How does reduction show that a syllogism in another figure is valid?

In syllogistic logic, reduction is the method of rearranging the terms of a syllogism to express it in a different figure, typically to demonstrate its validity by showing it conforms to the self-evident patterns of the first figure.12

Methods of Reduction

Aristotle and later logicians utilized two primary approaches to prove that a syllogism in the second, third, or fourth figure is valid by reducing it to an axiomatic first-figure mood (such as Barbara):

• Direct Reduction: This involves using rules of conversion (reversing the order of terms) and occasionally changing the order of premises to transform the argument into a first-figure mood. For example, a third-figure mood like Felapton can be reduced to the first-figure mood Ferio by converting its minor premise per accidens.3
• Indirect Reduction: When a syllogism cannot be directly reduced, it is proved via reductio ad absurdum. This method assumes the contradictory of the original conclusion and uses it alongside one of the original premises to derive a contradiction of the other premise within a first-figure syllogism. This proves that the original conclusion must be true if the premises are true.34

These procedures demonstrate that the validity of all categorical syllogisms rests upon the foundational, self-evident nature of the first figure.13