The third theistic argument I wish to discuss is the famous "ontological argument" first formulated by Anselm of Canterbury in the eleventh century. This argument for the existence of God has fascinated philosophers ever since Anselm first stated it. Few people, I should think, have been brought to belief in God by means of this argument; nor has it played much of a role in strengthening and confirming religious faith. At first sight Anselm's argument is remarkably unconvincing if not downright irritating; it looks too much like a parlor puzzle or word magic. And yet nearly every major philosopher from the time of Anselm to the present has had something to say about it; this argument has a long and illustrious line of defenders extending to the present. Indeed, the last few years have seen a remarkable flurry of interest in it among philosophers. What accounts for its fascination? Not, I think, its religious significance, although that can be underrated. Perhaps there are two reasons for it. First, many of the most knotty and difficult problems in philosophy meet in this argument. Is existence a property? Are existential propositions -- propositions of the form x exists -- ever necessarily true? Are existential propositions about what they seem to be about? Are there, in any respectable sense of "are," some objects that do not exist? If so, do they have any properties? Can they be compared with things that do exist? These issues and a hundred others arise in connection with Anselm's argument. And second, although the argument certainly looks at first sight as if it ought to be unsound, it is profoundly difficult to say what, exactly, is wrong with it. Indeed, I do not believe that any philosopher has ever given a cogent and conclusive refutation of the ontological argument in its various forms.
 At first sight, [Anselm's] argument smacks of trumpery and deceit; but suppose we look at it a bit more closely. Its essentials are contained in these words:
And assuredly that, than which nothing greater can be conceived, cannot exist in the understanding alone. For suppose it exists in the understanding alone; then it can be conceived to exist in reality; which is greater.
Therefore, if that, than which nothing greater can be conceived, exists in the understanding alone, the very being, than which nothing greater can be conceived, is one, than which a greater can be conceived. But obviously this is impossible. Hence there is no doubt that there exists a being, than which nothing greater can be conceived, and it exists both in the understanding and in reality.
 How can we outline this argument? It is best construed, I think, as a reductio ad absurdum argument. In a reductio you prove a given proposition p by showing that its denial, not-p, leads to (or more strictly, entails) a contradiction or some other kind of absurdity. Anselm's argument can be seen as an attempt to deduce an absurdity from the proposition that there is no God. If we use the term "God" as an abbreviation for Anselm's phrase "the being than which nothing greater can be conceived," then the argument seems to go approximately as follows: Suppose
(1) God exists in the understanding but not in reality. (reductio assumption)
(2) Existence in reality is greater than existence in the understanding alone. (premise)
(3) God's existence in reality is conceivable. (premise)
(4) If God did exist in reality, then He would be greater than He is. [from (1) and (2)]
(5) It is conceivable that there is a being greater than God is. [(3) and (4)]
(6) It is conceivable that there be a being greater than the being than which nothing greater can be conceived. [(5) by the definition of "God"]
 But surely (6) is absurd and self-contradictory; how could we conceive of a being greater than the being than which none greater can be conceived? So we may conclude that
(7) It is false that God exists in the understanding but not in reality.
 It follows that if God exists in the understanding, He also exists in reality; but clearly enough He does exist in the understanding, as even the fool will testify; therefore, He exists in reality as well.
 Now when Anselm says that a being exists in the understanding, we may take him, I think, as saying that someone has thought of or thought about that being. When he says that something exists in reality, on the other hand, he means to say simply that the thing in question really does exist. And when he says that a certain state of affairs is conceivable, he means to say, I believe, that this state of affairs is possible in our broadly logical sense, there is a possible world in which it obtains. This means that step (3) above may be put more perspicuously as
(3') It is possible that God exists
and step (6) as
(6') It is possible that there be a being greater than the being than which it is not possible that there be a greater.
 An interesting feature of this argument is that all of its premises are necessarily true if true at all. (1) is the assumption from which Anselm means to deduce a contradiction. (2) is a premise, and presumably necessarily true in Anselm's view; and (3) is the only remaining premise (the other items are consequences of preceding steps); it says of some other proposition (God exists) that it is possible. Propositions which thus ascribe a modality -- possibility, necessity, contingency -- to another proposition are themselves either necessarily true or necessarily false. So all the premises of the argument are, if true at all, necessarily true. And hence if the premises of this argument are true, then [provided that (6) is really inconsistent] a contradiction can be deduced from (1) together with necessary propositions; this means that (1) entails a contradiction and is, therefore, necessarily false.
