The Reducibility of Generalized Syllogisms with the Quantifiers
in Square {not all} and Square {most}

Jing Xu and Zhipeng Yu

doi:10.23880/phij-16000339

Philosophy International Journal Research Article 12 min read

The Reducibility of Generalized Syllogisms with the Quantifiers in Square {not all} and Square {most}

Jing Xu and Zhipeng Yu^*

^* Corresponding author

ISSN: 2641-9130 10.23880/phij-16000339 Received: October 14, 2024 Published: November 27, 2024

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Keywords

Generalized syllogism Square{not all} Square{most} Reducibility

Abstract

To explore the reducibility of non-trivial generalized syllogisms with the quantifiers in Square {not all} and Square {most}, this paper first gives the formalization of generalized syllogisms on the basis of set theory, and then proves the validity of the generalized syllogism EMO-3 by first-order logic and generalized quantifier theory; Finally, with the help of some reductive operations, the other 20 valid generalized syllogisms are deduced from the syllogism EMO-3. In other words, there are the reducible relationships between/among the 21 valid syllogisms. The reason for this is that any quantifier in a square can define the other three quantifiers. This research method is applicable to the study of syllogisms with quantifiers in other squares.

Introduction

Syllogism reasoning characterizes the semantic and reasoning nature of the quantifiers it involves, playing a significant part in natural language and human thinking [1]. Various kinds of syllogisms are frequently used in natural language, such as categorical syllogisms [2], generalized syllogisms [3, 4], Aristotelian modal syllogisms [5, 6], generalized modal syllogisms [7], and so forth. In generalized quantifier theory, a modern square consists of a quantifier and its three negation quantifiers [8]. Let Q be a generalized quantifier, and ØQ, QØ, and ØQØ respectively stand for its outer, inner and dual negative quantifier. Square {Q} = {Q,ØQ,QØ,ØQØ} shows the square composed of the four quantifiers. For example, Square {not all} = {not all, all, some, no}, and Square {most} = {most, at most half of the, fewer than half of the, at least half of the}. This paper concentrates on studying the reducibility of the generalized syllogisms which include at least one quantifiers in Square {not all} and Square {most}.

Preliminaries

Let g, t and y be the lexical variables in a generalized syllogism, and the sets composed of the three variables are G, T, and Y, respectively. Let D be the domain of the lexical variables, G∩Ybe the cardinality of the intersection of the set G and Y, and b, l, s and t be well-formed formulas (noted as wff). ‘⊢l’ means that the wff l is provable, and ‘b=defl’ that b can be defined by l.

The generalized syllogisms discussed in this paper only involves eight quantifiers in Square {not all} and Square {most}, which correspond to the following eight propositions: ‘Not all gs are ys’, ‘All gs are ys’, ‘Some gs are ys’, ‘No gs are ys’, ‘Most gs are ys’, ‘At most half of the gs are ys’, ‘Fewer than half of the gs are ys’, ‘At least half of the gs are ys’. They are abbreviated as not all(g, y), all(g, y), some(g, y), no(g, y), most(g, y), at most half of the(g, y), fewer than half of the(g, y), at least half of the(g, y), respectively. And they can be denoted by Proposition O, A, I, E, M, H, F, S, respectively.

Example 1: Major premise: No penguins are flying animals. Minor premise: Most penguins are animals living in Antarctica. Conclusion: Not all animals living in Antarctica are flying animals.

Let t, y, and g be variables representing a penguin, a flying animal and an animal living in the domain, respectively. Thus, this syllogism can be formalized as no (t, y)  most (t, g) ® not all (g, y) and abbreviated as EMO-3. In fact, there are countless instances of natural language corresponding to the generalized syllogism EMO-3.

Formal System of Generalized Syllogisms

The formal system of generalized syllogisms generally includes primitive symbols, basic axioms, formation rules, deductive rules, etc.

Primitive Symbols

Brackets: (, ) operators: ¬, ® quantifiers: not all, most lexical variables: g, t, y

Basic Axioms

A1: If b is a valid formula in first-order logic, then ⊢b. A2: ⊢no(t, y)∧most(t, g)®not all(g, y) (namely, the syllogism EMO-3).

Formation Rules

If Q is a quantifier, g and y are lexical variables, then Q(g, y) is a wff.
If b is a wff, so is Øb.
If b and l are wffs, so is b®l.
Only the formulas generated by the above three rules are wffs.

