Aristotle's Modal Syllogisms

Johnson, Fred

doi:10.1016/s1874-5857(04)80006-2

Cited by 13 publications

(11 citation statements)

References 23 publications

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“…Thomson (1993Thomson ( , 1997) also failed to provide a consistent explanation for Aristotelian modal syllogistic. Johnson (2004) and Malink (2013) provided anti-models for some invalid Aristotelian modal syllogisms. Zhang (2019Zhang ( , 2020 conducted a formal study of Aristotelian modal syllogisms from the perspective of modern logic.…”

Section: Introductionmentioning

confidence: 99%

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

Zhang¹,

Zhang²

2023

OJPP

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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.

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Section: Introductionmentioning

confidence: 99%

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

Zhang¹,

Zhang²

2023

OJPP

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“…There are many kinds of syllogisms in natural language, such as Aristotelian syllogisms (Hui, 2023), generalized syllogisms (Xiaojun, 2020a), Aristotelian modal syllogisms (Johnson, 2004), generalized modal syllogisms, and so on. This paper shall restrict attention entirely to how to obtain valid generalized modal syllogisms from valid generalized syllogisms.…”

Section: Introductionmentioning

confidence: 99%

How to Obtain Valid Generalized Modal Syllogisms from Valid Generalized Syllogisms

Xu¹,

Zhang²

2023

ASIR

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Making full use of the truth value definitions of sentences with quantification, possible world semantics and/or fuzzy logic, one can prove the validity of generalized modal syllogisms. This paper shows that the proof of the validity of a generalized modal syllogism can be transformed into that of its corresponding generalized syllogism, and that the generalized syllogism obtained by removing all modalities in any valid generalized modal syllogism is still valid. Therefore, the simplest way to screen out valid generalized modal syllogisms is to add modalities to valid generalized syllogisms, and then to delete all invalid syllogisms by means of the basic rules with which valid generalized modal syllogisms should meet. And then the remainders are valid. This paper illustrates how to obtain 12 valid generalized modal syllogisms by adding necessary modalities and/or possible modalities to any valid generalized syllogism. The two kinds of syllogisms discussed in this paper are composed of sentences with quantification which is the largest number of sentences in natural language. Hence, this innovative research can provide theoretical support for linguistics, logic, artificial intelligence, and among other fields.

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“…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

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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

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Aristotle's Modal Syllogisms

Cited by 13 publications

References 23 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

How to Obtain Valid Generalized Modal Syllogisms from Valid Generalized Syllogisms

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

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Aristotle's Modal Syllogisms

Cited by 13 publications

References 23 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

How to Obtain Valid Generalized Modal Syllogisms from Valid Generalized Syllogisms

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

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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