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Why did Russell and Whitehead introduce the axiom of reducibility in Principia Mathematica?

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logic: universal affirmative — Modern logic Representations of the universal affirmative, “All A's are B's” in modern logic.

In Principia Mathematica, Bertrand Russell and Alfred North Whitehead introduced the axiom of reducibility to address the limitations of their "ramified" theory of types.1

To avoid logical paradoxes, such as Russell’s paradox, they implemented a strict hierarchy of types to prevent "impredicative" definitions—those that define an object by referencing a collection that includes the object itself. However, this rigid structure proved too restrictive to derive the full scope of classical mathematics, particularly analysis.12

The axiom of reducibility was introduced as a necessary, albeit controversial, assumption to imply a partial collapse of this ramified hierarchy. By asserting that every higher-order function or property is equivalent to a simpler, predicative one, the axiom allowed Russell and Whitehead to recover the mathematical power required for their system to function, effectively rendering their enterprise impredicative despite their initial efforts to avoid such constructions.12