The Validity of Generalized Modal Syllogisms with the Generalized Quantifiers in Square{most}
generalized modal syllogisms; reducibility; modality; validity
Abstract
Due to the large number of generalized quantifiers in the English language, this paper only studies the fragment of generalized modal syllogistic that contains the quantifiers in Square{all} and Square{most}. On the basis of generalized quantifier theory, possible-world semantics, and set theory, this paper shows that there are reducible relations between/among the generalized modal syllogism £EM◇O-3 and at least the other 29 valid generalized modal syllogisms. This method can also be used to study syllogisms with other generalized quantifiers. The results obtained by means of formal deductive method have not only consistency, but also theoretical value for the development of inference theory in artificial intelligence.
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Isaac Ramos (2010) O SAGRADO E O PROFANO EM ALBERTO CAEIRO. 2(3), 26.
Edson Rodrigues, Diones Rodrigues, Lízia Bertuzzi, Marta Silva, Francieli Bueno, Ângela Wagner (2026) Entre o visível e o invisível: (re)leituras sobre percepções e (in)diferenças em AH/SD. 2(2), 1-12.
⊢no (2015) § 163 Macdonald's theorem on p-groups all of whose proper subgroups are of class at most 2. 113-115.
(2013) Million-Dollar Blocks. 187-206.
Walt Whitman (2023) The Most Inspiriting of all War'S Shows.
Unknown Title. 2.
Unknown Title. (3).
⊢all w)∧no(v, w)→at most half of the(v, z) (i.e. AEH-2.
(2020) Literary fact. (17).
(null) Table 1: Description of every rule included in the original Magenta rule-set, as well as every rule included in our Xota rule-set.. 2.
⊢¬not all(z, w)∧most(v, z)→◇¬no(v, w).
Unknown Title.
⊢all w)∧most(v, z)→◇some(v, w) (i.e. AM◇I-1.
Unknown Title. (2).
⊢all¬ ∧most(v, z)→◇some¬(z, w).
⊢all (v, D-w)∧most(v, z)→ ◇some(z, D-w)(i.e. AM◇I-3.
⊢all w)∧no(w, v)→at most half of the(v, z) (i.e. AEH-4.
⊢all w)∧most(v, z)→◇some(w, v) (i.e. MA◇I-4.
⊢no¬ ∧all¬(v, w)→at most half of the(v, z).
⊢no(z, D-w)∧all(v, D-w)→at most half of the(v, z) (i.e. EAH-2, by [and Definition 2).
⊢no¬ w)∧most(v, z)→◇not all¬(v, w).
⊢no(z, D-w)∧most(v, z)→◇not all(v, D-w) (i.e. EM◇O-1.
⊢no(D-w, z)∧all(v, D-w)→at most half of the(v, z) (i.e. EAH-1. (4).
⊢no(D-w, z)∧most(v, z)→◇not all(v, D-w) (i.e. EM◇O-2. (4).
(2020) ⊢all(v, D-w)∧fewer than half of the¬(v, z)→◇not all¬(D-w, z) (by.
⊢all(v, D-w)∧fewer than half of the(v, D-z)→◇not all(D-w, D-z) (i.e. FA◇O-3.
⊢all¬ D-w, z)∧all(v, D-w)→at least half of the¬.
Andrei Ustinov (2021) "Chinary" in M. Kuzmin's Journals. (19), 200-252.
⊢all(D-w, D-z)∧all(v, D-w)→at least half of the(v, D-z) (i.e. AAS-1.
(2020) ⊢all¬(D-w, z)∧fewer than half of the¬(v, z)→◇not all(v, D-w) (by.
⊢all(D-w, D-z)∧fewer than half of the(v, D-z)→◇not all(v, D-w) (i.e. AF◇O-2.
⊢all w)∧no(v, w)→◇at most half of the(v, z) (i.e. AE◇H-2.
⊢all w)∧no(w, v)→◇at most half of the(v, z) (i.e. AE◇H-4.
