甘岑逻辑演绎思想研究

发布时间：2018-11-21 12:54

【摘要】：《逻辑演绎研究》是甘岑提交哥廷根大学的就职论文,文中甘岑提出了两种逻辑演算系统,即“自然演绎”和“矢列式演算”系统。从自然演绎开始甘岑尝试构造一个接近数学实际推理的逻辑演算,而后针对自然演绎系统的不足进行了转化和改进,构造了矢列式演算系统,并证明了该逻辑演算系统的合理性。这篇论文呈现了甘岑的逻辑演绎思想,对逻辑演算进行了更深入的研究,对数理逻辑和证明论的发展具有深刻的影响。本文的研究目的在于通过呈现甘岑的逻辑演绎理论,并对他构造的逻辑演算进行比较研究,从而阐明甘岑逻辑演绎思想在现代逻辑史上的地位、影响和发展。文章第一部分介绍了数学中的公理系统和形式系统,进而阐述了弗雷格和希尔伯特借鉴数学的形式方法构造的逻辑演算形式系统.即“概念文字”和“希尔伯特式公理系统”,这一部分构成了甘岑逻辑演绎思想的理论基础。文章第二部分和第三部分是本文的重点和难点,系统地介绍了甘岑逻辑演绎思想的主要内容。第二部分阐述了甘岑的自然演绎系统,并对自然演绎系统与弗雷格-希尔伯特式公理系统进行比较分析,论述了甘岑自然演绎系统的优越性与存在的不足。第三部分阐述了甘岑的矢列式演算系统,介绍了证明“切割消除定理”的主要思想,并对甘岑的矢列式演算系统和其自然演绎系统进行比较,论述了矢列式演算在自然演绎基础上的改进,从而论证了甘岑构造的逻辑演算系统的合理性。第四部分是对甘岑逻辑演绎思想的总结与展望,总结了甘岑逻辑演绎思想的形成,并对其思想的影响和发展进行了论述。在公理系统的基础上,甘岑对逻辑演算进行了发展,构造了只包含规则,并且更加接近数学实际推理的自然演绎,这是甘岑构造逻辑演算的初步尝试,矢列式演算则是甘岑为了证明这样的逻辑演算的合理性所做出的改进。并且在“切割消除定理”的证明之后,甘岑论证了该定理的应用,证明了矢列式演算与自然演绎的等价性。此外,继甘岑之后国外众多逻辑学家和数学家们对甘岑的逻辑演绎思想进行了研究与发展。因此,甘岑的逻辑演绎思想在逻辑史上具有里程碑式的意义,并且至今仍具有重大的理论价值和研究意义。
[Abstract]:The study of logical deduction is an inaugural paper submitted by Ganzen to Gottingen University. In this paper, Ganzen puts forward two kinds of logic calculus systems, namely "natural deduction" and "vector calculus" system. Beginning with natural deduction, Ganzen tries to construct a logic calculus close to the actual mathematical reasoning, then transforms and improves the deficiency of natural deduction system, and constructs a vector calculus system. The rationality of the logic calculus system is proved. This paper presents Ganzen's logical deductive thought, and makes a deeper study of logic calculus, which has a profound influence on the development of mathematical logic and proof theory. The purpose of this paper is to illustrate the position, influence and development of Ganzen's logic deduction in the history of modern logic by presenting Ganzen's theory of logic deduction and comparing the logic calculus constructed by him. The first part of this paper introduces the axiomatic system and the formal system in mathematics, and then expounds the logical calculus formal system constructed by Frege and Hilbert using the mathematical formal method for reference. That is, concept writing and Hilbert-style axiom system, which constitute the theoretical basis of Ganzen's logical deduction thought. The second and third parts are the emphases and difficulties of this paper, and the main contents of Ganzen's logical deduction thought are introduced systematically. In the second part, the author expounds the natural deduction system of Ganzen, and compares the natural deduction system with the Freguer-Hilbert axiom system, and discusses the advantages and shortcomings of the natural deduction system of Ganzen. In the third part, the author expounds the vector calculus system of Ganzen, introduces the main idea of proving "cutting elimination theorem", and compares the vector calculus system of Ganzen with its natural deduction system. This paper discusses the improvement of vector calculus on the basis of natural deduction, thus proves the rationality of the logic calculus system constructed by Ganzen. The fourth part is the summary and prospect of Ganzen's logical deductive thought, summarizes the formation of Ganzen's logical deductive thought, and discusses its influence and development. On the basis of axiom system, Ganzen develops logical calculus and constructs a natural deduction which contains only rules and is closer to the actual reasoning of mathematics. This is a preliminary attempt to construct logic calculus by Ganzen. Vector calculus is an improvement made by Ganzen in order to prove the rationality of such logical calculus. After proving the "cutting elimination theorem", Ganzen proved the application of the theorem and proved the equivalence between vector calculus and natural deduction. In addition, after Ganzen, many foreign logicians and mathematicians have studied and developed Ganzen's logic deduction thought. Therefore, Ganzen's logic deductive thought has milestone significance in the history of logic, and still has great theoretical value and research significance.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：B812

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