Intuitively, if something is identical to Venus, it follows that something is identical to something. In the notation of MPL, the following is a correct application of the inference rule, Existential Generalization (EG):
1. (∃x)(x = Venus)
2. (∃y)(∃x)(x = y) 1, EG
(1) is contingently true: true, but possibly false. (2), however, is necessarily true. Ought we find this puzzling? That is one question. Now consider the negative existential, 'Vulcan does not exist.'
3. ~(∃x)( x = Vulcan)
4. (∃y)~(∃x)(x = y) 3, EG
(3) is contingently true while (4) is a logical contradiction, hence necessarily false. The inference is obviously invalid, having taken us from truth to falsehood. What went wrong?
Diagnosis A: "You can't existentially generalize on a vacuous term, and 'Vulcan' is a vacuous term."
The problem with this diagnosis is that whether a term is vacuous or not is an extralogical (extrasyntactic) question. Let 'a' be an arbitrary constant, and thus neither a place-holder nor a variable. Now if we substitute 'a' for 'Vulcan' we get:
3* ~(∃x)( x = a)
4. (∃y)~(∃x)(x = y) 3*, EG
The problem with this inference is with the conclusion: we don't know whether 'a' is vacuous or not. So I suggest
Diagnosis B: Singular existentials cannot be translated using the identity sign as in (1) and (3). This fact, pace van Inwagen, forces us to beat a retreat to the second-level analysis. We have to analyze 'Venus exists' in terms of
where 'V' is a predicate constant standing for the haecceity property, Venusity. Accordingly, what (5) says is that Venusity is instantiated. Similarly, 'Vulcan does not exist' has to be interpreted as saying that Vulcanity is not instantiated. Thus
where 'W' is a predicate constant denoting Vulcanity.
It is worth noting that we can existentially generalize (6) without reaching the absurdity of (4) by shifting to second-order logic and quantifying over properties:
That says that some property is such that it is not instantiated. There is nothing self-contradictory about (7).
But of course beating a retreat to the second-level analysis brings back the old problem of haecceities. Not to mention the circularity problem.
The thin theory is 'cooked' no matter how you twist and turn.