Is 'Existence' Univocal Because 'All' Is Univocal?

In this post, Bill Vallicella presents another argument by Peter van Inwagen for the univocity of existence and questions it (he posted and refuted the first argument here). I think Vallicella has a point. Still, I might grant that the Quinean like Van Inwagen can translate a singular existential statement so as to have the same form as a general existential statement and argue the conclusion still does not follow. Quine, in his famous paper "On What There Is," proposes that we treat for instance the relation '__ = Pegasus' as a single-place predicate '=Pegasus'; one can call this 'pegasizing' or formally 'P'. Then 'Pegasus exists' will just be 'something pegasizes', which will just be translated to, '(Ex)(Px)'. Even ignoring the problematic aspects here I would pose a different objection. I would accuse Van Inwagen's argument of being a 'non sequitur'. Vallicella states the argument thus:

(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.'

Therefore

(3) 'Exist(s)' is univocal.

Clearly, as is, this argument is not valid. To make it valid we need some further premise. I'm not sure what sort of plausible premise Van Inwagen is using to get to his conclusion, but maybe it is something like

(2.5) If two terms are interdefinable then each of the terms' uses share the same sense relation.

(Just a clarificatory point: Sense relations are things like 'univocity' or 'equivocity', and Van Inwagen thinks that all the uses of 'exists' are univocal.) How are we to understand 'interdefinable' here? Surely not as meaning that for each 'exists' statement there is a semantically identical 'every' statement, i.e. one with the exact same meaning, for that would be utterly question-begging. We must construe it then as something like 'for each 'exists' statement there is a logically equivalent 'every' statement'. The problem is that (2.5) is not obviously true on this interpretation. I'll explain.

I think we can admit that 'some' and 'every' are univocal, that these two are interdefinable in the sense that logically equivalent statements can be expressed in terms of each, but still say that 'some' doesn't fully capture the meaning of 'exists', and thus neither does 'every'. Of course, every 'some' statement is logically equivalent to another 'there exists' statement, but that does not imply they are semantically identical.

On the idea that 'exists' is analogical, the natural language quantifier 'there exists' has many senses, but all beings can be said to exist in one of those senses; thus the range of this quantifier includes all beings (regardless as to which sense of 'being' can be said of them). And since there are no non-existent beings, the range of the quantifier 'some' is over all beings. So the two quantifiers range over the same domain of discourse; and since for any 'some' statement there is a logically equivalent 'there exists' statement, it follows that we can translate logically equivalent statements involving either of them with the same symbol in predicate-logic, '(Ex)'. This is also why they are each logically equivalent to at least one 'all' statement. But it simply doesn't follow that they all share the same sense relation (univocal, equivocal, etc.). It is true that our 'some' quantifier ranges over only and all beings, but it ranges over them regardless as to which of the many analogous senses of 'being' can be said of them. So it's consistent with both 'some' and 'all' being univocal that 'being' or 'exists' are not.

Aristotelian-Thomism vs. Quineanism on Ontological Commitment

I recently got the Chalmers volume dealing with the foundations of ontology. I've been interested in this question for the last few months, and here are some of my thoughts on the matter.

Most philosophers these days accept the Quinean view that one is committed to whatever one quantifies over. If I say there are colors, i.e. (Ex)(x is a color),  and I think my saying is true, then I am committed to the existence of colors, end of story. Most contemporary debates as in the Chalmers volume are between those who hold to this Quinean view and a few people who want to revive some of the distinctions of Carnap.

From an Aristotelian-Thomistic perspective, the dominant Quinean view is incorrect for a few reasons. First of all it is wrong to say that metaphysics is just about finding what there is or listing an ontology. As this volume shows, many if not most philosophers these days accept this definition. Rather, ontology is just one part of metaphysics, the ultimate purpose of metaphysics being, in the spirit of Aristotle, the study of being qua being, i.e. the study of the fundamental nature of reality. This involves not only ontology, but also finding the essences of things and finding the various relations of ontological dependence these things stand in to each other.

Also, I think there is some confusion in the modern Quinean conception of existence, where being quantified over is taken to be being simpliciter (in Thomistic terms). On the contrary, there is being simpliciter, then the various diminished senses of being (being secundum quid, in Thomistic terms), as for example the being privations have or being in potentiality. Many of the things we quantify over might have being in one of these imperfect and diminished senses, but just because we quantify over them doesn't mean we have to conclude they have being simpliciter. This makes ontological questions quite trivial. For instance, of course there are numbers and numbers have being; one need only observe the fact that 2 is a number. The real question is finding out whether they have being simpliciter or being secundum quid, what categories of being they fall under, and in what relations of ontological dependence do they stand? Interestingly, in this volume, Kit Fine and Jonathan Schaffer seem to come to similar conclusions, though I don't think Fine is rigorous enough to be convincing to other philosophers already steeped in Quineanism. I'd like to write a paper on this at some point so as to make the Aristotelian position a bit more clear than Fine does.