Archive




Volume 8, Issue 6, December 2019, Page: 95-104
Screening out All Valid Aristotelian Modal Syllogisms
Xiaojun Zhang, Institute of Logic and Information, Sichuan Normal University, Chengdu, China
Received: Jan. 5, 2020;       Accepted: Jan. 15, 2020;       Published: Feb. 13, 2020
DOI: 10.11648/j.acm.20190806.12      View  94      Downloads  55
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.
Keywords
Generalized Quantifier Theory, Aristotelian Modal Syllogisms, Formalization, Validity, Possible Worlds
To cite this article
Xiaojun Zhang, Screening out All Valid Aristotelian Modal Syllogisms, Applied and Computational Mathematics. Vol. 8, No. 6, 2019, pp. 95-104. doi: 10.11648/j.acm.20190806.12
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
G. Patzig, Aristotle's Theory of the Syllogism, J. Barnes (trans.), Dordrecht: D. Reidel, 1969.
[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)
[22]
Xiaojun Zhang, Research on Generalized Quantifier Theory, Xiamen: Xiamen University Press, 2014. (in Chinese)
[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)
[24]
Xiaojun Zhang, Axiomatization of Aristotelian syllogistic logic based on generalized quantifier theory, Applied and Computational Mathematics, Vol 7, No. 3, 2018, pp. 167-172.
[25]
S. Peters, and D. Westerståhl, Quantifiers in Language and Logic, Oxford: Claredon Press, 2006.
[26]
Xiaojun Zhang, Research on Chinese Anaphora Resolution and its Reasoning Mode, Beijing: People’s Publishing House, 2018.
[27]
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.
[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.
Browse journals by subject