You define x = y by
(φ).φx ≡ φy . Q(x,y)
The ground of this definition should be that Q(x,y) is a tautology whenever “x” and “y” have the same meaning, and a contradiction, when they have different meanings.
     I will now try to show that this definition won't do nor any other that tries to make x = y a tautology or a contradiction.
     It is clear that Q(x,y) is a logical product. Let “a” and “b” be two names having different meanings. Then amongst the members of our product there will be one such that f(a) means p and f(b) means ~p. Let me call such a function a critical function fk. Now although we know that “a” and “b” have different meanings, still to say that they have not, cannot be nonsensical. For if it were, the negative proposition, i.e. that they have the same meaning, would be nonsensical too, for the negation of nonsense is nonsense. Now let us suppose, wrongly, that a = b, then, by
2)
substituting a for b in our logical product the critical function fk(a) becomes nonsensical (ambiguous) and, consequently, the whole product, too. On the other hand, let “c” and “d” be two names having the same meaning, then it is quite true that Q(c, d) becomes a tautology. But suppose now (wrongly) c ≠ d . Q(c, d) is a tautology still, for there is no critical function in our product. And even if it could be supposed (which it cannot) that c = d, surely a critical function fk (such that fk(c) means p fk(d) means ~p) cannot be supposed to exist, because this sign becomes meaningless. Therefore, if x = y were a tautology or a contradiction and correctly defined byQ(x,y), Q(a, b) would not be contradictory, but nonsensical (as this supposition, if it were the supposition that “a” and “b” had the same meaning, would make the critical function nonsensical). And therefore ~Q(a, b) would be nonsensical too, for the negation of nonsense is nonsense.
     In the case of c and d Q(c, d) remains tautologous, even if c and d could be supposed to be different (for in this case a critical function cannot be supposed to exist).
     The way out of all these troubles is to see that neither Q(x,y), also || although it is a very interesting function, nor any propositional function whatever, can be substituted for x = y.



   
     Your mistake becomes still clearer in its consequences; viz. when you try to say “there is an individual”. You are aware of the fact that the supposition of there being no individual makes
(x) . x = x E
“absolute nonsense”. But if E is to say “there is an individual”
3)
~E says: “there is no individual”. Therefore from ~E follows that E is nonsense. Therefore ~E must be nonsense itself, and therefore again so must be E.
     The case lies as before. E, according to your definition of the sign “ = ” may be a tautology right enough, but does not say “there is an individual”. Perhaps you will answer: of course it does not say “there is an individual” but it shows what we really mean when we say “there is an individual”. But this is not shown by E, but simply by the legitimate use of the symbol (x).. , and therefore just as well (and as badly) by the expression ~(x) . x = x. The same, of course, applies to your expressions “there are at least two individuals” and so on.