经典命题逻辑的切消注释

IF 0.3 4区 数学 Q1 Arts and Humanities Archive for Mathematical Logic Pub Date : 2021-11-26 DOI:10.1007/s00153-021-00800-8
Gabriele Pulcini
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引用次数: 3

摘要

在Schwichtenberg(《逻辑与数学基础研究》,第90卷,爱思唯尔出版社,第867–8951977页)中,Schwichtenberg对Tait的技术进行了微调(Tait在《不定语言的语法和语义》中,Springer,第204–236页,1968),以便为一阶经典逻辑提供Gentzen原始割消去程序的简化版本(Gallier在《计算机科学逻辑:自动定理证明的基础》中,Courier Dover Publications,伦敦,2015)。在这篇注释中,我们表明,仅限于经典命题逻辑的情况,Tait–Schwichtenberg算法允许进一步简化。这里提供的程序是在Kleene的序系统G4上实现的(Kleene在数理逻辑中,Wiley,纽约,1967;Smullyan在一阶逻辑中,Courier公司,伦敦,1995)。G4的逻辑规则的具体公式允许我们仅根据其结束序列的逻辑复杂性来提供无割证明的高度的边界。
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A note on cut-elimination for classical propositional logic

In Schwichtenberg (Studies in logic and the foundations of mathematics, vol 90, Elsevier, pp 867–895, 1977), Schwichtenberg fine-tuned Tait’s technique (Tait in The syntax and semantics of infinitary languages, Springer, pp 204–236, 1968) so as to provide a simplified version of Gentzen’s original cut-elimination procedure for first-order classical logic (Gallier in Logic for computer science: foundations of automatic theorem proving, Courier Dover Publications, London, 2015). In this note we show that, limited to the case of classical propositional logic, the Tait–Schwichtenberg algorithm allows for a further simplification. The procedure offered here is implemented on Kleene’s sequent system G4 (Kleene in Mathematical logic, Wiley, New York, 1967; Smullyan in First-order logic, Courier corporation, London, 1995). The specific formulation of the logical rules for G4 allows us to provide bounds on the height of cut-free proofs just in terms of the logical complexity of their end-sequent.

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来源期刊
Archive for Mathematical Logic
Archive for Mathematical Logic MATHEMATICS-LOGIC
CiteScore
0.80
自引率
0.00%
发文量
45
审稿时长
6-12 weeks
期刊介绍: The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.
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