On the thin theory, my existence is my identity-with-something. It follows that my nonexistence is my diversity-from-everything, and my merely possible nonexistence is my diversity from everything in one or more merely possible worlds. But -- and this I take it is Leo's point -- I needn't exist in merely possible world w for it to be true in w that I am diverse from everything in w. So w is not a world in which I am self-diverse, but simply a world in which I am diverse from everything in w. Had w been actual, I would not have been self-diverse; I would not have existed at all, i.e., I would not have been identical to any of the things that would have existed had w been actual.
To put it another way, on the thin theory, my actual existence is my self-identity, my identity with me. Opposing this reduction of singular existence to self-identity, I argued that if my existence is my self-identity, then the possibility of my nonexistence is the possibility of my being self-diverse -- which is absurd. Mollica's rejoinder in effect was that my possible nonexistence is not my possible self-diversity, but my possible diversity from everything distinct from me.
I could respond by saying that this objection begs the question by assuming the thin theory. But then Mollica could say that I am begging the question against him. Let me try a different tack.
If I am diverse from everything in w, but I don't exist in w, then something must represent me there. For part of what makes w w is that it lacks me. It is essential to w that it not contain me. But how express this fact if there is no representative of me in w? Now the only possible candifdate for a representative of me in possible worlds in which I do notr exist is my haecceity-property: identity-with-BV. If there is such a property, then it can go proxy for me in every possible world in which I do not exist, worlds which in part are defined by my nonexistence.
So it seems that Mollica's objection requires that there be haecceities such as identity-with-BV, and that these be properties that can exist unexemplified. But now two points.
First, there are no such haecceity properties for reasons given elsewhere, for example, here.
Second, if haecceities are brought into the picture, then we are back to the Fregean version of the thin theory according to which 'exists(s)' is a second-level property. But what I have been pounding on is the latest and most sophisticated version of the thin theory, that of van Inwagen. And we have seen that he rejects the view that 'exist(s)' is second-level.