[I am trying to understand the historical process of math, and also the principles in a larger scope than just learning the axioms. So I think this question is a good exercise in that direction.]
I will put a computational analogy here just to avoid jumping straight to $a \rlap {~\small+} \in A \iff a \in A, a \notin A$, and hopefully clarify my question.
Informally, if we allow $R = \{ x \mid x \not \in x \}$ to be understood as a process of building $R$ (or an algorithm in some sense), every set could be matched against the criterion sequentially, one at a time. When it comes to $R$ itself, up to that moment it qualifies to be included. After inclusion, it violates the criterion. So it should be removed. At that point, the process stay indefinitely cycling between inclusion and removal.
If such process is allowed to be run in parallel, i.e., all sets are matched against the criterion at once, we promptly get the finished $R$, except for $R$ as an element of itself, whose presence is undecidable. If we accept this as part of the life, $R$ would "exist" in $R$ with ambivalence - it would be "semi-contained" by $R$.
Therefore, $\rlap {~\small+} \in$ would be given by:
$a \rlap {~\small+} \in A \iff a \in A, a \notin A$
I suppose problems emerge when redefining cardinality... or the implications for the generality of this.