Research Article

Reducible Relationships between/among Aristotelian Modal Syllogisms Based on the Syllogism AAI-1

Jing Xu¹ and Long Wei²*

¹School of Marxism, Anhui Medical University, Hefei, China

²Department of Philosophy(Zhuhai), Sun Yat-Sen University, Zhu Hai, China

Submission: October 14, 2024; Published: November 27, 2024

*Corresponding author:Long Wei, Department of Philosophy (Zhuhai), Sun Yat-Sen University, Zhu Hai, China, 657703460@qq.com

How to cite this article:Jing Xu, Long W. Reducible Relationships between/among Aristotelian Modal Syllogisms Based on the Syllogism AAI −1 .Ann Soc Sci Manage Stud. 2024; 11(1): 555805. DOI: 10.19080/ASM.2024.11.555805

Abstract

In order to explore the reducible relationships between/among Aristotelian modal syllogisms, this paper first formally represents the syllogism AAI-1 and proves its validity in line with truth value definitions of (modal) propositions and some facts, then deduces the other 39 valid modal syllogisms form AAI-1by some reducible operations. The reason why different valid Aristotelian modal syllogisms are reducible is that there are transformable relations among four Aristotelian quantifiers, and that the modality  and ◇ can be mutually defined. This paper undoubtedly provides reasoning paradigm for studying other types of syllogisms.

Keywords:Aristotelian Modal Syllogisms; Aristotelian Quantifiers; Modalities; Reducible

Introduction

An Aristotelian modal syllogism can be regarded as an extension syllogism by adding at least one necessary modality () or/and possible modality (◇)to an Aristotelian syllogism [1]. Since two premises and one conclusion of an Aristotelian modal syllogism can be any of the following twelve propositions: Proposition A, E, I, O, A, E, I, O, ◇A, ◇E, ◇I and◇O. And the middle term of a syllogism has four different positions, thus there are (12*12*12*4-256=)6656 non-trivial Aristotelian modal syllogisms, where 256 is the number of Aristotelian syllogisms [2]. However, how to screen out valid Aristotelian modal syllogisms from the 6656 syllogisms?

This paper is inspired by previous works on Aristotelian syllogisms [3, 4] and Aristotelian modal syllogisms [5, 6]. This study attempts to derive other valid Aristotelian modal syllogisms based on the syllogism AAI-1, so as to explore reducible relationships between/among Aristotelian modal syllogisms.

Preliminary

For simplicity, we shall use Qas any of the four Aristotelian quantifiers (i.e. all, some, no and not all), and Q¬as the outer negative quantifier of Q and Q¬as the inner one. The letters , and dgh are lexical variables in propositions. The sets composed of the three variables are respectively D, G, and H, and U is the domain of a variable. Let ,,pqr and s be well-formed formulas (generally abbreviated to wffs). ‘p’ means that p is provable. The symbol ‘p=_defq’ means that p is defined by q. ‘iff’ is short for ‘if and only if’.

The propositions in Aristotelian syllogisms are: ‘All ds are hs’, ‘No ds are hs’, ‘Some ds are hs’, ‘Not all ds are hs’, namely Proposition A, E, I, O. And they can be formalized as tripartite structures: all(d, h), no(d, h), some(d, h), not all(d, h), respectively. The figures of syllogisms are defined as usual [7].

A non-trivial Aristotelian modal syllogism can be obtained by adding at least one modality (necessary modality  or/and possible modality ◇ ) to an Aristotelian syllogism. It can be illustrated by the following example:
i. Major premise: All female mammals give birth to offspring.
ii. Minor premise: All female gorillas are necessarily female mammals.
iii. Conclusion: Some female gorillas give birth to offspring.

Let g be a variable representing a female mammal in the domain, h be a variable representing an animal capable of giving birth to offspring in the domain, and d be a variable representing a female gorilla in the domain. So this syllogism can be formalized as all(g,h)∧all(d,g) → some(d,h), and AAI-1for short.

Where: e represents the base of the natural logarithm and equals 2.71828, B_1represents the annual growth rate, and is obtained through the first differentiation of the model, then dividing by Y as follows [8]:

Formal System of Aristotelian Modal Syllogistic

In order to establish a formal system of Aristotelian modal syllogistic, the following initial symbols, formation rules, relevant definitions, facts and reasoning rules are needed:

Initial Symbols

i. lexical variables: d, g, h
ii. quantifier: all
iii. modality: 
iv. unary negative operator: ¬
v. binary implication operator: →
vi. brackets: (, )

Formation Rules

i. If Q is a quantifier, d and h are lexical variables, then Q(d, h) is a wff.
ii. If p is a wff, then ¬p is a wff.
iii. If p is a wff, then p is a wff.
iv. If p and q are wffs, then p → q is a wff.
v. The set of all wffs is generated from (i) to (iv).

