id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
asir-56165	Shi, Haowei	Reducible Relationships between Generalized Syllogisms with the Quantifiers in Square{at least half of the}	2025	6	.pdf	application/pdf	2150	124	67	Fact 4 (subordination) : (4.1) ⊢all(p, n)→some(p, n); www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 9, No. 2, 2025 67 Published by SCHOLINK INC. (4.2) ⊢no(p, n)→not all(p, n); (4.3) ⊢all(p, n)→most(p, n); (4.4) ⊢most(p, n)→some(p, n); (4.5) ⊢all(p, n)→at least half of the(p, n); (4.6) ⊢at least half of the(p, n)→some(p, n); (4.7) ⊢at least half of the(p, n)→most(p, n); (4.8) ⊢at most half of the(p, n)→fewer than half of the(p, n); (4.9) ⊢fewer than half of the(p, n)→not all(p, n); (4.10) ⊢at most half of the(p, n)→not all(p, n); (4.11) ⊢no(p, n)→fewer than half of the(p, n); (4.12) ⊢no(p, n)→at most half of the(p, n). 3.3 Basic Axioms A1: If  is a valid proposition in classical logic, then ⊢. A2: ⊢at least half of the(t, n)all(t, p)→some(p, n) (i.e. the syllogism SAI-3).	cache/asir-56165.pdf	txt/asir-56165.txt