Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory

Xiaojun Zhang

doi:doi:10.11648/j.acm.20180703.23

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Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory

Xiaojun Zhang

Published in Applied and Computational Mathematics (Volume 7, Issue 3)

Received: 9 August 2018 Accepted: Published: 13 August 2018

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Abstract

Syllogistic reasoning is important due to the prominence of syllogistic arguments in human reasoning, and also to the role they have played in theory of reasoning from Aristotle onwards. Aristotelian syllogistic logic is a formal study of the meaning of four Aristotelian quantifiers and of their properties. This paper focuses on logical system based on syllogistic reasoning. It firstly formalized the 24 valid Aristotle’s syllogisms, and then has proven that the other 22 valid Aristotle’s syllogisms can be derived from the syllogisms ‘Barbara’ AAA-1 and ‘Celarent’ EAE-1 by means of generalized quantifier theory and set theory, so the paper has completed the axiomatization of Aristotelian syllogistic Logic. This axiomatization needs to make full use of symmetry and transformable relations between/among the monotonicity of the four Aristotelian quantifiers from the perspective of generalized quantifier theory. In fact, these innovative achievements and the method in this paper provide a simple and reasonable mathematical model for studying other generalized syllogisms. It is hoped that the present study will make contributions to the development of generalized quantifier theory, and to bringing about consequences to natural language information processing as well as knowledge representation and reasoning in computer science.

Published in	Applied and Computational Mathematics (Volume 7, Issue 3)
DOI	10.11648/j.acm.20180703.23
Page(s)	167-172
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Generalized Quantifier Theory, Aristotelian Syllogisms, Aristotelian Quantifiers, Axiomatization

References

[1]	N. Chater and M. Oaksford, the probability Heuristics model of syllogistic reasoning, Cognitive Psychology, Vol. 38, 1999, pp. 191-258.
[2]	G. Patzig, Aristotle's Theory of the Syllogism, J. Barnes (trans.), Dordrecht: D. Reidel, 1969.
[3]	L. S. Moss, Syllogistic logics with verbs, Journal of Logic and Computation, Vol. 20, No. 4, 2010, pp. 947-967.
[4]	P. Murinová P and V. Novák, A formal theory of generalized intermediate syllogisms, Fuzzy Sets and Systems, Vol. 186, 2012, pp. 47-80.
[5]	J. Endrullis, and L. S. Moss, Syllogistic logic with ‘Most’, in V. de Paiva et al. (eds. ), Logic, Language, Information, and Computation, 2015, pp. 124-139.
[6]	J. Łukasiewicz, Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, Clarendon Press, Oxford, 1951.
[7]	J. N. Martin, Aristotle’s natural deduction reconsidered, History and Philosophy of Logic, Vol. 18, No. 1 1997, pp. 1-15.
[8]	L. S. Moss, Completeness theorems for syllogistic fragments, in F. Hamm and S. Kepser (eds.), Logics for Linguistic Structures, Mouton de Gruyter, Berlin, 2008, pp. 143–173.
[9]	J. van Benthem, Questions about quantifiers, Journal of Symbol Logic, Vol. 49, No. 2, 1984, pp. 443-466.
[10]	D. Westerståhl, Aristotelian syllogisms and generalized quantifiers, Studia Logica, Vol. XLVII, No. 4, 1989, pp. 577-585.
[11]	Xiaojun Zhang, A Study of Generalized Quantifier Theory, Xiamen University Press, 2014. (in Chinese).
[12]	Xiaojun Zhang, A Study of Properties of Generalized Quantifiers, Ph.D. dissertation, Chinese Academy of Social Sciences, 2011. (in Chinese).
[13]	N. Ivanov, and D. Vakarelov, A system of relational syllogistic incorporating full Boolean reasoning, Journal of Logic, Language and Information, 2012 (12), pp. 433-459.
[14]	I. P. Hartmann, The relational syllogistic revisited, Perspectives on Semantic Representations for Textual Inference, CSLI Publications, 2014, pp. 195-227.
[15]	Xudong Hao, On the fourth figure of Aristotle’s syllogisms, Journal of Northwest University (Philosophy and Social Sciences), 2015 (6), pp. 142-146. (in Chinese).
[16]	L. S. Moss, Syllogistic Logic with Cardinality Comparisons, Springer International Publishing, 2016.
[17]	Shuai Li, and Xiaoming Ren, A New exploration on inductive logic of Aristotle, Journal of Chongqing University of Technology (Social Sciences), 2016 (7), pp. 6-11. (in Chinese).
[18]	Baoxiang Wu, Aristotel’s Syllogisms and its Extensions, Sichuan Normal University, Master’s Dissertation, 2017. (in Chinese).
[19]	D. Westerståhl, Quantifiers in formal and natural languages, in D. M. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. 14, 2007, pp. 227-242.
[20]	S. Peters, and D. Westerståhl, Quantifiers in Language and Logic, Claredon Press, Oxford, 2006.
[21]	Xiaojun Zhang, Sheng Li, Research on the formalization and axiomatization of traditional syllogisms, Journal of Hubei University (Philosophy and social sciences), No. 6, 2016, pp. 32-37. (in Chinese).

