A Fressellian replies again

The 'Fressellian' treatment of existence is a topic I keep coming back to, ever since I first encountered an earlier incarnation of Bill Vallicella's blog some years ago.  Somewhere I still have a draft of a response I wrote to one of the online papers I found at Bill's site.  I soon realised that Existence was Bill's subject and it would have been quite presumptuous for a rank amateur to be sending him unsolicited emails.  Since then Bill has been kind enough to respond to some of the comments I have made on his postings (though we often disagree), which encourages me to think that I have learned something about how to philosophize.  

Bill comments on my suggestion here that we can translate 'something exists' into the language of the predicate calculus by means of '∃x. Object(x)' where Object is the concept at the root of the Porphyrian tree of classification concepts.  Bill recounts how Frege offers a similar strategy using Self-identity as the  genus generalissimum, but this, he thinks, is open to a number of objections.  To say of something that it exists is not to say that it is self-identical, for if it ceases to exist does it cease to be self-identical? And there is a modal objection:  something might not have existed, but might it have been non-self-identical?  Bill generously links my name with Frege's on this idea but I would want to distance myself from Frege's use of Self-identity, since identity seems to me to be a linguistic relation over names rather than a relation over objects, and I'm not so sure that Bill's strictures against Self-identity as highest genus apply so well against Object.  If something ceases to exist it ceases to be an object, and if it might not have been in existence it might not have been an object, or so it seems to me.

Bill's challenge comes in two parts: first, to express 'something exists'; second, to reveal a property 'whose instantiation is the existence of something'.  He says he takes me to be saying something like the following, in his words,  'Just as the existence of cats is the being-instantiated of the concept cat, the existence of something is the being-instantiated of the concept Object.'   He thinks this cannot be right, 'since each cat [referring to his two cats] has its own existence, the existence of either cannot be the being-instantiated of any quidditative concept'.  In reply, I say that Bill is rather putting words into my mouth.  Words that I would not choose for myself, with my aversion for abstract nouns.  I would simply say that

There is a cat ⇔ ∃x. Cat(x) ⇔ The concept Cat is instantiated.

 and

There is an object ⇔ ∃x. Object(x) ⇔ The concept Object is instantiated.