Knowledge Mining about the Generalized Modal Syllogism E▯M◇F-2 with the Quantifiers in Square{fewer than half of the} and Square{no}

Volume 9, Issue 2, April 2024     |     PP. 37-46      |     PDF (270 K)    |     Pub. Date: May 7, 2024
DOI: 10.54647/computer520416    40 Downloads     6213 Views  

Author(s)

Mingwei Ma, School of Statistics, Capital University of Economics and Business, Beijing, China
Qing Cao, School of Philosophy, Anhui University, Hefei, China

Abstract
On the basis of set theory, generalized quantifier theory, and modal logic, this paper mainly focuses on the knowledge mining about generalized modal syllogism with the quantifiers in Square{fewer than half of the} and Square{no}. To this end, this paper firstly proves the validity of the non-trivial syllogism E▯M◇F-2, and deduces other 22 valid non-trivial generalized modal syllogisms based on relative reduction operations. The reason why syllogisms with different figures and forms can be mutually reduced is that any quantifier in Square{fewer than half of the} and Square{no} can define the other three quantifiers, and the necessary and possible modality are mutually dual. Since all the proofs in this article are deductive reasoning, their conclusions are consistent.

Keywords
knowledge mining; Square{fewer than half of the}; Square{no}; generalized modal syllogisms

Cite this paper
Mingwei Ma, Qing Cao, Knowledge Mining about the Generalized Modal Syllogism E▯M◇F-2 with the Quantifiers in Square{fewer than half of the} and Square{no} , SCIREA Journal of Computer. Volume 9, Issue 2, April 2024 | PP. 37-46. 10.54647/computer520416

References

[ 1 ] A. Chagrov and M. Zakharyaschev. Modal Logic, Oxford: Clarendon Press, 1997.
[ 2 ] P. R. Halmos. Naive Set Theory. New York: Springer-Verlag, 1974.
[ 3 ] F. Johnson. Aristotle’s modal syllogisms. Handbook of the History of Logic, I, 2004, pp.247-338.
[ 4 ] J. Łukasiewicz. Aristotle’s Syllogistic: From the Standpoint of Modern Formal Logic. Second edition, Oxford: Clerndon Press, 1957.
[ 5 ] P. Murinová, and V. Novák. A Formal Theory of Generalized Intermediate Syllogisms. Fuzzy Sets and Systems, Vol.186, 2012, pp.47-80.
[ 6 ] M. Malink. Aristotle’s Modal Syllogistic. Cambridge, MA: Harvard University Press, 2013.
[ 7 ] S. Peters, and D. Westerståhl. Quantifiers in Language and Logic. Oxford: Claredon Press, 2006.
[ 8 ] X. J. Zhang Axiomatization of Aristotelian syllogistic logic based on generalized quantifier theory. Applied and Computational Mathematics, Vol.7, No.3, 2018, pp.167-172.
[ 9 ] Y. J. Hao. The Reductions between/among Aristotelian Syllogisms Based on the Syllogism AII-3, SCIREA Journal of Philosophy, Vol.3, No.1, 2023, pp.12-22.
[ 10 ] L. H. Hao. The Validity of Generalized Modal Syllogisms Based on the Syllogism E▯M◇O-1, SCIREA Journal of Mathematics, Vol.9, No.1, 2024a, pp.11-22.
[ 11 ] L. H. Hao. Generalized Syllogism Reasoning with the Quantifiers in Modern Square{no} and Square{most}, Applied Science and Innovative Research, Vol.8, No.1, 2024b, pp.31-38.