 The resulting system is complete, in the sense that all and only truth-functionally valid sentences are derivable in the system. This despite the seeming defects of the system by modern standards, including the redundancy of one of the axioms, the use of defined symbols in expressions to which the rules of inference apply, and the use of defined symbols in the axioms. The derivations in ∗2 to ∗5 are abbreviated, but with an indication on the side of each line of what justifies it, and how any abbreviation can be undone. Theorems are proved primarily as needed in later numbers, but some were axioms, or important theorems of earlier versions of propositional logic, going back to The Principles of Mathematics. Aside from historical interest in their actual choices, however, the system of PM can be viewed as based on any standard system of propositional logic.