ABSTRACT
In showing that an axiom is non-independent, it is clearly essential that one should not make use of the axiom itself or of any thesis obtained from it, since either course would make the proof circular. Inspection of the proof of T56 will show that these conditions are satisfied. Since A4 is non-independent, the axiom-set for Principia Mathematica (PM) could be reduced by one. But no further reduction of this sort is possible; neither Al nor A2 nor A3 nor A5 is a consequence of the other three under Substitution and Detachment, and these four are therefore said to be independent axioms. The authors need some method for showing that the axiom cannot be a transform of the other axioms under the Rules of Transformation of the system. In proving the independence of the remaining axioms of PM, they shall always choose as the crucial property preserved by the rules some special case of Φ-validity.