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Xiaojun Zhang. (2018). Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory. Applied and Computational Mathematics, 7(3), 167-172. https://doi.org/10.11648/j.acm.20180703.23

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Xiaojun Zhang. Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory. Appl. Comput. Math. 2018, 7(3), 167-172. doi: 10.11648/j.acm.20180703.23

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Xiaojun Zhang. Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory. Appl Comput Math. 2018;7(3):167-172. doi: 10.11648/j.acm.20180703.23

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@article{10.11648/j.acm.20180703.23,
  author = {Xiaojun Zhang},
  title = {Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory},
  journal = {Applied and Computational Mathematics},
  volume = {7},
  number = {3},
  pages = {167-172},
  doi = {10.11648/j.acm.20180703.23},
  url = {https://doi.org/10.11648/j.acm.20180703.23},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180703.23},
  abstract = {Syllogistic reasoning is important due to the prominence of syllogistic arguments in human reasoning, and also to the role they have played in theory of reasoning from Aristotle onwards. Aristotelian syllogistic logic is a formal study of the meaning of four Aristotelian quantifiers and of their properties. This paper focuses on logical system based on syllogistic reasoning. It firstly formalized the 24 valid Aristotle’s syllogisms, and then has proven that the other 22 valid Aristotle’s syllogisms can be derived from the syllogisms ‘Barbara’ AAA-1 and ‘Celarent’ EAE-1 by means of generalized quantifier theory and set theory, so the paper has completed the axiomatization of Aristotelian syllogistic Logic. This axiomatization needs to make full use of symmetry and transformable relations between/among the monotonicity of the four Aristotelian quantifiers from the perspective of generalized quantifier theory. In fact, these innovative achievements and the method in this paper provide a simple and reasonable mathematical model for studying other generalized syllogisms. It is hoped that the present study will make contributions to the development of generalized quantifier theory, and to bringing about consequences to natural language information processing as well as knowledge representation and reasoning in computer science.},
 year = {2018}
}

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TY  - JOUR
T1  - Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory
AU  - Xiaojun Zhang
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PY  - 2018
N1  - https://doi.org/10.11648/j.acm.20180703.23
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AB  - Syllogistic reasoning is important due to the prominence of syllogistic arguments in human reasoning, and also to the role they have played in theory of reasoning from Aristotle onwards. Aristotelian syllogistic logic is a formal study of the meaning of four Aristotelian quantifiers and of their properties. This paper focuses on logical system based on syllogistic reasoning. It firstly formalized the 24 valid Aristotle’s syllogisms, and then has proven that the other 22 valid Aristotle’s syllogisms can be derived from the syllogisms ‘Barbara’ AAA-1 and ‘Celarent’ EAE-1 by means of generalized quantifier theory and set theory, so the paper has completed the axiomatization of Aristotelian syllogistic Logic. This axiomatization needs to make full use of symmetry and transformable relations between/among the monotonicity of the four Aristotelian quantifiers from the perspective of generalized quantifier theory. In fact, these innovative achievements and the method in this paper provide a simple and reasonable mathematical model for studying other generalized syllogisms. It is hoped that the present study will make contributions to the development of generalized quantifier theory, and to bringing about consequences to natural language information processing as well as knowledge representation and reasoning in computer science.
VL  - 7
IS  - 3
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Author Information