Deductive Rules

Rule 1 (subsequent weakening): From ⊢(lÙs®t) and ⊢(t®b) infer ⊢(lÙs®b). Rule 2 (antecedent strengthening): If ⊢(b®l) and ⊢(lÙs®t), then ⊢(bÙs®t). Rule 3 (antecedent strengthening): If ⊢(b®s) and ⊢(lÙs®t), then ⊢(lÙb®t). Rule 4 (anti-syllogism): From ⊢(lÙs®t) infer ⊢(¬tÙl®¬s).

Rule 5(anti-syllogism): From ⊢(lÙs®t) infer ⊢(¬tÙs®¬l).

Relevant Definitions

D1 (conjunction): (lÙs) = defØ(l®Øs); D2 (bi-condition): (l«s) = def (l®s)Ù(s®l); D3 (inner negation): (Q¬)(g, y) = def Q(g, D-y); D4 (outer negation): (¬Q)(g, y) = def It is not that Q(g, y); D5 (truth value): all (g, y) = def GÍY; D6 (truth value): some (g, y) = def G∩Y¹Æ; D7 (truth value): no (g, y) = def G∩Y=Æ; D8 (truth value): not all (g, y)=def G⊈Y; D9 (truth value): most (g, y) is true iff G∩Y>0.5G is true; D10 (truth value): at most half of the (g, y) is true iff G∩Y£0.5G; D11 (truth value): fewer than half of the (g, y) is true iffG∩Y<0.5G is true; D12 (truth value): at least half of the (g, y) is true iff G∩Y³0.5G is true.

Relevant Facts

Fact 1 (inner negation): (1.1) ⊢all(g, y)«noØ(g, y); (1.2) ⊢no(g, y)«allØ(g, y); (1.3) ⊢some(g, y)«not allØ(g, y); (1.4) ⊢not all(g, y)«someØ(g, y); (1.5) ⊢most(g, y)«fewer than half of theØ(g, y); (1.6) ⊢fewer than half of the(g, y)«mostØ(g, y); (1.7) ⊢at least half of the(g, y)«at most half of theØ(g, y); (1.8) ⊢at most half of the(g, y)«at least half of theØ(g, y).

Fact 2 (outer negation): (2.1) ⊢Øall(g, y)«not all(g, y); (2.2) ⊢Ønot all(g, y)«all(g, y); (2.3) ⊢Øno(g, y)«some(g, y); (2.4) ⊢Øsome(g, y)«no(g, y); (2.5) ⊢Ømost(g, y)«at most half of the(g, y); (2.6) ⊢Øat most half of the(g, y)«most(g, y); (2.7) ⊢Øfewer than half of the(g, y)«at least half of the(g, y); (2.8) ⊢Øat least half of the(g, y)«fewer than half of the(g, y).

Fact 3 (subordination): (3.1) ⊢all(g, y)→some(g, y); (3.2) ⊢no(g, y)→not all(g, y); (3.3) ⊢all(g, y)→most(g, y); (3.4) ⊢most(g, y)→some(g, y); (3.5) ⊢at least half of the(g, y)→some(g, y); (3.6) ⊢all(g, y)→at least half of the(g, y); (3.7) ⊢at most half of the(g, y)→not all(g, y); (3.8) ⊢fewer than half of the(g, y)→not all(g, y); (3.9) ⊢no(g, y)→fewer than half of the(g, y).

Fact 4 (symmetry): (4.1) ⊢some(g, y)«some(y, g);

(4.2) ⊢no(g, y)«no(y, g).

The above facts are elementary knowledge in generalized quantifier theory [9, 10] and first-order logic [11], so their proofs are omitted.

The Validity and Reducibility of Generalized Syllogisms Based on EMO-3

It is necessary to prove the validity of the generalized syllogism EMO-3 before discussing its reducible relationships between/among other syllogisms.

Theorem 1 (EMO-3): The generalized syllogism no(t, y)∧most(t, g)®not all(g, y) is valid.

