Define statements A and B:
A: “this statement” is Y
B: “this statement” is X
Assume I may combine statements using the logical operator “and”.
Define statement C:
C: A and B. (Meaning “C asserts A and B are both true”)
Replace the definitions of A and B into C:
C: (“this statement” is Y) and (“this statement” is X).
If “this statement” refers to “the current statement”, then "C is Y" and "C is X".
If Y and X were mutually exclusive, this contradicts: C is simultaneously something and its negation. In other words, "self-reference" and the the logical operator "and" can be problematic if the selves are distinct.
By limiting self-reference to the original definition you avoid that problem. In other words, disallow the notion of “the current statement” and only allow explicitly named labels. We could have defined A and B earlier as:
A: A is X
B: B is Y
Then, if X and Y are mutually-exclusive, C, having no claim of X or Y regarding itself, is consistent, since C asserts about A and B, not itself:
C: (A is X) and (B is Y)
Alternatively, by defining all self-reference qualities (Y and X earlier) to be equivalent, you also avoid the problem. (Meaning since X and Y are both self-referential qualities, then they must be the same thing.) “There’s only a single self” sort of thing. But then you cannot negate the self (which is a bit strange, now you have two types of truths, those that can be negated and those that cannot).
Another way to resolve the contradiction: the selves could relate to one another as a tree. Given that, when you assert X "and" Y in C from the example earlier, the "and" could be interpretted as intersect-ing (or unioning) parts of the tree. When you negate it, you are inverting the vertices in the set. Of course then the "self" is not a statement that is true or false, but a graph. So, perhaps, in general, if the "self" is not a true/false statement but some other mathematical object (such as a graph), then no contradiction.
How is self-reference usually handled in math?
A rigorous treatment of what you're doing needs a metalanguage to describe statements in an "object language". It can be shown that consistency of the object language is equivalent (if it's sufficiently rich) to its lacking a truth predicate for its own statements, so defining a uniform notion of truth has to be done in the metalanguage instead.