1. Kant's Objection
 The most famous and important objection to the ontological argument is contained in Immanuel Kant's Critique of Pure Reason. Kant begins his criticism as follows:
If, in an identical proposition, we reject the predicate while retaining the subject, contradiction results; and I therefore say that the former belongs necessarily to the latter. But if we reject the subject and predicate alike, there is no contradiction; for nothing is then left that can be contradicted. To posit a triangle, and yet to reject its three angles, is self-contradictory; but there is no contradiction in rejecting the triangle together with its three angles. The same holds true of the concept of an absolutely necessary being. If its existence is rejected, we reject the thing itself with all its predicates; and no question of contradiction can then arise. There is nothing outside it that would then be contradicted, since the necessity of the thing is not supposed to be derived from anything external; nor is there anything internal that would be contradicted, since in rejecting the thing itself we have at the same time rejected all its internal properties. "God is omnipotent" is a necessary judgment. The omnipotence cannot be rejected if we posit a Deity, that is, an infinite being; for the two concepts are identical. But if we say "There is no God," neither the omnipotence nor any other of its predicates is given; they are one and all rejected together with the subject, and there is therefore not the least contradiction in such a judgment...
For I cannot form the least concept of a thing which, should it be rejected with all its predicates, leaves behind a contradiction.
 One characteristic feature of Anselm's argument, as we have seen, is that if successful, it establishes that God exists is a necessary proposition. Here Kant is apparently arguing that no existential proposition -- one that asserts the existence of something or other -- is necessarily true; the reason, he says, is that no contra-existential (the denial of an existential) is contradictory or inconsistent. But in which of our several senses of inconsistent? What he means to say, I believe, is that no existential proposition is necessary in the broadly logical sense. And this claim has been popular with philosophers ever since. But why, exactly, does Kant think it's true? What is the argument? When we take a careful look at the purported reasoning, it looks pretty unimpressive; it's hard to make out an argument at all. The conclusion would apparently be this: if we deny the existence of something or other, we can't be contradicting ourselves; no existential proposition is necessary and no contra-existential is impossible. Why not? Well, if we say, for example, that God does not exist, then says Kant, "There is nothing outside it (i.e., God) that would then be contradicted, since the necessity of the thing is not supposed to be derived from anything external; nor is there anything internal that would be contradicted, since in rejecting the thing itself we have at the same time rejected all its internal properties."
 But how is this even relevant? The claim is that God does not exist can't be necessarily false. What could be meant, in this context, by saying that there's nothing "outside of" God that would be contradicted if we denied His existence? What would contradict a proposition like God does not exist is some other proposition -- God does exist, for example. Kant seems to think that if the proposition in question were necessarily false, it would have to contradict, not a proposition, but some object external to God -- or else contradict some internal part or aspect or property of God. But this certainly looks like confusion; it is propositions that contradict each other; they aren't contradicted by objects or parts, aspects or properties of objects. Does he mean instead to be speaking of propositions about things external to God, or about his aspects or parts or properties? But clearly many such propositions do contradict God does not exist; an example would be the world was created by God. Does he mean to say that no true proposition contradicts God does not exist? No, for that would be to affirm the nonexistence of God, an affirmation Kant is by no means prepared to make.
 So this passage is an enigma. Either Kant was confused or else he expressed himself very badly indeed. And either way we don't have any argument for the claim that contra-existential propositions can't be inconsistent. This passage seems to be no more than an elaborate and confused way of asserting this claim.
 The heart of Kant's objection to the ontological argument, however, is contained in the following passage:
"Being" is obviously not a real predicate; that is, it is not a concept of something which could be added to the concept of a thing. It is merely the positing of a thing, or of certain determinations, as existing in themselves. Logically, it is merely the copula of a judgment. The proposition "God is omnipotent" contains two concepts, each of which has its object -- God and omnipotence. The small word "is" adds no new predicate, but only serves to posit the predicate in its relation to the subject. If, now, we take the subject (God) with all its predicates (among which is omnipotence), and say "God is," or "There is a God," we attach no new predicate to the concept of God, but only posit it as an object that stands in relation to my concept. The content of both must be one and the same; nothing can have been added to the concept, which expresses merely what is possible, by my thinkings its object (through the expression "it is") as given absolutely. Otherwise stated, the real contains no more than the merely possible. A hundred real thalers does not contain the least coin more than a hundred possible thalers. For as the latter signify the concept and the former the object and the positing of the concept, should the former contain more than the latter, my concept would not, in that case, express the whole object, and would not therefore be an adequate concept of it. My financial position, however, is affected very differently by a hundred real thalers than it is by the mere concept of them (that is, of the possibility). For the object, as it actually exists, is not analytically contained in my concept, but is added to my concept (which is a determination of my state) synthetically; and yet the conceived hundred thalers are not themselves in the least increased through thus acquiring existence outside my concept.