Proof: Suppose that no(t, y) and most(t, g) are true, then T∩Y=Æ and T∩G>0.5T are true in terms of Definition D7 and D9, respectively. Thus it can be concluded that G⊈Y is true. This can be proven by reductio ad absurdum. Assuming G⊈Y is not true. That is to say, GÍY is true. Because we have obtained T∩Y=Æ, it follows that T∩G=Æ, which conflicts with T∩G>0.5T. So GÍY is not true. It means that G⊈Y is true. Hence not all(g, y) is true in virtue of Definition D8, just as expected. If the validity of one syllogism can be deduced from that of another one, it is said that there is a reducible relationship between the two syllogisms. Taking (2.1) in Theorem 2 as an example, it illustrates that the validity of generalized syllogism EMO-4 can be inferred from that of the syllogism EMO-3. In other words, there is a reducible relationship between syllogism EMO-3 and EMO-4. On these grounds, one can obtain Theorem 2 as follows.

Theorem 2: At least the following 20 valid syllogisms can be inferred from EMO-3: (2.1) ⊢EMO-3®EMO-4 (2.2) ⊢EMO-3®AEH-2 (2.3) ⊢EMO-3®AEH-2®AEH-4 (2.4) ⊢EMO-3®AMI-1 (2.5) ⊢EMO-3®AMI-1®MAI-4 (2.6) ⊢EMO-3®AMI-3 (2.7) ⊢EMO-3®AMI-3®MAI-3 (2.8) ⊢EMO-3®AEH-2®EAH-2 (2.9) ⊢EMO-3®AEH-2®EAH-2®EAH-1 (2.10) ⊢EMO-3®AEH-2®EAH-2®EMO-1 (2.11) ⊢EMO-3®AEH-2®EAH-2®EMO-1®EMO-2 (2.12) ⊢EMO-3®AEH-2®EAH-2®EAH-1®AAS-1 (2.13) ⊢EMO-3®AEH-2®EAH-2®EAH-1®AAS-1®AFO-2 (2.14) ⊢EMO-3®AEH-2®EAH-2®EAH-1®AAS-1®FAO-3 (2.15) ⊢EMO-3®AEH-2®AEO-2 (2.16) ⊢EMO-3®AEH-2®AEH-4®AEO-4 (2.17) ⊢EMO-3®AEH-2®EAH-2®EAO-2 (2.18) ⊢EMO-3®AEH-2®EAH-2®EAH-1®EAO-1 (2.19) ⊢EMO-3®AEH-2®EAH-2®EAH-1®AAS-1®AAI-1 (2.20) ⊢EMO-3®AMI-1®MAI-4®AAI-4 Proof:

[1] ⊢no(t, y)Ùmost(t, g)®not all(g, y) (i.e. EMO-3, Axiom A2) [2] ⊢no(y, t)Ùmost(t, g)®not all(g, y) (i.e. EMO-4, by [1] and Fact (4.2)) [3] ⊢Ønot all(g, y)Ùno(t, y)®Ømost(t, g) (by [1] and Rule 4) [4] ⊢all(g, y)Ùno(t, y)®at most half of the(t, g) (i.e. AEH-2, by [3], Fact (2.2) and (2.5)) [5] ⊢all(g, y)Ùno(y, t)®at most half of the(t, g) (i.e. AEH-4, by [4] and Fact (4.2)) [6] ⊢Ønot all(g, y)Ùmost(t, g)®Øno(t, y) (by [1] and Rule 5) [7] ⊢all(g, y)Ùmost(t, g)®some(t, y) (i.e. AMI-1, by [6], Fact (2.2) and (2.3)) [8] ⊢all(g, y)Ùmost(t, g)®some(y, t) (i.e. MAI-4, by [7] and Fact (4.1)) [9] ⊢allØ(t, y)Ùmost(t, g)®someØ(g, y) (by [1], Fact (1.2) and (1.4)) [10] ⊢all(t, D-y)Ùmost(t, g)®some(g, D-y) (i.e. AMI-3, by [9] and D3) [11] ⊢all(t, D-y)Ùmost(t, g)®some(D-y, g) (i.e. MAI-3, by [10] and Fact (4.1)) [12] ⊢noØ(g, y)ÙallØ(t, y)®at most half of the(t, g) (by [4], Fact (1.1) and (1.2)) [13] ⊢no(g, D-y)Ùall(t, D-y)®at most half of the(t, g) (i.e. EAH-2, by [12] and D3) [14] ⊢no(D-y, g)Ùall(t, D-y)®at most half of the(t, g) (i.e. EAH-1, by [13] and Fact (4.2)) [15] ⊢Øat most half of the(t, g)Ùno(g, D-y)®Øall(t, D-y) (by [13] and Rule 4) [16] ⊢most(t, g)Ùno(g, D-y)®not all(t, D-y) (i.e. EMO-1, by [15], Fact (2.1) and (2.6)) [17] ⊢most(t, g)Ùno(D-y, g)®not all(t, D-y) (i.e. EMO-2, by [16] and Fact (4.2)) [18] ⊢allØ(D-y, g)Ùall(t, D-y)®at least half of theØ(t, g) (by [14], Fact (1.2) and (1.8)) [19] ⊢all(D-y, D-g)Ùall(t, D-y)®at least half of the(t, D-g) (i.e. AAS-1, by [18] and D3) [20] ⊢Øat least half of the(t, D-g)Ùall(D-y, D-g)®Øall(t, D-y) (by [19] and Rule 4) [21] ⊢fewer than half of the(t, D-g)Ùall(D-y, D-g)®not all(t, D-y) (i.e. AFO-2, by [20], Fact (2.1) and (2.8)) [22] ⊢Øat least half of the(t, D-g)Ùall(t, D-y)®Øall(D-y, D-g) (by [19] and Rule 5) [23] ⊢fewer than half of the(t, D-g)Ùall(t, D-y)®not all(D-y, D-g) (i.e. FAO-3, by [22], Fact (2.1) and (2.8)) [24] ⊢at most half of the(t, g)®not all(t, g) (by Fact (3.7)) [25] ⊢all(g, y)Ùno(t, y)®not all(t, g) (i.e. AEO-2, by [4], [24] and Rule 1) [26] ⊢all(g, y)Ùno(y, t)®not all(t, g) (i.e. AEO-4, by [5], [24]

