I discussed one of the Peter van Inwagen's arguments here and found it wanting. He has a second argument: ". . . 'exists' is univocal owing to the interdefinability of 'there exists' and the obviously univocal 'all.' But this is a powerful argument, for, surely, 'all' means the same in 'All natural numbers have a successor' and 'All Greeks are mortal'?" (484). The argument could be put as follows:
'Every' is univocal.
'Exist(s)' and 'every' are interdefinable: 'Fs exist' is equivalent to 'It is not the case that everything is not an F.'
'Exist(s)' is univocal.
I accept this crisp little argument -- but with a restriction: 'exist(s)' is univocal across all affirmative and negative general existential sentences. But what about a singular existential such as 'Peter exists'? Does 'exist' in the latter have the same sense that it has in 'Rabbits exist'? I say it doesn't: 'exist(s)' is not univocal across all existence sentences, general and singular.
To warm up, what are we saying when we say that rabbits exist? On Frege's approach, we are saying that the concept rabbit is instantiated. So 'exist(s)' in general existentials means 'is instantiated.' But 'Peter exists' does not say that Peter is instantiated. So is it not spectacularly obvious that 'exist(s)' is not univocal across singular and general existentials?
But we needn't follow Frege is holding that 'exist(s)' is a second-level predicate. And van Inwagen does not follow him in this. Perhaps it would not be unfair to characterize van Inwagen as a half-way Fregean: he likes the notion that "existence is allied to number" but he does not take that characteristic Fregean thesis to entail that 'exist(s)' is a second-level predicate, i.e., a predicate of concepts, not objects. Van Inwagen could and would say something along these lines:
1. Rabbits exist: It is not the case that everything is not a rabbit. ~(x)~Rx.
2. Peter exists: It is not the case that everything is not identical to Peter. ~(x)~(x = Peter)
I will now try to show that, even on van Inwagen's preferred translations, there is still equivocity as between general and singular existentials. (1) and (2) are equivalent to
1*. Rabbits exist: Something is a rabbit. (Ex)Rx.
2*. Peter exists: Something is (identically) Peter. (Ex)(x = Peter).
Now it seems to me that we are still stuck with equivocation. The predicate in (1*) is 'something is (predicatively) ___.' The predicate in (2*) is 'something is (identically) ___.' Now the 'is' of predication is not the 'is' of identity. So the equivocation on 'exist(s)' remains in the form of an equivocation on 'is' as between the 'is' of predication and the 'is' of identity.
The equivocation ought to be obvious from the notation alone. The immediate juxtaposition of 'R' and 'x' in '(Ex)Rx' signifies that x is (predicatively) R. But in '(Ex)(x = Peter)' we find no such juxtaposition but a new sign, '=.'
My thesis, then, is that while 'exist(s)' is univocal across all general existentials, it is not univocal across all existentials. This reflects that fact that -- to switch over to material mode -- existence cannot be reduced to or eliminated in favor of any thin logical notion or combination of such notions.