1. 'Every' is univocal.
2. ''Exist(s)' and 'every' are interdefinable: 'Fs exist' is equivalent to 'It is not the case that everything is not an F.'
3. 'Exist(s)' is univocal.
He seems to regard this one as better, but it seems to me to be even more obviously bad. The first one just went imprecise at exactly the point at which it needed to be precise, but this one is just plain. Let's consider the underlying reason for (2) a moment:
(A) 'Fs exist' is equivalent to 'It is not the case that everything is not an F.'
This is in fact only true if it is understood in the following way:
(A') 'At least some Fs exist' is equivalent to 'It is not the case that everything that exists is not an F'.
If 'Fs exist' in (A) were indefinite in quantity, in the strict sense that it were impossible to say whether it were universal or particular, the claim would not be true; 'Fs exist' would be equivalent to a disjunction. Nor can we merely assume that it is particular, because, while rare, universals of this kind do exist, and are meaningful because they could be argued over, e.g.:
Every possible God exists.
Every possible world exists.
In the same way, 'everything' has to be taken to mean 'everything that exists', because if it means something else, e.g., 'everything that I am thinking might exist', the equivalence is lost.
All this is, of course, far more consistent with the way we usually think of this interdefinability, which is not in terms of the interdefinability of 'exists' and 'every' but in terms of the interdefinability of 'at least some' and 'every'. But when we look at the argument again, we see that (A') does not establish (2); 'exists' is found on both sides, and there's no way to cash out the 'exists' in 'everything that exists' in terms of another 'every' because that would in this case just lead to infinite regress -- the new 'every' could only maintain the equivalence if it were understood to mean 'everything that exists', i.e., 'It is not the case that everything that exists among everything that exists is not an F'.
If we were to put it in broadly Boolean algebraic terms rather than Fregean terms, this would all admit of the easy and obvious diagnosis that van Inwagen's (2) confuses two distinct logical functions, that of the logical domain (indicated by 'exists', although there are reasons to think that this word on its own leaves the domain insufficiently defined and that more background is needed) and that of the logical quantity (indicated by 'every', which tells us how much of the domain is included in the claim). Of course, van Inwagen is not himself operating in Boolean terms but Fregean terms, and Fregeans tend not to think one needs a precisely defined domain in the way that the old algebraic logicians did.