How to Deduce the Remaining 23 Valid Syllogisms from the Validity of the Syllogism <I>EIO-1</I>

Xiaojun Zhang; Hui Li; Yijiang Hao

doi:doi:10.11648/j.acm.20221106.11

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How to Deduce the Remaining 23 Valid Syllogisms from the Validity of the Syllogism EIO-1

Xiaojun Zhang, Hui Li, Yijiang Hao

Published in Applied and Computational Mathematics (Volume 11, Issue 6)

Received: 9 October 2022 Accepted: 1 November 2022 Published: 11 November 2022

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Abstract

Syllogistic reasoning plays an important role in human reasoning, and has been widely studied from Aristotle onward. In previous studies, when deriving all the other valid syllogisms, at least two valid syllogisms were taken as the basic axioms. While this paper derives all other valid syllogisms only from one valid syllogism. On the basis of generalized quantifier theory and set theory, this paper shows that the remaining 23 valid syllogisms can be derived only from the syllogism EIO-1 by making the best of the definitions of three negative quantifiers of Aristotelian quantifiers, the symmetry of Aristotelian quantifiers no and some, and several propositional reasoning rules such as anti-syllogism rules and the subsequent weakening rule, and so on. This paper syntactically provides a simple and reasonable mathematical model for studying other kinds of syllogisms, such as generalized syllogistic, rational syllogistic, Aristotelian modal syllogistic and generalized modal syllogistic. And this research shows that formalized logic has the characteristics of structuralism, that is, it studies not only the forms and laws of thinking, but also the structure of thinking objects and the relationship between structures. It is hoped that this formal and innovative research is not only beneficial to the further development of various syllogistic logics, but also to natural language information processing in computer science, and also to knowledge representation and knowledge reasoning in Artificial Intelligence.

Published in	Applied and Computational Mathematics (Volume 11, Issue 6)
DOI	10.11648/j.acm.20221106.11
Page(s)	160-164
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, Axioms, Aristotelian Quantifiers, Rules

References

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[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]	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.
[13]	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).
[14]	Yijiang Hao. Formal research on discourse reasoning in natural language. Journal of Hunan University of Science and Technology (Social Sciences Edition), 2016 (1): 33-37. (in Chinese).
[15]	Łukasiewicz J. Aristotle’s Syllogistic: From the Standpoint of Modern Formal Logic. Second edition, Oxford: Clerndon Press, 1957.
[16]	Shushan Cai. A formal system of Aristotle’s syllogism different from that of Łukasiewicz. Philosophical research, 1988 (4): 33- 41. (in Chinese).
[17]	Mengyao Huang, Xiaojun Zhang. Assertion or rejection of Łukasiewicz’s assertoric syllogism system ŁA. Journal of Chongqing University of Science and Technology (Social Sciences Edition), 2020 (2): 10-18. (in Chinese).
[18]	Xiaojun Zhang. Axiomatization of Aristotelian syllogistic logic based on generalized quantifier theory. Applied and Computational Mathematics. Vol. 7, No. 3, 2018, pp. 167-172.
[19]	S. Peters, and D. Westerståhl, Quantifiers in Language and Logic, Claredon Press, Oxford, 2006.
[20]	Beihai Zhou, Qiang Wang, Zhi Zheng. Aristotle’s division lattice and Aristotelian logic. Logic research, 2018 (2): 2-20. (in Chinese).
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Xiaojun Zhang, Hui Li, Yijiang Hao. (2022). How to Deduce the Remaining 23 Valid Syllogisms from the Validity of the Syllogism EIO-1. Applied and Computational Mathematics, 11(6), 160-164. https://doi.org/10.11648/j.acm.20221106.11

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Xiaojun Zhang; Hui Li; Yijiang Hao. How to Deduce the Remaining 23 Valid Syllogisms from the Validity of the Syllogism EIO-1. Appl. Comput. Math. 2022, 11(6), 160-164. doi: 10.11648/j.acm.20221106.11

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AMA Style

Xiaojun Zhang, Hui Li, Yijiang Hao. How to Deduce the Remaining 23 Valid Syllogisms from the Validity of the Syllogism EIO-1. Appl Comput Math. 2022;11(6):160-164. doi: 10.11648/j.acm.20221106.11

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@article{10.11648/j.acm.20221106.11,
  author = {Xiaojun Zhang and Hui Li and Yijiang Hao},
  title = {How to Deduce the Remaining 23 Valid Syllogisms from the Validity of the Syllogism EIO-1},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {6},
  pages = {160-164},
  doi = {10.11648/j.acm.20221106.11},
  url = {https://doi.org/10.11648/j.acm.20221106.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221106.11},
  abstract = {Syllogistic reasoning plays an important role in human reasoning, and has been widely studied from Aristotle onward. In previous studies, when deriving all the other valid syllogisms, at least two valid syllogisms were taken as the basic axioms. While this paper derives all other valid syllogisms only from one valid syllogism. On the basis of generalized quantifier theory and set theory, this paper shows that the remaining 23 valid syllogisms can be derived only from the syllogism EIO-1 by making the best of the definitions of three negative quantifiers of Aristotelian quantifiers, the symmetry of Aristotelian quantifiers no and some, and several propositional reasoning rules such as anti-syllogism rules and the subsequent weakening rule, and so on. This paper syntactically provides a simple and reasonable mathematical model for studying other kinds of syllogisms, such as generalized syllogistic, rational syllogistic, Aristotelian modal syllogistic and generalized modal syllogistic. And this research shows that formalized logic has the characteristics of structuralism, that is, it studies not only the forms and laws of thinking, but also the structure of thinking objects and the relationship between structures. It is hoped that this formal and innovative research is not only beneficial to the further development of various syllogistic logics, but also to natural language information processing in computer science, and also to knowledge representation and knowledge reasoning in Artificial Intelligence.},
 year = {2022}
}

