Screening out All Valid Aristotelian Modal Syllogisms

Xiaojun Zhang

doi:doi:10.11648/j.acm.20190806.12

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Screening out All Valid Aristotelian Modal Syllogisms

Xiaojun Zhang

Published in Applied and Computational Mathematics (Volume 8, Issue 6)

Received: 5 January 2020 Accepted: 15 January 2020 Published: 13 February 2020

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Abstract

It is easy to understand that whether a classical syllogism is valid. That whether a modal syllogism is valid is not so transparent. The prevailing view on Aristotelian modal syllogistic is that the syllogistic is incomprehensible due to its many faults and inconsistencies. Although adequate semantic analysis or reconstruction of the syllogistic have be given by many authors, it is far from obvious how to extend these results so as to consistently cover the whole modal syllogistic developed. The major aim of this paper is to overcome these difficulties, and screen out 384 Aristotelian valid modal syllogisms from 6656 Aristotelian modal syllogisms in natural language. They can be formalized by means of set theory and generalized quantifier theory, and their validity can be proved by possible world semantics and the truth definition of Aristotelian quantifiers defined in generalized quantifier theory. The basic steps of screening out all valid Aristotelian modal syllogisms are as follows: firstly one can get all possible modal syllogisms obtained by adding modal operators to 24 valid classical syllogisms, and secondly eliminate invalid modal syllogisms by characteristic rules of modal syllogisms. It is hoped that these innovative achievements will make contributions to promote the development of Aristotelian and generalized modal syllogistic, natural language information processing, and further research on knowledge representation and knowledge reasoning in computer science.

Published in	Applied and Computational Mathematics (Volume 8, Issue 6)
DOI	10.11648/j.acm.20190806.12
Page(s)	95-104
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), 2020. Published by Science Publishing Group

Keywords

Generalized Quantifier Theory, Aristotelian Modal Syllogisms, Formalization, Validity, Possible Worlds

References

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[2]	J. Martin, Aristotle’s natural deduction reconsidered, History and Philosophy of Logic Vol. 18, No. 1, 1997, pp. 1-15.
[3]	L. S. Moss, Completeness theorems for syllogistic fragments, in F. Hamm and S. Kepser (eds.), Logics for Linguistic Structures, Berlin: Mouton de Gruyter, 2008, pp. 143–173.
[4]	L. S. Moss, Syllogistic logics with verbs, Journal of Logic and Computation, Vol. 20, No. 4, 2010, pp. 947-967.
[5]	L. S. Moss, Syllogistic Logic with Cardinality Comparisons, Springer International Publishing, 2016.
[6]	P. Murinová, and V. Novák, A formal theory of generalized intermediate syllogisms, Fuzzy Sets and Systems, Vol. 186, No. 1, 2012, pp. 47-80.
[7]	N. Ivanov, and D. Vakarelov, A system of relational syllogistic incorporating full Boolean reasoning, Journal of Logic, Language and Information, Vol. 21, No. 4, 2012, pp. 433-459.
[8]	I. Pratt-Hartmann, The relational syllogistic revisited, Perspectives on Semantic Representations for Textual Inference, CSLI Publications, 2014, pp. 195-227.
[9]	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.
[10]	Baoxiang Wu, Aristotel’s Syllogisms and its Extensions, Sichuan Normal University, Master’s Dissertation, 2017. (in Chinese)
[11]	N. Chater, and M. Oaksford, The probability heuristics model of syllogistic reasoning, Cognitive Psychology, Vol. 38, No. 2, 1999, pp. 191-258.
[12]	M. Malink, A reconstruction of Aristotle’s modal syllogistic, History and Philosophy of Logic, Vol. 27, No. 2, 2006, pp. 95–141.
[13]	S. K. Thomason, Semantic Analysis of the Modal Syllogistic, Journal of Philosophical Logic, Vol. 26, No. 2, 1993, pp. 111–128.
[14]	S. K. Thomason, Relational model for the modal syllogistic, Journal of Philosophical Logic, Vol. 26, No. 2, 1997, pp. 129–1141.
[15]	P. Thom, The Logic of Essentialism: An Interpretation of Aristotle’s Modal Syllogistic, (Synthese Historical Library 43), Dordrecht: Kluwer, 1996.
[16]	F. Johnson, Models for modal syllogisms, Notre Dame Journal of Formal Logic, Vol. 30, No. 2, 1989, pp. 271-284.
[17]	F. Johnson, Aristotle’s modal syllogisms, Handbook of the History of Logic, Vol. 1, 2004, pp. 247-307.
[18]	M. Malink, Aristotle's Modal Syllogistic, Cambridge, MA: Harvard University Press, 2013.
[19]	J. van Benthem, Questions about quantifiers, Journal of Symbol Logic, Vol. 49, No. 2, 1984, pp. 443- 466.
[20]	D. Westerståhl, Aristotelian syllogisms and generalized quantifiers, Studia Logica, Vol. XLVII, No. 4, 1989, pp. 577-585.
[21]	Xiaojun Zhang, A Study of Properties of Generalized Quantifiers, PhD. dissertation, Chinese Academy of Social Sciences, 2011. (in Chinese)
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[23]	Xiaojun Zhang, and Sheng Li, Research on the formalization and axiomatization of classical syllogisms, Journal of Hubei University (Philosophy and social sciences), Vol. 43, No. 6, 2016, pp. 32-37. (in Chinese)
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[28]	A. Chagrov, and M. Zakharyaschev, Modal Logic, Oxford: Clarendon Press, 1997.
[29]	B. F. Chellas, Modal Logic: an Introduction, Cambridge: Cambridge University Press, 1980.

