Existence and Logical Form
There have been two main theories about the logical form of existence statements: the first-order predicate theory and the second-order predicate theory. One theory maintains that “exists” expresses a property that objects have; the other theory maintains that “exists” serves to make a statement about a predicate or concept or property, to the effect that it has instances. Both theories have their attractions and drawbacks, which I won’t discuss; my purpose is to present a new theory.
I call this theory “the intentional object theory”: it says that an ascription of existence predicates something of an intentional object (an object of thought or other mental representation), not of an ordinary object or a concept. We are talking about an intentional object when we speak of existence not a regular object or a concept (predicate, property); and we are ascribing a certain property to that object. For example, if I say “Sherlock Holmes does not exist” I am saying something about an object of thought—the subject matter of my thought. Similarly, if I say “Mick Jagger exists” I am speaking not of the existent Mick Jagger but of the intentional object corresponding to him—the object of thought I would still have even if he did not exist. I won’t go into the question of the meaning and legitimacy of talk of intentional objects; my aim is to make a proposal about logical form given that we can talk this way.  So let us introduce a piece of notation: we designate an intentional object by placing an asterisk after a name or other singular term for an object (existent or not), as in “Mick Jagger*”, to be read “the intentional object that corresponds to Mick Jagger”. The theory, then, is that statements of existence are about such objects, and only such objects; they are not about ordinary objects such as Sir Mick himself, or about predicates such as “lead singer of the Rolling Stones”.
But what do such statements say about Mick Jagger*? They say that this intentional object is actualized. So the analysis of “Mick Jagger exists” is “Mick Jagger* is actualized”. I am taking the term “actualized” as primitive but its intuitive meaning should be clear enough (we could also use “realized” or “exemplified”). One thing we must not do is analyze it as “corresponds to an object that exists”, since that introduces existence as a predicate of objects. Instead we are analyzing existence as applied to objects (misleadingly so) in terms of intentional objects being actualized: that is the primitive notion. Suppose we are debating the existence of unicorns, with you a believer and me a disbeliever, and you claim to have seen a unicorn last week: I might reply “That was just a figment of your imagination”. What I am saying is that your intentional object when it seemed to you that you saw a unicorn was not actualized—in contrast to your reported sightings of regular hornless horses. The logical form of my statement is: “Unicorns* are not actualized”. Whenever we think of an object, existent or non-existent, we have of an intentional object in mind, and that object can be said to be actualized or not. One way to conceive such objects is as possible objects (in some sense of “possible”); then we are saying of possible objects whether or not they are actualized (which is not to say “actual”). To put it in a way that is familiar, though potentially misleading, we introduce an ontology of intentional objects and then we analyze existence statements as predications of actualization of entities in this ontology. The metaphysics of this ontology is no doubt tricky and controversial, but I am steering clear of all that in order to make a logical point: existence statements are not about ordinary objects or about concepts; they are about intentional objects. If we use Meinong’s categories, we can put the position by saying that existence statements are always about subsistent entities, which can be said to be either actualized or not actualized, thus giving rise to existence and non-existence.
It is easy to get lost in the metaphysical fog here, so let me restate the thrust of the position in less fraught terms: existence is to be analyzed by means of a predicate of singular terms, not objects or concepts. When I say, “Mick Jagger exists” I am saying “The name ‘Mick Jagger’ is actualized”. This is not strictly correct, because names cannot be actualized in the sense intended, but it helps to get the logical point across: namely, that we have something else to play with aside from objects and concepts—we have names themselves. Indeed, one view of existence statements precisely is that they ascribe denotation to names, as in “The name ‘Mick Jagger’ denotes”. The intentional object theory is logically similar in that it uses a new predicate in application to something other than objects or concepts–the theory speaks of intentional objects being actualized and not of names denoting. The denotation theory is meta-linguistic; the actualization theory is, so to speak, meta-intentional. It uses an ontology of intentional objects to analyze existence statements by invoking a property of those objects, viz. actualization.
We thus allow that there is reference shift inside existence statements: names don’t refer to their ordinary reference but to intentional objects (compare Frege on indirect discourse). In “Mick Jagger exists” the name refers not to Sir Mick himself but to the intentional object Mick Jagger*. This solves an old problem: the problem of redundancy and self-contradiction. We are not referring to an object and then saying it exists (redundancy) or that it doesn’t exist (contradiction); we are referring to an intentional object, which is neutral between existence and non-existence, and then saying whether it is actualized or not. Suppose I am the subject of a psychological experiment in which the experimenter is feeding me visual impressions, some of which are veridical and some are not: some are of horses, some are of unicorns. She asks me to say whether I think the impression corresponds to anything real or not; for reasons of experimental protocol she prefers that I answer with the words, “That intentional object is actualized/not actualized”. Thus I explicitly refer to my intentional objects and predicate something of them, thereby expressing my existential beliefs. That is the gist of the theory I am proposing. 
If we try to reconstruct how the concept of existence came to be employed in the first place, the picture is that we noticed that some of our thoughts and percepts correspond to real things and some are about merely imaginary things. We wanted a way to distinguish between the two cases: so we took to saying that some of the things that enter our minds are actualized and some are not. Then we abbreviated this to speak directly of the existence of things, all along really meaning that objects of thought can be actualized (realized, exemplified) or not. Thus logical form became hidden in the simple sentences we use to talk about existence, such as “Mick Jagger exists”. There was always something funny and jarring about that form of words, but it was brief and convenient. The sentence doesn’t seem logically like “Mick Jagger is a good dancer” or “Mick Jagger is English”; and indeed it is really about something other than Mick the man. For Mick to exist is for the intentional object Mick* to be actualized, but nothing like this is true of his dancing ability or nationality. Vernacular statements of existence are misleading as to their true topic.
 I have discussed intentional objects in Logical Properties (Oxford: Oxford University Press, 2000), chapter 2, and in “The Objects of Intentionality” in Consciousness and Its Objects (Oxford: Oxford University Press, 2004).
 Imagine you are a brain in a vat: none of the things you experience actually exists, but you have an enormous range of merely intentional objects before your mind. Then to say of each non-existent thing that it does not exist is to say that the corresponding intentional object is not actualized—that it is merely intentional. If we now place the brain in a vat into a normal head so that the world is perceived, we can say that the subject’s intentional objects are actualized.