By whatever and by however many predicates we may think a thing -- even if we completely determine it -- we do not make the least addition to the thing when we further declare that this thing is. Otherwise it would not be exactly the same thing that exists, but something more than we had thought in the concept:and we could not, therefore, say that the object of my concept exists. If we think in a thing every feature of reality except one, the missing reality is not added by my saying that this defective thing exists.
 Now how, exactly is all this relevant to Anselm's argument? Perhaps Kant means to make point that we could put by saying that it's not possible to define things into existence. (People sometimes suggest that the ontological argument is just such an attempt to define God into existence.) And this claim is somehow connected with Kant's famous but perplexing dictum that being (or existence) is not a real predicate or property. But how shall we understand Kant here? What does it mean to say that existence isn't (or is) a real property?
 Apparently Kant thinks this is equivalent to or follows from what he puts variously as "the real contains no more than the merely possible"; "the content of both (i.e., concept and object) must be one and the same"; "being is not the concept of something that could be added to the concept of thing," and so on. But what does all this mean? And how does it bear on the ontological argument? Perhaps Kant is thinking along the following lines. In defining a concept -- bachelor, let's say, or prime number -- one lists a number of properties that are severally necessary and jointly sufficient for the concept's applying to something. That is, the concept applies to a given thing only if that thing has each of the listed properties, and if thing does have them all, then the concept in question applies to it. So, for example, to define the concept bachelor we list such properties as being unmarried, being male, being over the age of twenty-five, and the like. Take any one of these properties: a thing is a bachelor only if it has it, and if a thing has all of them, then it follows that it is a bachelor.
 Now suppose you have a concept C that has application contingently if at all. That is to say, it is not necessarily true that there are things to which this concept applies. The concept bachelor would be an example; the proposition there are bachelors, while true, is obviously not necessarily true. And suppose P1, P2, ... , Pn, are the properties jointly sufficient and severally necessary for something's falling under C. Then C can be defined as follows:
A thing x is an instance of C (i.e., C applies to x) if and only if x has P1, P2, ..., Pn.
 Perhaps Kant's point is this. There is a certain kind of mistake here we may be tempted to make. Suppose P1..., Pn are the defining properties for the concept bachelor. We might try to define a new concept superbachelor by adding existence to P1 ,...,Pn. That is, we might say
x is a superbachelor if and only if x has P1 - Pn, and x exists.
 Then (as we might mistakenly suppose) just as it is a necessary truth that bachelors are unmarried, so it is a necessary truth that superbachelors exist. And in this way it looks as if we've defined super-bachelors into existence.
 But of course this is a mistake, and perhaps that is Kant's point. For while indeed it is a necessary truth that bachelors are unmarried, what this means is that the proposition
(8) Everything that is a bachelor is unmarried
is necessarily true.
 Similarly, then,
(9) Everything that is a superbachelor exists
will be necessarily true. But obviously it doesn't follow that there are any superbachelors. All that follows is that
(10) All the superbachelors there are exist.
which is not really very startling. If it is a contingent truth, furthermore, that there are bachelors, it will be equally contingent that there are super-bachelors. We can see this by noting that the defining properties of the concept bachelor are included among those of superbachelor; it is a necessary truth, therefore, that every superbachelor is a bachelor. This means that
(11) There are some superbachelors
(12) There are some bachelors.
 But then if (12) is contingent, so is (11). Indeed, the concepts bachelor and superbachelor are equivalent in the following sense: it is impossible that there exists an object to which one but not the other of these two concepts applies. We've just seen that every superbachelor must be a bachelor. Conversely, however, every bachelor is a superbachelor: for every bachelor exists and every existent bachelor is a superbachelor. Now perhaps we can put Kant's point more exactly. Suppose we say that a property or predicate P is real only if there is some list of properties P1 to Pn such that the result of adding P to the list does not define a concept equivalent (in the above sense) to that defined by the list. It then follows, of course, that existence is not a real property or predicate. Kant's point, then, is that one cannot define things into existence because existence is not a real property or predicate in the explained sense.
