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How did Principia Mathematica attempt to derive arithmetic from logic?

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

Principia Mathematica, authored by Bertrand Russell and Alfred North Whitehead, was a monumental effort to demonstrate the school of logicism—the thesis that all of mathematics can be derived from pure logic without relying on specifically mathematical concepts.12

To achieve this, the authors proposed defining the number of a class as the collection of all classes equinumerous with it. This definition was intended to imply the standard axioms of arithmetic, such as the Peano Postulates. Because 19th-century mathematics had largely reduced other areas of analysis to arithmetic, the authors believed that successfully grounding arithmetic in logic would effectively ground all of mathematics.2

To avoid Russell’s paradox—which arises from self-referential definitions—they implemented a theory of types. This system restricted sets to containing only entities of a lower logical type. To further address "vicious circle" paradoxes, they employed a "ramified" theory of types and introduced the axiom of reducibility to ensure the derivation of standard mathematics remained possible.12