Xiaojun Zhang

Institute of Logic and Information, Sichuan Normal University, Chengdu, China; Fujian Provincial Key Laboratory of Brain-Like Intelligent Systems, Xiamen University, Xiamen, China

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Xiaojun Zhang. (2018). Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory. Applied and Computational Mathematics, 7(3), 167-172. https://doi.org/10.11648/j.acm.20180703.23

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ACS Style

Xiaojun Zhang. Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory. Appl. Comput. Math. 2018, 7(3), 167-172. doi: 10.11648/j.acm.20180703.23

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Xiaojun Zhang. Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory. Appl Comput Math. 2018;7(3):167-172. doi: 10.11648/j.acm.20180703.23

Copy | Download

@article{10.11648/j.acm.20180703.23,
  author = {Xiaojun Zhang},
  title = {Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory},
  journal = {Applied and Computational Mathematics},
  volume = {7},
  number = {3},
  pages = {167-172},
  doi = {10.11648/j.acm.20180703.23},
  url = {https://doi.org/10.11648/j.acm.20180703.23},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180703.23},
  abstract = {Syllogistic reasoning is important due to the prominence of syllogistic arguments in human reasoning, and also to the role they have played in theory of reasoning from Aristotle onwards. Aristotelian syllogistic logic is a formal study of the meaning of four Aristotelian quantifiers and of their properties. This paper focuses on logical system based on syllogistic reasoning. It firstly formalized the 24 valid Aristotle’s syllogisms, and then has proven that the other 22 valid Aristotle’s syllogisms can be derived from the syllogisms ‘Barbara’ AAA-1 and ‘Celarent’ EAE-1 by means of generalized quantifier theory and set theory, so the paper has completed the axiomatization of Aristotelian syllogistic Logic. This axiomatization needs to make full use of symmetry and transformable relations between/among the monotonicity of the four Aristotelian quantifiers from the perspective of generalized quantifier theory. In fact, these innovative achievements and the method in this paper provide a simple and reasonable mathematical model for studying other generalized syllogisms. It is hoped that the present study will make contributions to the development of generalized quantifier theory, and to bringing about consequences to natural language information processing as well as knowledge representation and reasoning in computer science.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory
AU  - Xiaojun Zhang
Y1  - 2018/08/13
PY  - 2018
N1  - https://doi.org/10.11648/j.acm.20180703.23
DO  - 10.11648/j.acm.20180703.23
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 167
EP  - 172
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20180703.23
AB  - Syllogistic reasoning is important due to the prominence of syllogistic arguments in human reasoning, and also to the role they have played in theory of reasoning from Aristotle onwards. Aristotelian syllogistic logic is a formal study of the meaning of four Aristotelian quantifiers and of their properties. This paper focuses on logical system based on syllogistic reasoning. It firstly formalized the 24 valid Aristotle’s syllogisms, and then has proven that the other 22 valid Aristotle’s syllogisms can be derived from the syllogisms ‘Barbara’ AAA-1 and ‘Celarent’ EAE-1 by means of generalized quantifier theory and set theory, so the paper has completed the axiomatization of Aristotelian syllogistic Logic. This axiomatization needs to make full use of symmetry and transformable relations between/among the monotonicity of the four Aristotelian quantifiers from the perspective of generalized quantifier theory. In fact, these innovative achievements and the method in this paper provide a simple and reasonable mathematical model for studying other generalized syllogisms. It is hoped that the present study will make contributions to the development of generalized quantifier theory, and to bringing about consequences to natural language information processing as well as knowledge representation and reasoning in computer science.
VL  - 7
IS  - 3
ER  -

Copy | Download