2. The Irrelevance of Kant's Objection
 If this is what he means, he's certainly right. But is it relevant to the ontological argument? Couldn't Anselm thank Kant for this interesting point and proceed merrily on his way? Where did he try to define God into being by adding existence to a list of properties that defined some concept? According to the great German philosopher and pessimist Arthur Schopenhauer, the ontological argument arises when "someone excogitates a conception, composed out of all sorts of predicates, among which, however, he takes care to include the predicate actuality or existence, either openly or wrapped up for decency's sake in some other predicate, such as perfection, immensity, or something of the kind." If this were Anselm's procedure -- if he had simply added existence to a concept that has application contingently if at all -- then indeed his argument would be subject to the Kantian criticism. But he didn't, and it isn't.
 The usual criticisms of Anselm's argument, then, leave much to be desired. Of course, this doesn't mean that the argument is successful, but it does mean that we shall have to take an independent look at it. What about Anselm's argument? Is it a good one? The first thing to recognize is that the ontological argument comes in an enormous variety of versions, some of which may be much more promising than others. Instead of speaking of the ontological argument, we must recognize that what we have here is a whole family of related arguments. (Having said this I shall violate my own directive and continue to speak of the ontological argument.)
3. The Argument Restated
 Let's look once again at our initial schematization of the argument. I think perhaps it is step (2)
(2) Existence in reality is greater than existence in the understanding alone
that is most puzzling here. Earlier we spoke of the properties in virtue of which one being is greater, just as a being, than another. Suppose we call them great-making properties. Apparently Anselm means to suggest that existence is a great-making property. He seems to suggest that a nonexistent being would be greater than in fact it is, if it did exist. But how can we make sense of that? How could there be a nonexistent being anyway? Does that so much as make sense?
 Perhaps we can put this perspicuously in terms of possible worlds. You recall that an object may exist in some possible worlds and not others. There are possible worlds in which you and I do not exist; these worlds are impoverished, no doubt, but are not on that account impossible. Furthermore, you recall that an object can have different properties in different worlds. In the actual world Paul I. Zwier is not a good tennis player; but surely there are worlds in which he wins the Wimbledon Open. Now if a person can have different properties in different worlds, the he can have different degrees of greatness in different worlds. In the actual world Raquel Welch has impressive assets; but there is a world RW which she is fifty pounds overweight and mousey. Indeed, there are worlds in which she does not so much as exist. What Anselm means to be suggesting, I think, is that Raquel Welch enjoys very little greatness in those worlds in which she does not exist. But of course this condition is not restricted to Miss Welch. What Anselm means to say, most generally, is that for any being x and worlds W and W', if x exists in W but not in W',then x's greatness in W exceeds x's greatness in W'. Or, more modestly, perhaps he means to say that if a being does not exist in a world W (and there is a world which x does exist), then there is at least one world in which the greatness of x exceeds the greatness of x in W. Suppose Raquel Welch does not exist some world W. Anselm means to say that then at least one possible world in which she has degree of greatness that exceeds the degree greatness she has in that world W. (It is plausible indeed, to go much further and hold that she no greatness at all in worlds in which she does not exist.)
 But now perhaps we can restate the whole argument in a way that gives us more insight into its real structure. Once more, use the term "God" to abbreviate the phrase "the being than which it is not possible that there be a greater." Now suppose
(13) God does not exist in the actual world.
Add the new version of premise (2):
(14) For any being x and world W, if x does not exist in W, then there is a world W' such that the greatness of x in W' exceeds the greatness of x in W.
Restate premise (3) in terms of possible worlds
(15) There is a possible world in which God exists.
 And continue on:
(16) If God does not exist in the actual world, then there is a world W' such that the greatness of God in W' exceeds the greatness of God in the actual world. [from (14)]
(17) So there is a world W' such that the greatness of God in W' exceeds the greatness of God in the actual world. [(13) and (16)]
(18) So there is a possible being x and a world W' such that the greatness of x in W' exceeds the greatness of God in actuality. [(17)]
(19) Hence it's possible that there be a being greater than God is. [(18)]
(20) So it's possible that there be a being greater than the being than which it's not possible that there be a greater. [(19), replacing "God" by what it abbreviates]
 But surely
(21) It's not possible that there be a being greater than the being than which it's not possible that there be a greater.
 So (13) [with the help of premises (14) and (15)] appears to imply (20), which, according to (21), is necessarily false. Accordingly, (13) is false. So the actual world contains a being than which it's not possible that there be a greater -- that is, God exists.
