Reduction between the Syllogism OAO-3 and the Remaining 23 Valid Syllogisms

Abstract

With the help of the definitions of three negative quantifiers of Aristotelian quantifiers (i.e. all, no, some and not all), the symmetry of no and some, and some basic inference rules in propositional logic, one can deduce the remaining 23 valid syllogisms only from the syllogism OAO-3. In other words, there is reducible relations between/among different forms and different figures of valid traditional syllogisms. And these reducible relations actually reflect the transformation relations between the monotonicity of the four Aristotelian quantifiers. This paper provides a computational level of reasoning for syllogistic logic and an important theoretical basis for knowledge representation and knowledge reasoning in computers.

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.…”