  Going beyond his basic syllogistic, Aristotle reduced the 3rd and 4th first figure syllogisms to second figure syllogisms, thus de facto reducing all syllogisms to Barbara and Celarent; and later on in the Prior Analytics he invokes a type of cut-rule by which a multi-premise syllogism can be reduced to two or more basic syllogisms. From a modern perspective, Aristotle’s system can be understood as a sequent logic in the style of natural deduction and as a fragment of first-order logic. It has been shown to be sound and complete if one interprets the relations expressed by the categorical sentences set-theoretically as a system of non-empty classes as follows: AaB is true if and only if the class A contains the class B. AeB is true if and only if the classes A and B are disjoint. AiB is true if and only if the classes A and B are not disjoint. AoB is true if and only if the class A does not contain the class B. It is generally agreed, though, that Aristotle’s syllogistic is a kind of relevance logic rather than classical. The vexing textual question what exactly Aristotle meant by ‘syllogisms’ has received several rival interpretations, including one that they are a certain type of conditional propositional form. Most plausibly, perhaps, Aristotle’s complete and incomplete syllogisms taken together should be understood as formally valid premise-conclusion arguments; and his complete and completed syllogisms taken together as (sound) deductions.