Why are Baroco and Bocardo proved through reductio ad absurdum?

In the study of syllogistic logic, the moods Baroco and Bocardo are considered exceptional because they cannot be reduced to the axiomatic first-figure moods through direct methods like conversion or the reordering of premises. Instead, they must be proved indirectly using reductio ad absurdum.1

The process for these two moods involves the following steps:

• Assumption: One assumes the truth of the premises of the syllogism and the contradictory of its conclusion.1
• Construction: These are used to form a new syllogism in Barbara (a first-figure mood).1
• Contradiction: The conclusion of this new Barbara syllogism will contradict one of the original premises of the Baroco or Bocardo mood.1
• Conclusion: Because the resulting contradiction proves that the assumption (the contradictory of the original conclusion) must be false, it follows by logical necessity that the original conclusion is true.1

This method of indirect reduction demonstrates that, given the validity of Barbara, the premises of Baroco and Bocardo validly imply their respective conclusions.1