⊢¬◇at most half of the(v, z)∧all(z, w)→¬no(v, w).
Matt Barton (2019) Half-Life 2. 319-322.
⊢most ∧all(z, w)→◇some(v, w) (i.e. AM◇I-1.
⊢¬at most half of the(v, z)∧no(v, w)→◇¬all(z, w).
⊢most ∧no(v, w)→◇not all(z, w) (i.e. EM◇O-3.
⊢no¬ ∧all¬(v, w)→◇at most half of the(v, z).
⊢no(z, D-w)∧all(v, D-w)→◇at most half of the(v, z) (i.e. EA◇H-2, by [and Definition 2).
⊢¬◇at most half of the(v, z)∧all(z, w)→¬no.
⊢¬at most half of the(v, z)∧all(z, w)→◇¬no.
⊢most ∧all(z, w)→◇some(w, v) (i.e. MA◇I-4.
⊢¬◇at most half of the(v, z)∧no(w, v)→¬all(z, w).
Kaoru Takara, Eisuke Fujita, Munenari Inoguchi, Satoshi Iizuka (2024) Special Issue on NIED Frontier Research on Science and Technology for Disaster Risk Reduction and Resilience 2024. 19(5), 715-716.
⊢most ∧no¬(z, w)→◇not all¬(v, w).
Jerry Burger (2011) Is That All There Is? Reaction to the That's-Not-All Procedure. 213-216.
⊢most (2014) OXFORD, All Souls College Library. 166-166.
Kate Potts (2020) 'All Things are Words of Some Strange Tongue': Dictionary Definition Form in Contemporary American Poetry. 238-256.
⊢no(D-w, z)∧all(v, D-w)→◇at most half of the(v, z) (i.e. EA◇H-1. (4).
⊢most(v, z)∧no(D-w, z)→◇not all(v, D-w) (i.e. EM◇O-2. (4).
⊢all(D-w, D-z)∧all(v, D-w)→◇at least half of the(v, D-z) (i.e. AA◇S-1.
(2020) Decision aids quickly and accurately rule out heart attack for almost half of all patients tested.
⊢¬at least half of the(v, D-z)∧all(D-w, D-z)→◇¬all(v, D-w).
(2020) ⊢fewer than half of the(v, D-z)∧all(D-w, D-z)→◇not all(v, D-w) (i.e. AF◇O-2.
⊢¬◇at least half of the(v, D-z)∧all(v, D-w)→¬all(D-w, D-z).
(2020) ⊢¬at least half of the(v, D-z)∧all(v, D-w)→◇¬all(D-w, D-z).
⊢fewer than half of the(v, D-z)∧all(v, D-w)→◇not all(D-w, D-z) (i.e. FA◇O-3.
J Łukasiewicz (1957) Aristotle's Syllogistic: From the Standpoint of Modern Formal Logic.
G Patzig (1969) Aristotle's Theory of the Syllogism.
Paul Halmos (1974) Naive Set Theory.
F Chellas (1980) Modal Logic: an Introduction.
F Johnson (2004) Aristotle's modal syllogisms. I, 247-338.
Petra Murinová, Vilém Novák (2012) A formal theory of generalized intermediate syllogisms. 186(1), 47-80.
J Endrullis, L Moss (2015) Syllogistic logic with 'Most. 124-139.
Xiaojun Zhang, Baoxiang Wu (2021) Research on Chinese Textual Reasoning.
Long Wei (2023) Formal System of Categorical Syllogistic Logic Based on the Syllogism AEE-4. 13(01), 97-103.
Hui Li (2023) Reduction between Categorical Syllogisms Based on the Syllogism EIO-2. 7(1), p30.
Cheng Zhang, Xiaojun Zhang (2023) Reduction between Aristotelian Modal Syllogisms Based on the Syllogism ◇I□A◇I-3. 13(02), 145-154.
Jing Xu, Xiaojun Zhang (2023) The Reducibility of Generalized Modal Syllogisms Based on AM◇I-1. 3(1).
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2024-02-19
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