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APA Style

Xiaojun Zhang. (2020). Screening out All Valid Aristotelian Modal Syllogisms. Applied and Computational Mathematics, 8(6), 95-104. https://doi.org/10.11648/j.acm.20190806.12

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ACS Style

Xiaojun Zhang. Screening out All Valid Aristotelian Modal Syllogisms. Appl. Comput. Math. 2020, 8(6), 95-104. doi: 10.11648/j.acm.20190806.12

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AMA Style

Xiaojun Zhang. Screening out All Valid Aristotelian Modal Syllogisms. Appl Comput Math. 2020;8(6):95-104. doi: 10.11648/j.acm.20190806.12

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@article{10.11648/j.acm.20190806.12,
  author = {Xiaojun Zhang},
  title = {Screening out All Valid Aristotelian Modal Syllogisms},
  journal = {Applied and Computational Mathematics},
  volume = {8},
  number = {6},
  pages = {95-104},
  doi = {10.11648/j.acm.20190806.12},
  url = {https://doi.org/10.11648/j.acm.20190806.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20190806.12},
  abstract = {It is easy to understand that whether a classical syllogism is valid. That whether a modal syllogism is valid is not so transparent. The prevailing view on Aristotelian modal syllogistic is that the syllogistic is incomprehensible due to its many faults and inconsistencies. Although adequate semantic analysis or reconstruction of the syllogistic have be given by many authors, it is far from obvious how to extend these results so as to consistently cover the whole modal syllogistic developed. The major aim of this paper is to overcome these difficulties, and screen out 384 Aristotelian valid modal syllogisms from 6656 Aristotelian modal syllogisms in natural language. They can be formalized by means of set theory and generalized quantifier theory, and their validity can be proved by possible world semantics and the truth definition of Aristotelian quantifiers defined in generalized quantifier theory. The basic steps of screening out all valid Aristotelian modal syllogisms are as follows: firstly one can get all possible modal syllogisms obtained by adding modal operators to 24 valid classical syllogisms, and secondly eliminate invalid modal syllogisms by characteristic rules of modal syllogisms. It is hoped that these innovative achievements will make contributions to promote the development of Aristotelian and generalized modal syllogistic, natural language information processing, and further research on knowledge representation and knowledge reasoning in computer science.},
 year = {2020}
}

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TY  - JOUR
T1  - Screening out All Valid Aristotelian Modal Syllogisms
AU  - Xiaojun Zhang
Y1  - 2020/02/13
PY  - 2020
N1  - https://doi.org/10.11648/j.acm.20190806.12
DO  - 10.11648/j.acm.20190806.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 95
EP  - 104
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20190806.12
AB  - It is easy to understand that whether a classical syllogism is valid. That whether a modal syllogism is valid is not so transparent. The prevailing view on Aristotelian modal syllogistic is that the syllogistic is incomprehensible due to its many faults and inconsistencies. Although adequate semantic analysis or reconstruction of the syllogistic have be given by many authors, it is far from obvious how to extend these results so as to consistently cover the whole modal syllogistic developed. The major aim of this paper is to overcome these difficulties, and screen out 384 Aristotelian valid modal syllogisms from 6656 Aristotelian modal syllogisms in natural language. They can be formalized by means of set theory and generalized quantifier theory, and their validity can be proved by possible world semantics and the truth definition of Aristotelian quantifiers defined in generalized quantifier theory. The basic steps of screening out all valid Aristotelian modal syllogisms are as follows: firstly one can get all possible modal syllogisms obtained by adding modal operators to 24 valid classical syllogisms, and secondly eliminate invalid modal syllogisms by characteristic rules of modal syllogisms. It is hoped that these innovative achievements will make contributions to promote the development of Aristotelian and generalized modal syllogistic, natural language information processing, and further research on knowledge representation and knowledge reasoning in computer science.
VL  - 8
IS  - 6
ER  -

