In "Being, Existence, and Ontological Commitment" (in Metametaphysics: New Essays on the Foundations of Ontology, eds. Chalmers et al., Oxford 2009, pp. 472-506), Peter van Inwagen argues that 'exists' is univocal: it does not have "different meanings when applied to objects in different categories." (482) This post will examine one of his arguments, an argument found on p. 482. All quotations are from this page.
Van Inwagen begins by noting that number words such as 'six' or 'forty-three' do not "mean different things when they are used to count objects of different sorts." Surely he is correct: "If you have written thirteen epics and I own thirteen cats, the number of your epics is the number of my cats." So the first premise of the argument is the indisputable:
1. Number-words are univocal in sense: they mean the same regardless of the sorts of object they are used to count.
Van Inwagen takes his second premise straight from Frege:
2. "But existence is closely allied to number."
How so? Well, to say that unicorns do not exist is equivalent to saying that the number of unicorns is zero, and to say that horses exist is equivalent to saying that the number of horses is one or more. Surely that is true for both affirmative and negative general existentials. Whether it is true for singular existentials is a further question.
Van Inwagen proceeds: "The univocacy [univocity] of number and the the intimate connection between number and existence should convince us that existence is univocal." The conclusion of the argument, then, is:
3. Existence is univocal.
The first thing to notice about this argument is that it is not even valid. Trouble is caused by the fudge-phrase 'closely allied to' and van Inwagen's shift from 'exists' to existence. But repairs are easily made, and charity demands that we make them. Here is a valid argument that van Inwagen could have given:
1. Number-words are univocal
2*. 'Exist(s)' is a number-word
3*. 'Exist(s)' is univocal.
The latter argument is plainly valid in point of logical form: the conclusion follows from the premises. It is the argument van Inwagen should have given. Unfortunately the argument is unsound. Although (1) is indisputably true, (2*) is false.
Consider my cat Max Black. I joyously exclaim, 'Max exists!' My exclamation expresses a truth. Compare 'Cats exist.' Now I agree with van Inwagen that the general 'Cats exist' is equivalent to 'The number of cats is one or more.' But it is perfectly plain that the singular 'Max exists' is not equivalent to 'The number of Max is one or more.' For the right-hand-side of the equivalence is nonsense, hence necessarily neither true nor false.
This question makes sense: 'How many cats are there in BV's house?' But this question makes no sense: 'How many Max are there in BV's house?' Why not? Well, 'Max' is a proper name (Eigenname in Frege's terminology) not a concept-word (Begriffswort in Frege's terminology). Of course, I could sensibly ask how many Maxes there are hereabouts, but then 'Max' is not a proper name, but a stand-in for 'person/cat named "Max" .' The latter phrase is obviously not a proper name.
Van Inwagen's argument strikes me as very bad, and I am puzzled why he is seduced by it. (Actually, I am not puzzled: van Inwagen is in lock-step with Quine; perhaps the great prestige of the latter has the former mesmerized.) Here is my counterargument:
4. 'Exists' sometimes functions as a first-level predicate, a predicate of specific (named) individuals.
5. Number-words never function as predicates of specific (named) individuals
6. 'Exists' is not a number-word.
7. The (obvious) univocity of number-words is not a good reason to think that 'exists' is univocal.
Of course, there is much more to say -- in subsequent posts. For example if you deny (4), why is that denial more reasonable than the denial of (2*)?