Whitehead's (Badly) Emended Principia

摘要

There are many wonderful puzzles concerning Principia Mathematica, but none are more striking than those arising from the crisis that befell Whitehead in November of 1910. Volume 1 appeared in December of 1910. Volume 2 on cardinal numbers and Russell's relation arithmetic might have appeared in 1911 but for Whitehead's having halted the printing. He discovered that inferences involving the typically ambiguous notation 'Nc'alpha' for the cardinal number of alpha (i.e. the class of all classes similar with alpha) might generate fallacies. When the volume appeared in 1912, it was extensively emended by Whitehead and accompanied by a Prefatory Statement of Symbolic Conventions. This paper endeavors to recover from Whitehead's bad emendations-including his bewildering thesis that since 'alpha is an element of Nc'alpha' is 'true whenever significant,' proves alpha is an element of Nc'alpha a is to be accepted. It is supposedly a fallacy to apply Modus Ponens and infer proves there exists! Nc 'alpha from proves alpha is an element of Nc'alpha and proves alpha is an element of. Nc'alpha superset of there exists! Nc'alpha.