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The Normal-Form Decision Method in the Combined Calculus

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Abstract

The original decision criterion and method of the combined calculus, presented by D. Hilbert and W. Ackermann, and applied by later logicians, are illuminating, but also go seriously awry and lead the universality and preciseness of the combined calculus to be damaged. The main error is that they confuse the two levels of the combined calculus in the course of calculating. This paper aims to resolve the problem through dividing the levels of the combined calculus, introducing a mixed operation mode, and therefore finding a normal-form decision method approximated by that in sentential logic. Finally, this paper provides a more precise proof of Hauber’s theorem by using the normal-form decision method of the combined calculus.

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References

  • He Y, Peng Y (1983) Introduction to logic. East China Normal University Press, Shanghai

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  • Hilbert D, Ackermann W (1949) Grundzüge der theoretischen Logik, 3rd German edn. Springer, Berlin

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  • Hilbert D, Ackermann W (1950) Principles of mathematical logic. The English version translated from German (2nd edition) by L. Hammond, G. Leckie, F. Steinhardt, edited with notes by R. E. Luce. Rutegers University

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  • Ma L (2003) On the combined calculus. The Science Press, Beijing

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Acknowledgements

This article is dedicated to the memory of the late Chinese logician Muyong Yu, who is the first to discover George Klaus and other logician’s mistakes in their operation of the combined calculus, and then doubt the theory of Hilbert and Ackermann. This article attempts to improve Hilbert and Ackermann’s work, and is fully affirmed and encouraged by professors Muyong Yu, Yuxin Zhen and Jianjun Zhang. This work was supported by the Chinese National Social Science Fund under Grant 17BZX004. The author thinks the anonymous reviewers on an early version of this article, and thanks my graduate students for helping me searching first-hand materials.

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  1. Huaqiao University, Xiamen, China

    Lei Ma

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  1. Lei Ma
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Correspondence to Lei Ma.

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Ma, L. The Normal-Form Decision Method in the Combined Calculus. Axiomathes 28, 461–489 (2018). https://doi.org/10.1007/s10516-018-9376-4

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  • DOI: https://doi.org/10.1007/s10516-018-9376-4

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