Syllogistic reasoning is important due to the prominence of syllogistic arguments in human reasoning, and also to the role they have played in theory of reasoning from Aristotle onwards. Aristotelian syllogistic logic is a formal study of the meaning of four Aristotelian quantifiers and of their properties. This paper focuses on logical system based on syllogistic reasoning. It firstly formalized the 24 valid Aristotle' syllogisms, and then has proven that the other 22 valid Aristotle's syllogisms can be derived from the syllogisms 'Barbara' AAA-1 and 'Celarent' EAE-1 by means of generalized quantifier theory and set theory, so the paper has completed the axiomatization of Aristotelian syllogistic Logic. This axiomatization needs to make full use of symmetry and transformable relations between/among the monotonicity of the four Aristotelian quantifiers from the perspective of generalized quantifier theory. In fact, these innovative achievements and the method in this paper provide a simple and reasonable mathematical model for studying other generalized syllogisms. It is hoped that the present study will make contributions to the development of generalized quantifier theory, and to bringing about consequences to natural language information processing as well as knowledge representation and reasoning in computer science.
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