 Now where, if anywhere, can we fault this argument? Step (13) is the hypothesis for reductio, the assumption to be reduced to absurdity, and is thus entirely above reproach. Steps (16) through (20) certainly look as if they follow from the items they are said to follow from. So that leaves only (14), (15), and (20). Step (14) says only that it is possible that God exists. Step (15) also certainly seems plausible: if a being doesn't even exist in a given world, it can't have much by way of greatness in that world. At the very least it can't have its maximum degree of greatness -- a degree of greatness that it does not excel in any other world -- in a world where it doesn't exist. And consider (20): surely it has the ring of truth. How could there be a being greater than the being than which it's not possible that there be a greater? Initially, the argument seems pretty formidable.
4. Its Fatal Flaw
 But there is something puzzling about it. We can see this if we ask what sorts of things (14) is supposed to be about. It starts off boldly: "For any being x and world W, ..." So (14) is talking about worlds and beings. It says something about each world-being pair. And (16) follows from it, because (16) asserts of God and the actual world something that according to (14) holds of every being and world. But then if (16) follows from (14), God must be a being. That is, (16) follows from (14) only with the help of the additional premise that God is a being. And doesn't this statement -- that God is a being -- imply that there is or exists a being than which it's not possible that there be a greater? But if so, the argument flagrantly begs the question; for then we can accept the inference from (14) to (16) only if we already know that the conclusion is true.
 We can approach this same matter by a slightly different route. I asked earlier what sorts of things (14) was about; the answer was: beings and worlds. We can ask the same or nearly the same question by asking about the range of the quantifiers -- "for any being," "for any world -- in (14). What do these quantifiers range over? If we reply that they range over possible worlds and beings -- actually existing beings -- then the inference to (16) requires the additional premise that God is an actually existing being, that there really is a being than which it is not possible that there be a greater. Since this is supposed to be our conclusion, we can't very gracefully add it as a premise. So perhaps the quantifiers don't range just over actually existing beings. But what else is there? Step (18) speaks of a possible being -- a thing that may not in fact exist, but could exist. Or we could put it like this. A possible being is a thing that exists in some possible world or other; a thing x for which there is a world W, such that if W had been actual, x would have existed. So (18) is really about worlds and possible beings. And what it says is this: take any possible being x and any possible world W. If x does not exist in W, then there is a possible world W' where x has a degree of greatness that surpasses the greatness that it has in W. And hence to make the argument complete perhaps we should add the affirmation that God is a possible being.
 But are there any possible beings -- that is, merely possible beings, beings that don't in fact exist? If so, what sorts of things are they? Do they have properties? How are we to think of them? What is their status? And what reasons are there for supposing that there are any such peculiar items at all?
 These are knotty problems: Must we settle them in order even to consider this argument? No. For instead of speaking of possible beings and the worlds in which they do or don't exist, we can speak of properties and the worlds in which they do or don't have instances, are or are not instantiated or exemplified. Instead of speaking of a possible being named by the phrase, "the being than which it's not possible that there be a greater," we may speak of the property having an unsurpassable degree of greatness -- that is, having a degree of greatness such that it's not possible that there exist a being having more. And then we can ask whether this property is instantiated in this or other possible worlds. Later on I shall show how to restate the argument this way. For the moment please take my word for the fact that we can speak as freely as we wish about possible objects; for we can always translate ostensible talk about such things into talk about properties and the worlds in which they are or are not instantiated.
 The argument speaks, therefore, of an unsurpassably great being -- of a being whose greatness is not excelled by any being in any world. This being has a degree of greatness so impressive that no other being in any world has more. But here we hit the question crucial for this version of the argument. Where does this being have that degree of greatness? I said above that the same being may have different degrees of greatness in different worlds; in which world does the possible being in question have the degree of greatness in question?
 All we are really told, in being told that God is a possible being, is this: among the possible beings there is one that in some world or other has a degree of greatness that is nowhere excelled.
 And this fact is fatal to this version of the argument. I said earlier that (21) has the ring of truth; a closer look (listen?) reveals that it's more of a dull thud. For it is ambiguous as between
(21') It's not possible that there be a being whose greatness surpasses that enjoyed by the unsurpassably great being in the worlds where its greatness is at a maximum
(21'') It's not possible that there be a being whose greatness surpasses that enjoyed by the unsurpassably great being in the actual world.
