The Reducibility of the Generalized Syllogism MMI-4 with the Quantifiers in Square{most} and Square{some}

Volume 9, Issue 4, August 2024     |     PP. 84-92      |     PDF (260 K)    |     Pub. Date: August 16, 2024
DOI: 10.54647/mathematics110491    36 Downloads     2600 Views  

Author(s)

Haiping Wang, School of Philosophy, Anhui University, China
Jiaojiao Yuan, Party School of Zigong Municipal Committee of the Communist Party of China

Abstract
This paper firstly proves the validity of the generalized syllogism MMI-4 with the quantifiers in Square{most} and Square{some}, and then making full use of the relevant definitions, facts, and reasoning rules to infer the other 20 valid generalized ones from the syllogism MMI-4. In other words, there are reducible relationships between/among these valid generalized syllogisms. The reason for this is because any quantifier in Square{some} can define the other three quantifiers, and so can any quantifier in Square{most}. This study has important theoretical value for natural language information processing.

Keywords
generalized quantifiers; generalized syllogisms; reducibility; validity

Cite this paper
Haiping Wang, Jiaojiao Yuan, The Reducibility of the Generalized Syllogism MMI-4 with the Quantifiers in Square{most} and Square{some} , SCIREA Journal of Mathematics. Volume 9, Issue 4, August 2024 | PP. 84-92. 10.54647/mathematics110491

References

[ 1 ] Łukasiewicz, J. (1957). Aristotelian Syllogistic: From the Standpoint of Modern Formal Logic. Second edition, Oxford: Clerndon Press.
[ 2 ] Zhang, X. J., and Li S. (2016). Research on the formalization and axiomatization of traditional syllogisms, Journal of Hubei University (Philosophy and Social Sciences), 6: 32-37. (in Chinese)
[ 3 ] Hao, Y. J. (2023). The Reductions between/among Aristotelian Syllogisms Based on the Syllogism AII-3, SCIREA Journal of Philosophy, 3(1): 12-22.
[ 4 ] Johnson, F. Aristotle’s modal syllogisms[J]. Handbook of the History of Logic, 2004(1): 247-338.
[ 5 ] Malink, M. (2013). Aristotle’s Modal Syllogistic, Cambridge, MA: Harvard University Press.
[ 6 ] Zhang, C. (2023). Formal Research on Aristotelian Modal Syllogism from the Perspective of Mathematical Structuralism, Doctoral Dissertation, Anhui University, 2023. (in Chinese)
[ 7 ] Ivanov, N., & Vakarelov, D. (2012). A system of relational syllogistic incorporating full Boolean reasoning. Journal of Logic, Language and Information, (21), 433-459.
[ 8 ] Endrullis, J. and Moss, L. S. (2015). “Syllogistic Logic with ‘Most’.” In: V. de Paiva et al. (eds. ), Logic, Language, Information, and Computation (pp. 124-139).
[ 9 ] Hamilton, A. G. (1978). Logic for Mathematicians. Cambridge: Cambridge University Press.
[ 10 ] Peters, S. and Westerståhl, D. (2006). Quantifiers in Language and Logic, Claredon Press, Oxford.