Bill Vallicella's otherwise excellent discussion of 'thin' theories of existence in a series of posts at the Maverick Philosopher website has so far ignored the difference between theories where the existence of a singular referent is presupposed, and theories where it is asserted.
Consider 'Socrates is wise'. This affirms wisdom of Socrates. Does it also affirm existence of him? Does it then consist of two propositions, one of which affirms existence, the other of which affirms wisdom? Or does it consist of just one, that affirms wisdom without affirming existence? The latter view, sometimes called the 'Strawsonian' or 'presupposition' view, involves the following problem.
If Socrates does not exist, the negation of 'Socrates is wise', i.e. 'It is not the case that Socrates is wise' must be true. But if 'Socrates is wise' does not assert the existence of Socrates, neither does its negation (for a negation can deny no more than the corresponding affirmation affirmed). In which case the negation expresses exactly the same thing as if Socrates had existed, and wisdom was being denied of him, i.e. expresses exactly what 'Socrates is non-wise' expresses. But then it affirms non-wisdom of something, and so requires the existence of Socrates. But that cannot be, if Socrates does not exist.
But there is no such problem with the view that existence, as well as wisdom, is asserted. Then the negation of 'Socrates is wise' is the negation of a conjunction, and so is equivalent to the disjunction 'There is no such person as Socrates, or wisdom does not apply to Socrates'. There are different facts corresponding to Socrates' not existing, and Socrates existing but not having wisdom, and so the falsity of 'Socrates is wise' does not imply that a non-wise Socrates exists.
 For an early defence of assertionism, see my translation of chapters 12 and 14 of the Ockham's Summa Logicae II here. For a later one that should be familiar to almost everyone, see Russell's Theory of Descriptions.