and Rule 1) [27] ⊢no(g, D-y)Ùall(t, D-y)®not all(t, g) (i.e. EAO-2, by [13], [24] and Rule 1) [28] ⊢no(D-y, g)Ùall(t, D-y)®not all(t, g) (i.e. EAO-1, by [14], [24] and Rule 1) [29] ⊢at least half of the(t, D-g)®some(t, D-g) (by Fact (3.5)) [30] ⊢all(D-y, D-g)Ùall(t, D-y)®some(t, D-g) (i.e. AAI-1, by [19], [29] and Rule 1) [31] ⊢all(t, g)®most(t, g) (by Fact (3.3)) [32] ⊢all(g, y)Ùall(t, g)®some(y, t) (i.e. AAI-4, by [8], [31] and Rule 3) It can be seen that the above 14 valid generalized syllogisms be deduced in line with the validity of the generalized syllogism EMO-3 from Step 1 to Step 23. Due to the fact that Aristotelian syllogisms are special cases of generalized syllogisms [12], the above 6 valid Aristotelian syllogisms also can be derived from Step 24 to Step 32 in terms of subsequent weakening rule and antecedent strengthening rule. So far, a total of 20 valid syllogisms have been obtained from the syllogism EMO-3. If one similarly continues the above reductive operations, the other valid syllogisms can be deduced from the syllogism EMO-3.

Conclusion and Future Work

To explore the reducibility of non-trivial generalized syllogisms with the quantifiers in Square{not all} and Square{most}, this paper first gives the formalization of generalized syllogisms on the basis of set theory, and then proves the validity of the generalized syllogism EMO-3 by first-order logic and generalized quantifier theory; Finally, with the help of some reductive operations, the other 20 valid generalized syllogisms are deduced from the syllogism EMO-3. In other words, there are the reducible relationships between/among the above 21 valid syllogisms. The reason for this is that any quantifier in a square can define the other three quantifiers. This research contributes to breaking through the existing research paradigms in linguistics and logic, integrating the latest research methods and findings, and providing macro-level ideas and concrete approaches for constructing a truly automated natural language processing system, as well as enhancing interpersonal dialogue and human-computer interaction. Any quantifier and its three negation quantifiers can form a modern square. Can the research method in this paper be applied to study other generalized syllogisms with quantifiers in other squares? These questions are meaningful and worthy of further exploration.

Acknowledgement

This work was supported by the Quality Engineering

Project of Anhui Medical University under Grant No. 2022xjjyxm03.

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