Screening out All Valid Aristotelian Modal Syllogisms

Abstract

It is easy to understand that whether a classical syllogism is valid. That whether a modal syllogism is valid is not so transparent. The prevailing view on Aristotelian modal syllogistic is that the syllogistic is incomprehensible due to its many faults and inconsistencies. Although adequate semantic analysis or reconstruction of the syllogistic have be given by many authors, it is far from obvious how to extend these results so as to consistently cover the whole modal syllogistic developed. The major aim of this paper is to overcome these difficulties, and screen out 384 Aristotelian valid modal syllogisms from 6656 Aristotelian modal syllogisms in natural language. They can be formalized by means of set theory and generalized quantifier theory, and their validity can be proved by possible world semantics and the truth definition of Aristotelian quantifiers defined in generalized quantifier theory. The basic steps of screening out all valid Aristotelian modal syllogisms are as follows: firstly one can get all possible modal syllogisms obtained by adding modal operators to 24 valid classical syllogisms, and secondly eliminate invalid modal syllogisms by characteristic rules of modal syllogisms. It is hoped that these innovative achievements will make contributions to promote the development of Aristotelian and generalized modal syllogistic, natural language information processing, and further research on knowledge representation and knowledge reasoning in computer science.

How to Deduce the Other 91 Valid Aristotelian Modal Syllogisms from the Syllogism IAI-3

“…A, E, I, O, A, E, I, O, A, E, I and O, and the middle term has four different positions (that is, four figures), there are (1212124256=) 6656 Aristotelian modal syllogisms in natural language, of which 256 represents the number of Aristotelian syllogisms. Zhang (2020b) screened out 384 valid syllogisms from 6656 Aristotelian modal syllogisms. Through the way of reduction, this paper profoundly reveals the logical relations between/among 91 valid Aristotelian modal syllogisms.…”

“…All of the above facts are the general knowledge of generalized quantifier theory (Zhang, 2018) or modal logic (Zhang, 2020b), thus their proofs are omitted here.…”

Formal System of Categorical Syllogistic Logic Based on the Syllogism <i>AEE-4</i>

“…Syllogisms are important forms of reasoning in natural language and logic from Aristotle onwards. There are various syllogisms in natural language, such as categorical syllogisms (Moss, 2008), modal syllogisms (Zhang, 2020a(Zhang, , 2020b, generalized syllogisms (Murinová & Novák, 2012), relational syllogisms (Pratt-Hartmann, 2009, syllogisms with adjectives (Moss, 2011), and so on. Among them, categorical syllogisms have a long history of research and are widely used in human reasoning (Chen, 2000).…”

“…Categorical syllogisms involve sentences of the following four forms: all xs are y, no xs are y, some xs are y, and not all xs are y. This article focuses on the time-honored categorical syllogistic logic which has been discussed from different perspectives since Aristotle, for example by Łukasiewicz (1957), Corcoran (1972), van Benthem (1984, Westerståhl (1989), Martin (1997), and Zhang (2016, 2020a, 2020b, 2021, and so on. The reason why categorical syllogistic logic is widely studied is that it is a common form of reasoning in natural languages.…”

Reduction between Categorical Syllogisms Based on the Syllogism EIO-2

“…In natural language, there are various syllogisms, such as categorical syllogisms (Łukasiewicz, 1957;Moss, 2008;Westerstå hl, 1989), generalized syllogisms (Murinová , & Nová k, 2012;Endrullis & Moss, 2015), modal syllogisms (Johnson, 2004;Zhang, 2020aZhang, , 2020b, relational syllogisms, syllogisms with verbs (Moss, 2010), and syllogisms with Boolean operations (Ivanov & Vakarelov, 2012), and so on.…”