 There is an important difference between these two. The greatest possible being may have different degrees of greatness in different worlds. Step (21') points to the worlds in which this being has its maximal greatness; and it says, quite properly, that the degree of greatness this being has in those worlds is nowhere excelled. Clearly this is so. The greatest possible being is a possible being who in some world or other has unsurpassable greatness. Unfortunately for the argument, however, (21') does not contradict (20). Or to put it another way, what follows from (13) [together with (14) and (15)] is not the denial of (21'). If that did follow, then the reductio would be complete and the argument successful. But what (20) says is not that there is a possible being whose greatness exceeds that enjoyed by the greatest possible being in a world where the latter's greatness is at a maximum; it says only that there is a possible being whose greatness exceeds that enjoyed by the greatest possible being in the actual world -- where, for all we know, its greatness is not at a maximum. So if we read (21) as (21'), the reductio argument falls apart.
 Suppose instead we read it as (21''). Then what it says is that there couldn't be a being whose greatness surpasses that enjoyed by the greatest possible being in Kronos, the actual world. So read, (21) does contradict (20). Unfortunately, however, we have no reason, so far, for thinking that (21'') is true at all, let alone necessarily true. If, among the possible beings, there is one whose greatness in some world or other is absolutely maximal -- such that no being in any world has a degree of greatness surpassing it -- then indeed there couldn't be a being that was greater than that. But it doesn't follow that this being has that degree of greatness in the actual world. It has it in some world or other but not necessarily in Kronos, the actual world. And so the argument fails. If we take (21) as (21'), then it follows from the assertion that God is a possible being; but it is of no use to the argument. If we take it as (21''), on the other hand, then indeed it is useful in the argument, but we have no reason whatever to think it true. So this version of the argument fails.
5. A Modal Version of the Argument
 But of course there are many other versions; one of the argument's chief features is its many-sided diversity. The fact that this version is unsatisfactory does not show that every version is or must be. Professors Charles Hartshorne and Norman Malcolm claim to detect two quite different versions of the argument in Anselm's work. In the first of these versions existence is held to be a perfection or a great-making property; in the second it is necessary existence. But what could that amount to? Perhaps something like this. Consider a pair of beings A and B that both do in fact exist. And suppose that A exists in every other possible world as well -- that is, if any other possible world has been actual, A would have existed. On the other hand, B exists in only some possible worlds; there are worlds W such that had any of them been actual, B would not have existed. Now according to the doctrine under consideration, A is so far greater than B. Of course, on balance it may be that A is not greater than B; I believe that the number seven, unlike Spiro Agnew, exists in every possible world; yet I should be hesitant to affirm on that account that the number seven is greater than Agnew. Necessary existence is just one of several great-making properties, and no doubt Agnew has more of some of these others than does the number seven. Still, all this is compatible with saying that necessary existence is a great-making property. And given this notion, we can restate the argument as follows:
(22) It is possible that there is a greatest possible being.
(23) Therefore, there is a possible being that in some world W' or other has a maximum degree of greatness -- a degree of greatness that is nowhere exceeded.
(24) A being B has the maximum degree of greatness in a given possible world W only if B exists in every possible world.
 (22) and (24) are the premises of this argument; and what follows is that if W' had been actual, B would have existed in every possible world. That is, if W' had been actual, B's nonexistence would have been impossible. But logical possibilities and impossibilities do not vary from world to world. That is to say, if a given proposition or state of affairs is impossible in at least one possible world, then it is impossible in every possible world. There are no propositions that in fact are possible but could have been impossible; there are none that are in fact impossible but could have been possible. Accordingly, B's nonexistence is impossible in every possible world; hence it is impossible in this world; hence B exists and exists necessarily.
6. A Flaw in the Ointment
 This is an interesting argument, but it suffers from at least one annoying defect. What it shows is that if it is possible that there be a greatest possible being (if the idea of a greatest possible being is coherent) and if that idea includes necessary existence, then in fact there is a being that exists in every world and in some world has a degree of greatness that is nowhere excelled. Unfortunately it doesn't follow that the being in question has the degree of greatness in question in Kronos, the actual world. For all the argument shows, this being might exist in the actual world but be pretty insignificant here. In some world or other it has maximal greatness; how does this show that it has such greatness in Kronos?
