Reduction between Categorical Syllogisms Based on the Syllogism EIO-2

Li, Hui

doi:10.22158/asir.v7n1p30

Cited by 5 publications

(5 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Aristotelian syllogisms characterize the semantic and inferential properties of the following four Aristotelian quantifiers: all, no, some and not all, which are type <1, 1> quantifiers (Zhang, 2018). The proposition containing a type <1, 1> quantifier Q can be formalized into a tripartite structure like Q(S, P) (Li, 2023). In this paper, S, M and P refer to the set of objects represented by the lexical variables of categorical propositions; p, q, r and s are propositional variables.…”

Section: Preliminariesmentioning

confidence: 99%

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism &#9671;I&#9633;A&#9671;I-3

Zhang¹,

Zhang²

2023

OJPP

View full text Add to dashboard Cite

In order to provide a consistent explanation for Aristotelian modal syllogistic, this paper reveals the reductions between the Aristotelian modal syllogism IAI-3 and the other valid modal syllogisms. Specifically, on the basis of formalizing Aristotelian modal syllogisms, this paper proves the validity of IAI-3 by means of the truth value definition of (modal) categorical propositions. Then in line with classical propositional logic and modal logic, generalized quantifier theory and set theory, this paper deduces the other 47 valid Aristotelian modal syllogisms from the modal syllogism IAI-3. This study shows that the reasons why these syllogisms are reducible are: 1) any of Aristotelian quantifier can be defined by the other three Aristotelian quantifiers; 2) the Aristotelian quantifiers some and no have symmetry; 3) the possible modality  and necessary modality  can be mutually defined. This formal study of Aristotelian modal syllogistic not only conforms to the needs of formalization transformation of various information in the era of artificial intelligence, but also provides a unified mathematical research paradigm for other kinds of syllogistic.

show abstract

Section: Preliminariesmentioning

confidence: 99%

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism &#9671;I&#9633;A&#9671;I-3

Zhang¹,

Zhang²

2023

OJPP

View full text Add to dashboard Cite

show abstract

“…Aristotelian syllogisms characterize the semantic and inferential properties of the following four Aristotelian quantifiers: all, no, some and not all, which are type <1, 1> quantifiers [18]. The proposition containing a type <1, 1> quantifier Q can be formalized into a tripartite structure like Q(S, P) [19]. In this paper, S, M and P refer to the set of objects represented by the lexical variables of categorical propositions; p, q, r and s are propositional variables.…”

Section: Preliminariesmentioning

confidence: 99%

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism ◊I�A◊I-3

Zhang¹,

Zhang²

2023

J Arti Inte & Cloud Comp

View full text Add to dashboard Cite

In order to provide a consistent explanation for Aristotelian modal syllogistic, this paper reveals the reductions between the Aristotelian modal syllogism ◊I�A◊I-3 and the other valid modal syllogisms. Specifically, on the basis of formalizing Aristotelian modal syllogisms, this paper proves the validity of ◊I�A◊I-3 by means of the truth value definition of (modal) categorical propositions. Then in line with the symmetry of Aristotelian quantifiers some and no, the definition of inner and outer negations of Aristotelian quantifiers, and some rules in classical propositional logic, this paper deduces the other 47 valid Aristotelian modal syllogisms from the modal syllogism ◊I�A◊I-3. The reason why these syllogisms are reducible is that: (1) any of Aristotelian quantifier can be defined by the other three Aristotelian quantifiers; (2) the Aristotelian quantifiers some and no have symmetry; (3) the possible modality ◊ and necessary modality £ can be mutually defined. This formal study of Aristotelian modal syllogistic not only conforms to the needs of formalization transformation of various information in the era of artificial intelligence, but also provides a unified mathematical research paradigm for other kinds of syllogistic.

show abstract

“…Syllogistic reasoning is one of the important forms of reasoning in natural language and human thinking. There are various kinds of syllogisms in natural language, such as Aristotelian syllogisms (Hui, 2023), Aristotelian modal syllogisms (Johnson, 2004), generalized syllogisms (Murinová & Novák, 2012;Endrullis et al, 2015), generalized modal syllogisms, and so on. This paper shall restrict attention mainly to the reducibility of generalized modal syllogisms.…”

Section: Introductionmentioning

confidence: 99%

The Reducibility of Generalized Modal Syllogisms Based on ☐AM◊I-1

Zhang

2023

Philosophy

View full text Add to dashboard Cite

There is the reducibility between the generalized modal syllogism AMI-1 and the other 20 valid generalized modal syllogisms. This paper first proves the validity of the generalized modal syllogism based on the truth value definitions of sentences with quantification, set theory and modal logic, then derives the other 20 valid generalized modal syllogisms from the syllogism  AM  I-1 in line with some facts and inference rules. The reason why these syllogisms are reducible is that: (1) any of the Aristotelian quantifiers can be defined by the other three Aristotelian quantifiers; (2) any of the four generalized quantifiers in this paper (that is, most, at most half of the, fewer than half of the and at least half of the) can be defined by the other three generalized quantifiers; (3) the Aristotelian quantifiers some and no have symmetry; (4) a necessary modality  and a possible modality  can be mutually defined.And the process of these reductions are ultimately presented in a structured formalization way.Thus, this paper provides a fragmentary research approach for other generalized modal syllogisms including four generalized quantifiers with transformation relations. There are many generalized modal syllogisms in natural language. Therefore, this study has practical

show abstract

Reduction between Categorical Syllogisms Based on the Syllogism EIO-2

Cited by 5 publications

References 7 publications

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism &#9671;I&#9633;A&#9671;I-3

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism &#9671;I&#9633;A&#9671;I-3

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism ◊I�A◊I-3

The Reducibility of Generalized Modal Syllogisms Based on ☐AM◊I-1

Contact Info

Product

Resources

About

Reduction between Categorical Syllogisms Based on the Syllogism EIO-2

Cited by 5 publications

References 7 publications

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism &amp;#9671;I&amp;#9633;A&amp;#9671;I-3

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism &amp;#9671;I&amp;#9633;A&amp;#9671;I-3

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism ◊I�A◊I-3

The Reducibility of Generalized Modal Syllogisms Based on ☐AM◊I-1

Contact Info

Product

Resources

About

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism ◇I□A◇I-3

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism ◇I□A◇I-3