Page:Wittgenstein - Tractatus Logico-Philosophicus, 1922.djvu/145

Rh but "$$(\exists x).f(x,x)$$"; and not "$$(\exists x,y).f(x,y).\sim x=y$$"  but "$$(\exists x,y).f(x,y)$$".

(Therefore instead of Russell's "$$(\exists x,y).f(x,y)$$": "$$(\exists x,y).f(x,y)\or(\exists x).f(x,x)$$".)

Instead of "$$(x).fx\supset x=a$$" we therefore write e.g. "$$(\exists x).fx.\supset.fa:\sim (\exists x,y).fx.fy$$"

And the proposition "only one $$x$$ satisfies $$f$$" reads: "$$(\exists x).fx:\sim (\exists x,y).fx.fy$$". The identity sign is therefore not an essential constituent of logical notation.

And we see that apparent propositions like: "$$a=a$$", "$$a=b.b=c.\supset a=c$$", "$$(x).x=x$$", "$$(\exists x).x=a$$", etc. cannot be written in a correct logical notation at all.

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

This is the place to solve all the problems which 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.

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

(It is nonsense to place the hypothesis $$p\supset p$$ before a proposition in order to ensure that its arguments have the right form, because the hypothesis for a non-proposition as argument becomes not false but meaningless, and because the proposition itself becomes senseless for  Rh