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

: 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.

Generalized modal syllogism includes both generalized quantifiers and modalities.There are two kinds of quantifiers in natural language, that is, Aristotelian quantifiers (i.e.all, some, no, and not all) and generalized quantifiers (such as, most, both, fewer than half of the).The former is a special case of the latter, while the latter is an extension of the former.In other words, Aristotelian quantifiers are trivial generalized quantifiers.One can obtain a generalized modal syllogism by adding at least one and at most three non-overlapping modalities (that is, necessary modality  or possible modality ◇) to a generalized syllogism (Hao, 2024b).

Knowledge Representation of Generalized Modal Syllogisms
In this paper, b, t, and z denote lexical variables, and D represents the domain.The sets composed of b, t, and z are denoted as B, T, and Z, respectively.Let  、  、  and  be well-formed formulas (shorted as wff).'B∩Z'represents the cardinality for the intersection of the set B and Z. '⊢' says that the formula is provable, and '=def ' states that  can be defined by .Others are similar.The operators in the paper such as, , , ,  are symbols in modal logic (Chagrov and Zakharyaschev, 1997) and set theory (Halmos).
The generalized modal syllogisms studied in this paper involves the following 24 propositions: (1) all(b, z), some (b, z), no (b, z), not all (b, z), fewer than half of the (b, z), at least half of the (b, z), most (b, z), at most half of the most (b, z), which are respectively abbreviated as Proposition A, I, E, O, F, S, M, and H. (2) all(b, z), some(b, z), no(b, z), not all(b, z), fewer than half of the(b, z),  at least half of the(b, z),  most(b, z),  at most half of the most(b, z), which are respectively abbreviated as Proposition A, I, b, O, F, S, M, and H; (3)◇all(b, z), ◇some(b, z), ◇no(b, z), ◇not all(b, z), ◇fewer than half of the(b, z), ◇at least half of the (b, z), ◇most(b, z), ◇at most half of the most (b, z), which are respectively abbreviated as Proposition ◇A, ◇I, ◇E, ◇O, ◇F, ◇S, ◇M, and ◇H.
A non-trivial generalized modal syllogism contains at least one non-trivial generalized quantifier and one modality.This paper mainly studies knowledge mining about the generalized modal syllogism EM◇F-2 with the quantifiers in Square{fewer than half of the} and Square {no}.An instance of the syllogism EM◇F-2 in natural language is as follows: Major premise: No cat is a dog.
Minor premise: Most pet animals are necessarily dogs.
Conclusion: Fewer than half of pet animals are possibly cats.
Let z be a lexical variable that stands for cats in the domain, t be a lexical variable that denotes dogs in the domain, and b be a lexical variable that represents pet animals in the domain.Then this syllogism can be formalized as 'no(z, t)most (b, t)◇fewer than half of the(b, z)', which is abbreviated as EM◇F-2.Others are similar.

Generalized Modal Syllogism System
For any quantifier Q, there are three kinds of negative quantifiers, that is, outer negation Q, inner negation Q, and dual negation Q.Any quantifier Q and its three negative ones can form a Square{Q}={Q, Q, Q , Q}.In other words, any quantifier in Square{Q} can define the other three quantifiers.For example, Square{fewer than half of the}={fewer than half of the, at least half of the, most, at most half of the}, and Square{no}={no, some, all, not all}.The generalized modal syllogisms studied in this paper only involves 8 quantifiers in Square{fewer than half of the} and Square{no}.The definable relationship between these quantifiers can be found in the following Fact 1 and Fact 2.

Formation Rules
(1) If Q is a quantifier, b and z are lexical variables, then Q(b, z) is a wff.
(2) If  is a wff, then so are  and .
(3) If  and  are wffs, then so is .
(4) Only the formulas obtained by the above rules are wffs.

Basic Axioms
A1: If  is a valid formula in first-order logic, then ⊢.

Conclusion
On the basis of set theory, generalized quantifier theory, and modal logic, this paper mainly focuses on the knowledge mining about generalized modal syllogisms with the quantifiers in Square{fewer than half of the} and Square{no}.For this purpose, this paper firstly proves the -45 -Can this research method provide a unified research paradigm for knowledge mining of other generalized modal syllogisms involving other quantifiers (such as at most 1/3 of the, fewer than 3/5 of the) ?Is the generalized syllogism fragment system studied in this paper sound and complete?These problems require further discussion.

3. 5
Relevant Definitions D1: ()=def(); D2: () =def ()(); D3: (Q)(b, z)=def Q(b, Dz); D4: (Q)(b, z)=def It is not that Q(b, z); D5: ◇Q(b, z)=def Q(b, z); D6: no(b, z) is true when and only when B∩Z= is true in any real world; D7: most(b, z) is true when and only when B∩Z0.5B is true in any possible world; D8: ◇fewer than half of the(b, z) is true when and only when B∩Z0.5B is true in at least one possible world.
validity of the non-trivial syllogism E  M◇F-2 in Theorem 1, and then deduces other 22 valid non-trivial generalized modal syllogisms based on relative reduction operations (such as facts and definitions) in Theorem 2. 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.