As a consequence of their remarks, the presentation of this paper has been essentially improved.