 But perhaps we can repair the argument. J. N. Findlay once offered what can only be called an ontological disproof of the existence of God. Findlay begins by pointing out that God, if He exists, is an "adequate object of religious worship." But such a being, he says, would have to be a necessary being; and, he adds, this idea is incredible "for all who share a contemporary outlook." "Those who believe in necessary truths which aren't merely tautological think that such truths merely connect the possible instances of various characteristics with each other; they don't expect such truths to tell them whether there will be instances of any characteristics. This is the outcome of the whole medieval and Kantian criticism of the ontological proof." I've argued above that "the whole medieval and Kantian criticism" of Anselm's argument may be taken with a grain or two of salt. And certainly most philosophers who believe that there are necessary truths, believe that some of them do tell us whether there will be instances of certain characteristics; the proposition there are no married bachelors is necessarily true, and it tells us that there will be no instances whatever of the characteristic married bachelor. Be that as it may what is presently relevant in Findlay's piece is this passage:
Not only is it contrary to the demands and claims inherent in religious attitudes that their object should exist "accidentally"; it is also contrary to these demands that it should possess its various excellences in some merely adventitious manner. It would be quite unsatisfactory from the religious stand point, if an object merely happened to be wise, good, powerful, and so forth, even to a superlative degree. ... And so we are led on irresistibly, by the demands inherent in religious reverence, to hold that an adequate object of our worship must possess its various excellences in some necessary manner.
 I think there is truth in these remarks. We could put the point as follows. In determining the greatness of a being B in a world W, what counts is not merely the qualities and properties possessed by B in W; what B is like in other worlds is also relevant. Most of us who believe in God think of Him as a being than whom it's not possible that there be a greater. But we don't think of Him as a being who, had things been different, would have been powerless or uninformed or of dubious moral character. God doesn't just happen to be a greatest possible being; He couldn't have been otherwise.
 Perhaps we should make a distinction here between greatness and excellence. A being's excellence in a given world W, let us say, depends only upon the properties it has in W; its greatness in W depends upon these properties but also upon what it is like in other worlds. Those who are fond of the calculus might put it by saying that there is a function assigning to each being in each world a degree of excellence; and a being's greatness is to be computed (by someone unusually well informed) by integrating its excellence over all possible worlds. Then it is plausible to suppose that the maximal degree of greatness entails maximal excellence in every world. A being, then, has the maximal degree of greatness in a given world W only if it has maximal excellence in every possible world. But maximal excellence entails omniscience, omnipotence, and moral perfection. That is to say, a being B has maximal excellence in a world W only if B has omniscience, omnipotence, and moral perfection in W -- only if B would have been omniscient, omnipotent, and morally perfect if W had been actual.
7. The Argument Restated
 Given these ideas, we can restate the present version of the argument in the following more explicit way.
(25) It is possible that there be a being that has maximal greatness.
(26) So there is a possible being that in some world W has maximal greatness.
(27) A Being has maximal greatness in a given world only if it has maximal excellence in every world.
(28) A being has maximal excellence in a given world only if it has omniscience, omnipotence, and moral perfection in that world.
 And now we no longer need the supposition that necessary existence is a perfection; for obviously a being can't be omnipotent (or for that matter omniscient or morally perfect) in a given world unless it exists in that world. From (25), (27), and (28) it follows that there actually exists a being that is omnipotent, omniscient, and morally perfect; this being, furthermore, exists and has these qualities in every other world as well. For (26), which follows from (25), tells us that there is a possible world W', let's say, in which there exists a being with maximal greatness. That is, had W' been actual, there would have been a being with maximal greatness. But then according to (27) this being has maximal excellence in every world. What this means, according to (28), is that in W' this being has omniscience, omnipotence, and moral perfection in every world. That is to say, if W' had been actual, there would have existed a being who was omniscient and omnipotent and morally perfect and who would have had these properties in every possible world. So if W' had been actual, it would have been impossible that there be no omnipotent, omniscient, and morally perfect being. But while contingent truths vary from world to world, what is logically impossible does not. Therefore, in every possible world W it is impossible that there be no such being; each possible world W is such that if it had been actual, it would have been impossible that there be no such being. And hence it is impossible in the actual world (which is one of the possible worlds) that there be no omniscient, omnipotent, and morally perfect being. Hence there really does exist a being who is omniscient, omnipotent, and morally perfect and who exists and has these properties in every possible world. Accordingly these premises, (25), (27), and (28), entail that God, so thought of, exists. Indeed, if we regard (27) and (28) as consequences of a definition -- a definition of maximal greatness -- then the only premise of the argument is (25).
 But now for a last objection suggested earlier. What about (25)? It says that there is a possible being having such and such characteristics. But what are possible beings? We know what actual beings are -- the Taj Mahal, Socrates, you and I, the Grand Teton -- these are among the more impressive examples of actually existing beings. But what is a possible being? Is there a possible mountain just like Mt. Rainier two miles directly south of the Grand Teton? If so, it is located at the same place as the Middle Teton. Does that matter? Is there another such possible mountain three miles east of the Grand Teton, where Jenny Lake is? Are there possible mountains like this all over the world? Are there also possible oceans at all the places where there are possible mountains? For any place you mention, of course, it is possible that there be a mountain there; does it follow that in fact there is a possible mountain there?
