Knowledge Reasoning about the Aristotelian Syllogism IAI-4

Volume 9, Issue 2, April 2024     |     PP. 23-30      |     PDF (234 K)    |     Pub. Date: April 1, 2024
DOI: 10.54647/mathematics110479    40 Downloads     37872 Views  

Author(s)

Siyi Yu, School of Philosophy, Anhui University, Hefei, China
Xiaojun Zhang, School of Philosophy, Anhui University, Hefei, China

Abstract
On the basis of set theory, propositional logic and generalized quantifier theory, this paper indicates that the other 23 valid syllogisms can be only derived from the syllogism IAI-4. These derivations use the symmetry of quantifiers no and some, the definitions of inner and outer negation of Aristotelian quantifiers, deductive rules of propositional logic, and some relevant facts, and so on. Moreover, this paper establishes a concise formalized axiomatic system for Aristotelian syllogistic logic and puts forward a research paradigm for the study of other syllogistic. This formal method aligns with the idea of knowledge reasoning and knowledge mining in artificial intelligence.

Keywords
Aristotelian syllogisms; Aristotelian quantifiers; symmetry; reducibility

Cite this paper
Siyi Yu, Xiaojun Zhang, Knowledge Reasoning about the Aristotelian Syllogism IAI-4 , SCIREA Journal of Mathematics. Volume 9, Issue 2, April 2024 | PP. 23-30. 10.54647/mathematics110479

References

[ 1 ] J. Łukasiewicz, (1957), Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, Oxford: Clarendon Press.
[ 2 ] Shushan Cai, (1988), A formal system of Aristotle’s syllogism different from that of Łukasiewicz, Philosophical research, 4: 33-41. (in Chinese)
[ 3 ] Xiaojun Zhang, Sheng Li, (2016), Research on the formalization and axiomatization of traditional syllogisms, Journal of Hubei University (Philosophy and social sciences), 6: 32-37. (in Chinese)
[ 4 ] Beihai Zhou, Qiang Wang, Zhi Zheng, (2018), Aristotle’s division lattice and Aristotelian logic. Logic research, 2: 2-20. (in Chinese)
[ 5 ] Xiaojun Zhang, (2018), Axiomatization of Aristotelian syllogistic logic based on generalized quantifier theory. Applied and Computational Mathematics, 7(3): 167-172.
[ 6 ] Xiaojun Zhang, Baoxiang Wu, (2021), Research on Chinese Textual Reasoning, Beijing: People’s Publishing House. (in Chinese)
[ 7 ] Xiaojun Zhang, Hui Li, and 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.
[ 8 ] Cheng Zhang, (2022), The Remaining 23 Valid Aristotelian Syllogisms can be Deduced only from the Syllogism IAI-3, SCIREA Journal of Computer, 7(5), 85-95.
[ 9 ] Hui Li, (2023), Reduction between categorical syllogisms based on the syllogism EIO-2. Applied Science and Innovative Research, (7), 30-37.
[ 10 ] Long Wei, (2023), Formal system of categorical syllogistic logic based on the syllogism AEE-4. Open Journal of Philosophy, (13), 97-103.