Existence, Extension and the Ontological Argument

Note: This is a paper I wrote as an undergraduate, which used to be on my old (pre Y2K) website. I thought I had transported it here, but apparently hadn't got to this one yet. The topic came up in another discussion online, so I here it is for reference.

     Nearly everyone is aware that pi, the ratio of a circle's circumference to its diameter, is an irrational number. That is, it cannot be expressed as the quotient of two whole numbers, no matter what those whole numbers are. In other words, although we've only actually calculated the value of pi to a few sextillion digits, we can categorically state that the string of digits will stretch out to infinity without ever settling into a permanent pattern of repetition. Indeed, we can even show that this is true despite the fact that no one has ever drawn, or will ever draw a perfect circle. It follows implicitly from the theoretical definition of a perfect circle; for pi to be rational, a circle would have to have a finite number of sides, which is incompatible with the meaning of the word circle.

     St. Anselm's ontological argument for the existence of God hinges upon the assumption that existence is a predicable quality in the same sense that perfect, circle, and irrational are. In order for the argument to hold water, we must accept that the quality of existence follows implicitly from the concept of That Than Which Nothing Greater Can Be Conceived, just as the irrationality of pi follows implicitly from the definition of a perfect circle.
     In order to answer the question of existence's predicability, we must first articulate what it is we mean by predicability, and then see whether or not it is something which may be said of existence, i.e. is predicability predicable of existence?
     At first glance, we might well be inclined to assert that it is. Grammatically, at least, it is incontrovertibly a predicate, since it is a verb, and an intransitive one at that. One may intelligibly say of something that it exists, and perhaps more importantly, one may intelligibly say that something does not exist.
     However, the objection has been raised by the likes of Parmenides that being is in fact not a deniable trait. That is to say, it is contradictory to define something as non-existent. Anything which can be described can be described as "something which..."; to include non-existence among its characteristics is equivalent to describing it as "nothing which...". We might invent a class of objects called "not-beings", which have all the characteristics of beings except existence, and have the explicit quality of not-existence. Such definitions are inherently paradoxical, since by definition they cannot exist (or they'll be just ordinary beings, not non-beings) and they cannot fail to exist (since by failing to exist, they satisfy their definitions and therefore exist). This paradox is, by the way, of the same sort that Bertrand Russell used to attack set theory, i.e. by introducing the set of all sets that are not members of themselves.
     Very well, then. We have shown that the denial of existence cannot intelligibly be predicated of a thing. From this, it would seem to follow that the assertion of existence is obligatory. Does this mean that everything that can be spoken of or referred to in any way must necessarily exist? That dragons and minotaurs and those ghastly creatures of chaos, the unicorns, exist? Even more bizarre, that round squares and honest politicians exist?
     The last two we can deal with easily enough by the self-denial paradox; round squares are by definition logical impossibilities, just as are non-beings. And, just as the logical impossibility of self-denial allowed us to dispose of the denial of existence, we may handily dispose of the notion that predicated existence necessitates actual existence, or indeed that it has any connection whatsoever to actuality.
     What do we mean, then, when we say that something does not exist? Surely it is not nonsensical to say, "Dragons do not exist." How can we reconcile the implicit being in the definition of everything with the denial of being in the statement "Dragons do not exist"?
     Bertrand Russell did it by arguing that what we call existence is not a quality of the things themselves, but of the ideas of those things. That is, when we say that dragons do not exist, what we really mean is that of all the things there are, none of them are referred to by the word dragon or its attendant definition. (That definition, of course, must implicitly include being, if the definition is to be intelligible or possible.) Likewise, when we say that dragons DO exist, we really mean that the word dragon extends to something it intends. In other words, "Dragons exist" is a convenient shorthand for "The term dragon has an extension."
     Of course, this just puts Anselm's argument into different terms. Rather than saying that the maximally perfect being must exist, now, he would say that the intension of a maximally perfect being implies an extension. This seems a preposterous claim; how can an intension necessitate extension? And yet, there are simple examples. The word word, by its intension, cannot possibly fail to have extension. How might That Than Which Nothing Greater Can Be Conceived necessarily imply its own extension?
     Let us assume the existence, i.e., the extension of something. What it is precisely is unimportant. In fact, we might just point to some lump of matter, and label it with the convenient word, This. (If we wish, we can take nothing for granted, and take This to mean ourselves, after Descartes.) Its intension is very simple; it means whatever we happen to be pointing at, and thus is identical with its extension. Now, conceiving such an entity is trivially easy, as is accepting its existence. Since any whole is at least as great as any of its parts, it should also be possible to conceive the whole formed by This and some other thing. As long as there exists something outside of This, it is possible to conceive of something greater, namely [This + something else].
Following this to its ultimate conclusion, we find that That Than Which Nothing Greater Can Be Conceived intends simply the totality of everything which can be conceived. (Given that being is implicitly predicated of anything intelligibly conceived, it is trivially true that being is predicable of TTWNGCBC.) Is it sensible to say that TTWNGCBC lacks extension?
     It is possible, of course, to conceive of things which lack extension. However, some things are their own extensions, like the word word. The intension of TTWNGCBC says nothing specifically about physical existence; it only refers to something conceived. Nor does it necessarily have to be conceived; it need only be conceivable. Therefore TTWNGCBC need only be a potential idea to have extension. It is impossible to conceive of anything greater that the totality of all things conceivable, so there is nothing implicitly paradoxical about the intension of TTWNGCBC. Ergo, it is a potential idea, and is its own extension.

     TTWNGCBC therefore unquestionably exists, inasmuch as any idea may be said to exist. Anselm has chosen to call TTWNGCBC by the name God. I think a more elegant name would be simply Being, or perhaps Existence. In this way, Existence is most certainly not a predicate, but a noun, and the extent to which we say things exist depends in fact upon the extent to which their various predicates partake in Existence. Thus, dragons partake in Being by being large, fire-breathing, carnivorous and imaginary, while toads partake in Being by being small, fly-eating, amphibious and real.