 These are some questions that arise when we ask ourselves whether there are merely possible beings that don't in fact exist. And the version of the ontological argument we've been considering seems to make sense only on the assumption that there are such things. The earlier versions also depended on that assumption; consider for example, this step of the first version we considered:
(18) So there is a possible being x and a world W' such that the greatness of x in W' exceeds the greatness of God in actuality.
 This possible being, you recall, was God Himself, supposed not to exist in the actual world. We can make sense of (18), therefore, only if we are prepared to grant that there are possible beings who don't in fact exist. Such beings exist in other worlds, of course; had things been appropriately different, they would have existed. But in fact they don't exist, although nonetheless there are such things.
 I am inclined to think the supposition that there are such things -- things that are possible but don't in fact exist -- is either unintelligible or necessarily false. But this doesn't mean that the present version of the ontological argument must be rejected. For we can restate the argument in a way that does not commit us to this questionable idea. Instead of speaking of possible beings that do or do not exist in various possible worlds, we may speak of properties and the worlds in which they are or are not instantiated. Instead of speaking of the possible fat man in the corner, noting that he doesn't exist, we may speak of the property being a fat man in the corner, noting that it isn't instantiated (although it could have been). Of course, the property in question, like the property being a unicorn, exists. It is a perfectly good property which exists with as much equanimity as the property of equininity, the property of being a horse. But it doesn't happen to apply to anything. That is, in this world it doesn't apply to anything; in other possible worlds it does.
8. The Argument Triumphant
 Using this idea we can restate this last version of the ontological argument in such a way that it no longer matters whether there are any merely possible beings that do not exist. Instead of speaking of the possible being that has, in some world or other, a maximal degree of greatness, we may speak of the property of being maximally great or maximal greatness. The premise corresponding to (25) then says simply that maximal greatness is possibly instantiated, i.e., that
(29) There is a possible world in which maximal greatness is instantiated.
 And the analogues of (27) and (28) spell out what is involved in maximal greatness:
(30) Necessarily, a being is maximally great only if it has maximal excellence in every world
(31) Necessarily, a being has maximal excellence in every world only if it has omniscience, omnipotence, and moral perfection in every world.
 Notice that (30) and (31) do not imply that there are possible but nonexistent beings -- any more than does, for example,
(32) Necessarily, a thing is a unicorn only if it has one horn.
 But if (29) is true, then there is a possible world W such that if it had been actual, then there would have existed a being that was omnipotent, omniscient, and morally perfect; this being, furthermore, would have had these qualities in every possible world. So it follows that if W had been actual, it would have been impossible that there be no such being. That is, if W had been actual,
(33) There is no omnipotent, omniscient, and morally perfect being
would have been an impossible proposition. But if a proposition is impossible in at least one possible world, then it is impossible in every possible world; what is impossible does not vary from world to world. Accordingly (33) is impossible in the actual world, i.e., impossible simpliciter. But if it is impossible that there be no such being, then there actually exists a being that is omnipotent, omniscient, and morally perfect; this being, furthermore, has these qualities essentially and exists in every possible world.
 What shall we say of this argument? It is certainly valid; given its premise, the conclusion follows. The only question of interest, it seems to me, is whether its main premise -- that maximal greatness is possibly instantiated -- is true. I think it is true; hence I think this version of the ontological argument is sound.
 But here we must be careful; we must ask whether this argument is a successful piece of natural theology, whether it proves the existence of God. And the answer must be, I think, that it does not. An argument for God's existence may be sound, after all, without in any useful sense proving God's existence. Since I believe in God, I think the following argument is sound:
Either God exists or 7 + 5 = 14
It is false that 7 + 5 = 14
Therefore God exists.
 But obviously this isn't a proof; no one who didn't already accept the conclusion, would accept the first premise. The ontological argument we've been examining isn't just like this one, of course, but it must be conceded that not everyone who understands and reflects on its central premise -- that the existence of a maximally great being is possible -- will accept it. Still, it is evident, I think, that there is nothing contrary to reason or irrational in accepting this premise. What I claim for this argument, therefore, is that it establishes, not thetruth of theism, but its rational acceptability. And hence it accomplishes at least one of the aims of the tradition of natural theology.