Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing.
I write therefore not “f(a,b).a=b” but “f(aa)” (or “f(bb)”). And not “f(a,b).~a=b”, but “f(a,b)”.
And analogously: not “(∃x,y).f(x,y).x=y”, but “(∃x).f(x,x)”; and not “(∃x,y).f(x,y).~x=y”, but “(∃x,y).f(x,y)”.
(Therefore instead of Russell’s “(∃x,y).f(x,y)”: “(∃x,y).f(x,y).∨.(∃x).f(x,x)”.)
Instead of “(x):fx⊃x=a” we therefore write e.g. “(∃x).fx.⊃.fa:~(∃x,y).fx.fy”.
And if the proposition “only one x satisfies f()” reads: “(∃x).fx:~(∃x,y).fx.fy”.
The identity sign is therefore not an essential constituent of logical notation.
And we see that the apparent propositions like: “a=a”, “a=b.b=c.⊃a=c”, “(x).x=x”. “(∃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 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.
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 archetype Proposition, Thing, etc. So Russell in the Principles 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 occupied 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 argument 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 hypothesis attached for this purpose.)
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?
In the general propositional form, propositions occur in a proposition only as bases of the truth-operations.
At first sight it appears as if there were also a different way in which one proposition could occur in another.
Especially in certain propositional forms of psychology, like “A thinks, that p is the case,” or “A thinks p”, etc.
Here it appears superficially as if the proposition p stood to the object A in a kind of relation.
(And in modern epistemology (Russell, Moore, etc.) those propositions have been conceived in this way.)
But it is clear that “A believes that p,” “A thinks p,” “A says p,” are of the form “ ‘p’ says p”: and here we have no coordination of a fact and an object, but a coordination of facts by means of a coordination of their objects.
This shows that there is no such thing as the soul—the subject, etc.—as it is conceived in superficial psychology. A composite soul would not be a soul any longer.
The correct explanation of the form of the proposition “A judges p” must show that it is impossible to judge a nonsense. (Russell’s theory does not satisfy this condition.)
To perceive a complex means to perceive that its constituents are combined in such and such a way.
This perhaps explains that the figure
can be seen in two ways as a cube; and all similar phenomena. For we really see two different facts.