Here is an argument adapted from Peter van Inwagen for the univocity of 'exist(s)' across general and singular existentials.
a. Number-words are univocal.
b. 'Exist(s)' is a number-word.
c. 'Exist(s)' is univocal.
(a) is plainly true. The words 'six' and 'forty-nine' have the same sense regardless of what we are counting. As van Inwagen puts it, "If you have written thirteen epics and I own thirteen cats, the number of your epics is the number of my cats."
(b) captures the Fregean claim that ". . . existence is analogous to number. Affirmation of existence is in fact nothing but denial of the number nought." (Foundations of Arithmetic, p. 65)
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 maintains "The univocacy [univocity] of number and the the intimate connection between number and existence should convince us that existence is univocal."
I am not convinced.
Consider my cat Max Black. I exclaim, 'Max exists!' My exclamation expresses a truth. Contrast the singular 'Max exists' with the general 'Cats exist.' 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 being used as a proper name, but as a stand-in for 'person/cat named "Max" .' The latter phrase is obviously not a proper name.
And so I deny the univocity of 'exist(s)' across general and singular existentials.
Andrew Bailey lodges the following objection to what I maintain:
You note that "‘Max exists’ is not equivalent to ‘The number of Max is one or more’", and that seems right.
But why think "The number of Max is one or more" is the way to say of Max that he exists using number-words? Why not, instead, "At least one thing is Max"? My suggestion, note, would align closely with the way one would ordinarily translate "Max exists" into the predicate logic: -- 'Ex(x=Max)' -- a statement of logic one might render in English as "there is at least one thing that is identical to Max".
Dr. Bailey is of course right that 'Max exists' can be translated into standard first-order predicate logic in the way he indicates and that this is equivalent in 'canonical English' to 'There is at least one thing that is identical to Max.' Bailey's rebuttal seems to be the following: Just as we can express 'Cats exist' as 'At least one thing is a cat,' we can express 'Max exists' as 'At least one thing is Max.'
But this response is unavailing. Note that the 'is' in 'is a cat' is not the 'is' of identity, but the 'is' of predication, while the 'is' in 'is Max' is the 'is of identity. So if Bailey tries to secure the univocity of 'exist(s)' in this way, he does so by exploiting an equivocation on 'is.'
Another possible rebuttal would be by invoking haecceities. One might argue that there is no equivocation on 'is' because both of the following feature the 'is' of predication:
At least one thing is a cat
At least one thing is Max-identical.
On the second approach one secures the univocity of 'exist(s)' but at the expense of those metaphysical monstrosities known as haecceity properties. The haecceity H of x is a property x cannot fail to instantiate, alone instantiates in the actual world, and that nothing distinct from x instantiates in any possible world. If Max has such such a property -- call it Maxity -- then this property captures Max's haecceitas or thisness, where 'thisness' is to be understood as irreducible and nonqualitative. If there is such a property, then it is the property of identity-with-Max or Max-identity.
So if you want to maintain the univocity of 'exist(s)' across general and singular existentials, you must either conflate the 'is' of identity' with the 'is' of predication, or embrace haecceity properties.
Gentlemen, pick your poison!