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Author Information

Xiaojun Zhang

Institute of Logic and Information, Sichuan Normal University, Chengdu, China

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Plain Text BibTeX RIS

APA Style

Xiaojun Zhang. (2020). Screening out All Valid Aristotelian Modal Syllogisms. Applied and Computational Mathematics, 8(6), 95-104. https://doi.org/10.11648/j.acm.20190806.12

Copy | Download

ACS Style

Xiaojun Zhang. Screening out All Valid Aristotelian Modal Syllogisms. Appl. Comput. Math. 2020, 8(6), 95-104. doi: 10.11648/j.acm.20190806.12

Copy | Download

AMA Style

Xiaojun Zhang. Screening out All Valid Aristotelian Modal Syllogisms. Appl Comput Math. 2020;8(6):95-104. doi: 10.11648/j.acm.20190806.12

Copy | Download

@article{10.11648/j.acm.20190806.12,
  author = {Xiaojun Zhang},
  title = {Screening out All Valid Aristotelian Modal Syllogisms},
  journal = {Applied and Computational Mathematics},
  volume = {8},
  number = {6},
  pages = {95-104},
  doi = {10.11648/j.acm.20190806.12},
  url = {https://doi.org/10.11648/j.acm.20190806.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20190806.12},
  abstract = {It is easy to understand that whether a classical syllogism is valid. That whether a modal syllogism is valid is not so transparent. The prevailing view on Aristotelian modal syllogistic is that the syllogistic is incomprehensible due to its many faults and inconsistencies. Although adequate semantic analysis or reconstruction of the syllogistic have be given by many authors, it is far from obvious how to extend these results so as to consistently cover the whole modal syllogistic developed. The major aim of this paper is to overcome these difficulties, and screen out 384 Aristotelian valid modal syllogisms from 6656 Aristotelian modal syllogisms in natural language. They can be formalized by means of set theory and generalized quantifier theory, and their validity can be proved by possible world semantics and the truth definition of Aristotelian quantifiers defined in generalized quantifier theory. The basic steps of screening out all valid Aristotelian modal syllogisms are as follows: firstly one can get all possible modal syllogisms obtained by adding modal operators to 24 valid classical syllogisms, and secondly eliminate invalid modal syllogisms by characteristic rules of modal syllogisms. It is hoped that these innovative achievements will make contributions to promote the development of Aristotelian and generalized modal syllogistic, natural language information processing, and further research on knowledge representation and knowledge reasoning in computer science.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Screening out All Valid Aristotelian Modal Syllogisms
AU  - Xiaojun Zhang
Y1  - 2020/02/13
PY  - 2020
N1  - https://doi.org/10.11648/j.acm.20190806.12
DO  - 10.11648/j.acm.20190806.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 95
EP  - 104
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20190806.12
AB  - It is easy to understand that whether a classical syllogism is valid. That whether a modal syllogism is valid is not so transparent. The prevailing view on Aristotelian modal syllogistic is that the syllogistic is incomprehensible due to its many faults and inconsistencies. Although adequate semantic analysis or reconstruction of the syllogistic have be given by many authors, it is far from obvious how to extend these results so as to consistently cover the whole modal syllogistic developed. The major aim of this paper is to overcome these difficulties, and screen out 384 Aristotelian valid modal syllogisms from 6656 Aristotelian modal syllogisms in natural language. They can be formalized by means of set theory and generalized quantifier theory, and their validity can be proved by possible world semantics and the truth definition of Aristotelian quantifiers defined in generalized quantifier theory. The basic steps of screening out all valid Aristotelian modal syllogisms are as follows: firstly one can get all possible modal syllogisms obtained by adding modal operators to 24 valid classical syllogisms, and secondly eliminate invalid modal syllogisms by characteristic rules of modal syllogisms. It is hoped that these innovative achievements will make contributions to promote the development of Aristotelian and generalized modal syllogistic, natural language information processing, and further research on knowledge representation and knowledge reasoning in computer science.
VL  - 8
IS  - 6
ER  -

Copy | Download