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AU  - Yijiang Hao
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AB  - Syllogistic reasoning plays an important role in human reasoning, and has been widely studied from Aristotle onward. In previous studies, when deriving all the other valid syllogisms, at least two valid syllogisms were taken as the basic axioms. While this paper derives all other valid syllogisms only from one valid syllogism. On the basis of generalized quantifier theory and set theory, this paper shows that the remaining 23 valid syllogisms can be derived only from the syllogism EIO-1 by making the best of the definitions of three negative quantifiers of Aristotelian quantifiers, the symmetry of Aristotelian quantifiers no and some, and several propositional reasoning rules such as anti-syllogism rules and the subsequent weakening rule, and so on. This paper syntactically provides a simple and reasonable mathematical model for studying other kinds of syllogisms, such as generalized syllogistic, rational syllogistic, Aristotelian modal syllogistic and generalized modal syllogistic. And this research shows that formalized logic has the characteristics of structuralism, that is, it studies not only the forms and laws of thinking, but also the structure of thinking objects and the relationship between structures. It is hoped that this formal and innovative research is not only beneficial to the further development of various syllogistic logics, but also to natural language information processing in computer science, and also to knowledge representation and knowledge reasoning in Artificial Intelligence.
VL  - 11
IS  - 6
ER  -

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Author Information

Xiaojun Zhang

School of Philosophy, Anhui University, Hefei, China
Hui Li

School of Philosophy, Anhui University, Hefei, China
Yijiang Hao

Institute of Philosophy, Chinese Academy of Social Sciences, Beijing, China

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APA Style

Xiaojun Zhang, Hui Li, Yijiang Hao. (2022). How to Deduce the Remaining 23 Valid Syllogisms from the Validity of the Syllogism EIO-1. Applied and Computational Mathematics, 11(6), 160-164. https://doi.org/10.11648/j.acm.20221106.11

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ACS Style

Xiaojun Zhang; Hui Li; Yijiang Hao. How to Deduce the Remaining 23 Valid Syllogisms from the Validity of the Syllogism EIO-1. Appl. Comput. Math. 2022, 11(6), 160-164. doi: 10.11648/j.acm.20221106.11

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AMA Style

Xiaojun Zhang, Hui Li, Yijiang Hao. How to Deduce the Remaining 23 Valid Syllogisms from the Validity of the Syllogism EIO-1. Appl Comput Math. 2022;11(6):160-164. doi: 10.11648/j.acm.20221106.11

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@article{10.11648/j.acm.20221106.11,
  author = {Xiaojun Zhang and Hui Li and Yijiang Hao},
  title = {How to Deduce the Remaining 23 Valid Syllogisms from the Validity of the Syllogism EIO-1},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {6},
  pages = {160-164},
  doi = {10.11648/j.acm.20221106.11},
  url = {https://doi.org/10.11648/j.acm.20221106.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221106.11},
  abstract = {Syllogistic reasoning plays an important role in human reasoning, and has been widely studied from Aristotle onward. In previous studies, when deriving all the other valid syllogisms, at least two valid syllogisms were taken as the basic axioms. While this paper derives all other valid syllogisms only from one valid syllogism. On the basis of generalized quantifier theory and set theory, this paper shows that the remaining 23 valid syllogisms can be derived only from the syllogism EIO-1 by making the best of the definitions of three negative quantifiers of Aristotelian quantifiers, the symmetry of Aristotelian quantifiers no and some, and several propositional reasoning rules such as anti-syllogism rules and the subsequent weakening rule, and so on. This paper syntactically provides a simple and reasonable mathematical model for studying other kinds of syllogisms, such as generalized syllogistic, rational syllogistic, Aristotelian modal syllogistic and generalized modal syllogistic. And this research shows that formalized logic has the characteristics of structuralism, that is, it studies not only the forms and laws of thinking, but also the structure of thinking objects and the relationship between structures. It is hoped that this formal and innovative research is not only beneficial to the further development of various syllogistic logics, but also to natural language information processing in computer science, and also to knowledge representation and knowledge reasoning in Artificial Intelligence.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - How to Deduce the Remaining 23 Valid Syllogisms from the Validity of the Syllogism EIO-1
AU  - Xiaojun Zhang
AU  - Hui Li
AU  - Yijiang Hao
Y1  - 2022/11/11
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221106.11
DO  - 10.11648/j.acm.20221106.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 160
EP  - 164
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221106.11
AB  - Syllogistic reasoning plays an important role in human reasoning, and has been widely studied from Aristotle onward. In previous studies, when deriving all the other valid syllogisms, at least two valid syllogisms were taken as the basic axioms. While this paper derives all other valid syllogisms only from one valid syllogism. On the basis of generalized quantifier theory and set theory, this paper shows that the remaining 23 valid syllogisms can be derived only from the syllogism EIO-1 by making the best of the definitions of three negative quantifiers of Aristotelian quantifiers, the symmetry of Aristotelian quantifiers no and some, and several propositional reasoning rules such as anti-syllogism rules and the subsequent weakening rule, and so on. This paper syntactically provides a simple and reasonable mathematical model for studying other kinds of syllogisms, such as generalized syllogistic, rational syllogistic, Aristotelian modal syllogistic and generalized modal syllogistic. And this research shows that formalized logic has the characteristics of structuralism, that is, it studies not only the forms and laws of thinking, but also the structure of thinking objects and the relationship between structures. It is hoped that this formal and innovative research is not only beneficial to the further development of various syllogistic logics, but also to natural language information processing in computer science, and also to knowledge representation and knowledge reasoning in Artificial Intelligence.
VL  - 11
IS  - 6
ER  -

Copy | Download