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Knowledge Representation and Knowledge Reasoning in Square{most} and Square{all}
Feifei Yang and Xiaojun Zhang*
School of Philosophy, Anhui University, Hefei, China
Submission: March 24 , 2024; Published: July 03, 2024
*Corresponding author: Xiaojun Zhang, School of Philosophy, Anhui University, Hefei, China, Email: 591551032@qq.com
How to cite this article: Feifei Yang and Xiaojun Zhang*. Knowledge Representation and Knowledge Reasoning in Square{most} and Square{all}. Ann Soc Sci Manage Stud. 2024; 10(4): 555793. DOI: 10.19080/ASM.2024.10.555793
Abstract
Making much of reduction operations, this paper firstly proves the validity of the generalized syllogism 1,EMO− and then infers the other 14 valid generalized ones from EMO−1. That is to say that these syllogisms are reducible. The reason why these syllogisms are reducible is that the quantifiers in Square{all} can be mutually defined, and that so can the quantifiers in Square{most}. All proofs in this paper are deductive reasoning. All proofs in this paper are deductive reasoning, therefore this work is consistent. It is hoped that this paper not only benefits the study of modern logic, but also further development of inference machines.
Keywords: Aristotelian Quantifiers; Generalized Quantifiers; Generalized Syllogisms; Validity; Non-Trivial Generalized Syllogisms
Introduction
Generally speaking, there are lots of generalized quantifiers in natural language [1]. Noun phrases as well as their determiners are generalized quantifiers (such as his bag, most, both, fewer than half of the, infinitely many). In particular, Aristotelian quantifiers are special cases of generalized quantifiers, the latter being extensions of the former [2]. Hence, generalized syllogisms are extensions of Aristotelian syllogisms. The generalized quantifiers except Aristotelian ones are non-trivial generalized quantifiers, thus a non-trivial generalized syllogism includes at least one of them [3,4]. There are many domestic and international works on classical syllogisms, such as Aristotelian syllogisms [5-7] and Aristotelian modal syllogisms [8-10].
But there are few works on generalized syllogisms. This paper aims to promote their study. Since there are many generalized quantifiers in English language, this paper specifically concentrates on the generalized syllogisms with the generalized quantifiers in Square{most} which is composed of the following: most, at most half of the, fewer than half of the and at least half of the. Any one of them is the other three outer, inner, or dual negative quantifiers. Any of them can define the other three. Similarly, so can the quantifiers in Square{all} which is constituted of Aristotelian quantifiers (i.e. all, not all, some, no).
Preliminaries
In this paper, let g, r and u be lexical variables, and the sets composed of g, r and u be G, R, and U, respectively. ‘|G∩U|’ indicates the cardinality for the intersection of G and U. And D
represents the domain of these variables. Q stands for any of the quantifiers in Square{all}, and Q¬ and ¬Q for its inner and outer negative quantifier, respectively. Let β,δ,φ and λ be well-formed formulas (shortened to wff). 
states that the wff β can be proved, and def‘δ=defλ’ that d can be defined by λ. The others are similar. All logical connectives (such as ¬,∧,→,↔) in this paper are common operators in first-order logic [11].
The generalized syllogisms studied in the paper merely involve the quantifiers as follows: most, at least half of the, fewer than half of the, at most half of the, all, some, no, not all, and a non-trivial one at least involves one of the first four. Consequently, these syllogisms only involve the following propositions: all(g,u), not all(g,u), some(g,u), no(g,u), most(g,u), at least half of the(g,u),at most half of the(g,u), fewer than half of the(g,u). They are respectively abbreviated as Proposition A, O, I, E, M, S, H, and F. For example, the first figure syllogism no(r,u) ∧ most(g,r) → not all(g,u) is abbreviated as EMO-1. Its instance is as follows:
Major premise: No student is a teacher.
Minor premise: Most of people in the classroom are students.
Conclusion: Not all people in the classroom are teachers.
Generalized Syllogism System with the Quantifier ‘most
This system includes the following parts: primitive symbols, relevant definitions, deductive rules and facts, and so on.
Primitive Symbols
i. brackets: (, )
ii. variables: g, r, u
iii. connectives: ¬,→
iv. quantifiers: most, all
Formation Rules
i. If Q is a quantifier, g and u are variables, then Q(g, u) is a wff.
ii. If λ is a wff, then so is ¬λ .
iii. If λ and δ are wffs, then so is λ →δ .
iv. Only the formulas constructed by the above rules are wffs.
Basic Axioms
A1: If β is a valid proposition in classical logic, then 
.
A2: 
no(r,u) ∧most (g, r )→not all (g,u) (that
is, the syllogism EMO-1).
Rules of Deduction
Rule 1 (subsequent weakening): If 
and 
then 
Rule 2 (anti-syllogism): If 
then 
Rule 3 (anti-syllogism): If 
then 
Relevant Definitions
D1 (conjunction): (β ∧δ) = def¬( β →¬δ );
D2 (bicondition):
D3 (inner negation): 
D4 (outer negation): 
It is not that
Q(g,u);
D5 (truth value): 
D6 (truth value): 
D8 (truth value): 
D9 (truth value): 
D10 (truth value): 
is true iff 
is true;
D11 (truth value): most (g,u) is true iff 
is true;
D12 (truth value): at most half of the(g,u) is true iff 
D13 (truth value): fewer than half of the(g,u) is
true iff 
is true.
Relevant Facts
Fact 1(inner negation):
Fact 2(outer negation):
Fact 3 (symmetry):
Fact 4(Subordination):
The above facts can be proved in generalized quantifier theory [12,13] or first-order logic [11].
The Reducible Relationships between/among Generalized Syllogisms
The validity of the syllogism EMO −1 is proved in Theorem 1. The following (2.1) ‘EMO −1→ EAH − 2’ illustrates the syllogisms EAH − 2 derived from EMO −1. In other words, EAH − 2 is valid. It can be concluded that there is a reducible relationship between the two syllogisms. Other situations are similar to this.
Theorem 1 (EMO-1): The syllogism no(r,u) ∧most (g, r )→not all (g,u) is valid.
Proof: Suppose that no(r,u) and most (g, r ) are true, then R∩U =φ and |G∩R |> 0.5 |G| are true according to Definition D8 and D11, respectively. Hence it shows that |G∩U |≤ 0.5 |G|. And it shows that at most half of the(g,u) is true in the light of Definition D11. Thus, it gives that not all (g,u) is true in line with Fact (4.7).
Theorem 2: There are at least the following 14 valid generalized syllogisms obtained from EMO −1:
Proof:
So far, through the above 23 steps of reduction operations, the above 14 valid generalized syllogisms in Theorem 2 have been inferred from the valid syllogism EMO −1.
Conclusion and Future Work
All in all, making much of reduction operations, this paper firstly uses relevant definitions, rules and facts to prove the validity of the generalized syllogism EMO −1, and then infers the other 14 valid generalized ones from EMO −1. That is to say that these syllogisms are reducible. The reason why these syllogisms have reducibility is that the quantifiers in Square{all} can be mutually defined, and that so can the quantifiers in Square{most}. All proofs in this paper are deductive reasoning, therefore this work is consistent.
In the same way, one can derive more valid syllogisms by continuing to use reduction operations. Moreover, their validity can be proven similar to Theorem 1. As a matter of fact, there are (8´8´8´4-4´4´4´4=) 1972 non-trivial generalized syllogisms merely involving the quantifiers in Square {all} and Square{most}. How can one effectively filter out all the valid ones among them? This question deserves further discussion.
Acknowledgement
This work was supported by the National Social Science Fund of China under Grant No. 22FZXB092.
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