Deductive Reasoning Based on the Valid Aristotelian Syllogism AEE-2

This paper firstly formalizes Aristotelian syllogisms based on the tripartite structure of categorical propositions, and then uses the truth definitions of categorical propositions to prove the validity of the Aristotelian syllogism AEE-2. Then, the remaining 23 valid syllogisms are derived from the syllogism AEE-2 with the help of relevant facts, inner and outer negation definitions of quantifiers, and deductive rules. In other words, this paper reveals the reducible relationship between/among these 24 syllogisms and establishes a succinct formal reason system for Aristotelian syllogistic. The deductive reasoning not only ensures consistency in its results, but also provides a concise mathematical paradigm for other types of syllogisms.