Relevant Reasoning Rules

Rule 1 (subsequent weakening): If  ( p ∧ q → r) and  (r → s), then  ( p ∧ q → s).
Rule 2 (anti-syllogism): If  ( p ∧ q → r), then  (¬r ∧ p → ¬q).
Rule 3 (anti-syllogism): If  ( p ∧ q → r), then  (¬r ∧ p → ¬p).

Relevant Definitions

Definition 1 (bi-conditional): ( p ↔ q) =def ( p → q) ∧ (q → p);
Definition 2 (conjunction): ( p ∧ q) =def ¬( p → ¬q);
Definition 3 (inner negation): Q¬(d, h) =def Q(d,U − h);
Definition 4 (outer negation): ¬Q(d, h) =def It is not that Q(d, h);
Definition 5 (possibility): ◇p =_def ¬¬p;
Definition 6 (truth value definition): all(d, h) is true iff D ⊆ H is true in any real world;
Definition 7 (truth value definition): all(d, h) is true iff D ⊆ H is true in any possible world.

Relevant Facts

According to Definition 3 and Definition 4, Fact 1 and Fact 2 can be obtained respectively. Fact 1 (inner negation):
(1.1)  all (d, h) ↔ no¬(d, h);
(1.2)  no (d, h) ↔ all¬(d, h);
(1.3)  some (d, h) ↔ not all¬(d, h);
(1.4)  not all (d, h) ↔ some¬(d, h).
Fact 2 (outer negation):
(2.1)  ¬not all (d, h) ↔ all (d, h);
(2.2)  ¬all (d, h) ↔ not all (d, h);
(2.3)  ¬no (d, h) ↔ some (d, h);
(2.4)  ¬some (d, h) ↔ no (d, h).

The following Fact 3 to Fact 8 can be obtained in the light of modal logic [8] or possible world semantics [9] or generalized quantifier theory [10, 11]. So their proofs are omitted here.
Fact 3 (dual):
(3.1) ¬Q(d, h) ↔◇¬Q(d, h);
(3.2) ¬◇Q(d, h) ↔¬Q(d, h).
Fact 4: Q(d, h) →Q(d, h).
Fact 5: Q(d, h) →◇Q(d, h).
Fact 6:  Q(d, h) →◇Q(d, h).
Fact 7:
(7.1)  all (d, h) → some (d, h);
(7.2)  no (d, h) → not all (d, h).
Fact 8 (symmetry of some and no):
(8.1)  some (d, h) ↔ some (h, d );
(8.2)  no (d, h) ↔ no (h, d ).

Reduction between the Modal Syllogism AAI −1and Other Valid Modal Syllogisms

The following Theorem 1 proves that the Aristotelian modal syllogism AAI-1 is valid. Theorem 2 says that the other 39 valid syllogisms can be deduced from AAI-1. For example, in the Theorem (2.1),  AAI-1→AAI-4 ’ means that the validity of the syllogism AAI-4 can be inferred from that of the syllogism AAI-1. Thus, there is a reducible relationship between the two syllogisms. Other cases are similar.

Theorem 1 (AAI-1) : The syllogism all(g,h) ∧all(d, g)→some(d, h) is valid.

Proof: Assume that all(g, h) and all(d, g) are true, then G ⊆ H is true in any real world and D ⊆ G is true in any possible world according to Definition 6 and Definition 7, respectively. Because all real worlds are possible worlds. Now it follows that D ⊆ H is true in any real world. So all(d, h) is true in virtue of Definition 6. Then some(d, h) is true in terms of Fact (7.1). Hence, the syllogism all(g,h) ∧all(d, g)→some(d, h) is valid, just as desired.

Theorem 2: The following 39 valid modal syllogisms can be deduced from the syllogism AAI-1:

Proof:

At this point, the above 39 valid Aristotelian modal syllogisms are derived from the syllogism AAI −1by means of some definitions, reasoning rules and facts, etc. This proof process illustrates that there are reducibility between/among these 40 syllogisms. More valid modal syllogisms can be similarly obtained from the syllogism AAI −1.

Conclusion and Future Work

This paper includes the following works: (1) Formally representing Aristotelian modal syllogisms. (2) Proving the validity of the Aristotelian modal syllogism AAI −1in line with the truth value definitions of propositions and some facts. (3) Deducing the other 39 valid modal syllogisms form AAI −1by the symmetry of Aristotelian quantifiers some and no, the duality of  and◇, inner and outer negation of Aristotelian quantifiers, and so on. All in all, the reason why different Aristotelian modal syllogisms are reducible is that there are transformable relations among four Aristotelian quantifiers, and that the necessary modality and the possible one can be mutually defined.

Can one similarly derive other valid syllogisms from any valid Aristotelian modal syllogism, such as Can the method be applied to study other types of syllogisms, such as relational syllogisms, generalized modal syllogisms? These questions are worthy of in-depth research.

Acknowledgement

This work was supported by Humanities and Social Sciences Research Project of the Chinese Ministry of Education under Grant No.19YJC740123.