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5.535

So all problems disappear which are connected with such pseudo-propositions.

This is the place to solve all the problems with arise through Russell's "Axiom of Infinity".

What the axiom of infinity is meant to say would be expressed in language by the fact that there is an infinite number of names with different meanings.

5.5351

There are certain cases in which one is tempted to use expressions of the form "a=a" or "p  p" As, for instance, when one would speak of the archtype Proposition, Thing, etc. So Russell in the Princples of Mathematics has rendered the nonsense "p is a proposition" in symbols by "p  p" and has put it as hypothesis before certain propositions to show that their places for arguments could only be occurpied by propositions.

(It is nonsense to place the hypothesis p  p before a proposition in order to ensure that its arguments have the right form, because the hypotheses for a non-proposition as arugment becomes not false but meaningless, and because the proposition itself becomes senseless for arguments of the wrong kind, and therefore it survives the wrong arguments no better and no worse than the senseless hypthesis attached for this purpose.)

5.5352

Similarly it was proposed to express "There are no things" by "~(x) . x=x". But even if this were a proposition -- would it not be true if indeed "There were things", but these were not identical with themselves?