'Horses exist' is an example of an affirmative general existential sentence. What is the status of the predicate '___ exist' in such a sentence? One might maintain that 'exist(s)' is a second-level predicate; one might maintain that it is a first-level distributive predicate; one might maintain that it is a first-level non-distributive (collective) predicate.
1. Frege famously maintained that 'exist(s)' is a second-level predicate, a predicate of concepts only, and never a first-level predicate, a predicate of objects. Russell followed him in this. A consequence of this view is that 'Horses exist' is not about what it seems to be about, and does not say what it seems to say. It seems to be about horses, and seems to say of them that they exist. But on Frege's analysis the sentence is about the concept horse and says of it, not that it exists, but that it has one or more instances.
Paradoxically, the sentence ''Horses exist' on Frege's analysis says about a non-horse something that cannot be true of a horse or of any concrete thing!
For an interesting comparison, consider 'Horses surround my house.' Since no horse could surround my house, it is clear that the sentence is not about each of the horses that surround my house. What then is it about? One will be tempted to reach for some such singularist analysis as: 'A set of horses surrounds my house.' But this won't do since no such abstract object as a set could surround anything. So if the sentence is really about a set of horses then it cannot say what it appears to say. It must be taken to say something different from what it appears to say. So what does 'Horses surround my house' say about a set if it is about a set?
One might be tempted to offer this translation: 'A set of horses is such that its members are surrounding my house.' But this moves us in a circle, presupposing as it does that we already understand the irreducibly plural predication 'Horses surround my house.' After all, if the members of a set of horses surround my house that is no different from horses surrounding my house.
The circularity here is structurally similar to that of the Fregean analysis. If 'Horses exist' is about a concept, and says of that concept that it has instances, then of course those instances are horses that exist. So the attempt to remove existence from individuals and make of it a property of concepts ends up reinstating existence as a 'property' of individuals.
Pursuing the analogy a bit further, the refusal to grant that there are irreducibly plural predications such as 'Horses surround my house' is like the refusal to grant that there are irreducibly first-level existence sentences.
2. Pursuing the analogy still further, is it possible to construe the predicate in 'Horses exist' as a non-distributive first-level predicate like the predicate in 'Horses surround my house'? First some definitions.
A predicate F is distributive just in case it is analytic that whenever some things are F, then each is F. Thus a distributive predicate is one the very meaning of which dictates that if it applies to some things, then it applies to each of them. 'Blue' is an example. If some things are blue, then each of them is blue.
If a predicate is not distributive, then it is non-distributive (collective). If some Occupy-X nimrods have the building surrounded, it does not follow that each such nimrod has the building surrounded. If some students moved a grand piano into my living room, it does not follow that each student did. If bald eagles are becoming extinct, it does not follow that each bald eagle is becoming extinct. Individual animals die, but no individual animal ever becomes extinct. If the students come from many different countries, it does not follow that each comes from many different countries. If horses have an interesting evolutionary history, it does not follow that each horse has an interesting evolutionary history.
I will assume for the purposes of this post that 'Horses surround my house' and 'Horses have an interesting evolutionary history' are irreducibly plural predications. (That they are plural is obvious; that they are irreducibly plural is not. For arguments see Thomas McKay, Plural Predication.) And of course they are first-level as well: they are about horses, not about concepts or properties or propositional functions. Now is 'Horses exist' assimilable to 'Horses surround my house' or is it assimilable to 'Horses are four-legged'? The predicate in the later is a distributive first-level predicate, whereas the predicate in 'Horses surround my house' is a non-distributive first-level predicate.
I am assuming that the 'Fressellian' second-level analysis is out, but I won't repeat the arguments I have given ad nauseam elsewhere.
I do not understand how 'exist(s)' could be construed as a non-distributive predicate. For if it is non-distributive, then it is possible that some things exist without it being the case that each of them exists. And that I do not understand. If horses exist, then each horse exsts.
Peter van Inwagen seems (though it not clear to me) to be saying that 'exists(s)' is a non-distributive first-level predicate. He compares 'Horses exist' to 'Horses have an interesting evolutionary history.' 'Horses exist,' he tells us, is equivalent to 'The number of horses is not zero.' ("Being, Existence, and Ontological Commitment," p. 483) But he denies that 'exists(s)' is second-level. To say that the number of horses is not zero is to predicate of horses that they number more than zero. (483) It is not to predicate of the concept horse that the cardinality of its extension is more than zero.
Now we cannot say of a horse that it surrounds a house or has an interesting evolutionary history. We can say that of horses, but not of a horse. Can we say of a horse that it numbers more than zero? We can of course say of horses that they number more than zero. But I don't see how we can sensibly say of an individual horse that it numbers more than zero. Perhaps Frege was wrong to think that number words can only be predicates of concepts which are ones-over-many. Perhaps all one needs is the many, the plurality. But it seems one needs at least that to swerve as logical subject. If this is right, and to exist is to number more than zero, then we cannot sensibly say of an individual that it exists. We can say this of individuals but not of an individual. But surely we can say of an individual horse that it exists. So I conclude that 'exist(s)' cannot be a first-level non-distributive predicate.
3. And so one is driven to the conclusion that 'exist(s)' is a first-level distributive predicate. 'Horses exist' says of each individual horse that it exists. But isn't this equally objectionable? The vast majority of horses are such that I have no acquaintance with them at all. So how can my use of 'Horses exist' be about each horse?
It is at this juncture that Frege gets his revenge:
We must not think that I mean to assert something of an African chieftain from darkest Africa who is wholly unknown to me, when I say 'All men are mortal.' I am not saying anything about either this man or that man, but I am subordinating the concept man to the concept of what is mortal. In the sentence 'Plato is mortal' we have an instance of subsumption, in the sentence 'All men are mortal' one of subordination. What is being spoken about here is a concept, not an individual thing. (Posthumous Writings, p. 213)
Plato falls under the concept man; he does not fall within it. The concept mortal does not fall under the concept man -- no concept is a man -- but falls within it. When I say that all men are mortal I am not talking about individual men, but about the concept man, and I am saying that this concept has as part of its content the subconcept mortal.
Similarly, my utterance of 'Horses exist' cannot be about each horse; it is about the concept horse, and says that it has instances -- which is the view I began by rejecting and for god reason.
We seem to have painted ourselves into an aporetic corner. No exit. Kein Ausgang. A-poria.