the ontological argument thesis statement

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4 The Ontological Argument

Brian Leftow is the Nolloth Professor of Philosophy of the Christian Religion at the University of Oxford.

• Published: 02 September 2009
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The term “ontological argument” was Kant's name for one member of a family of arguments that began with Anselm of Canterbury. These arguments all try to prove God's existence a priori, via reasoning about the entailments of a particular description of God. The description almost always involves God's greatness or perfection. Where it does not, the argument has a premise justified by God's greatness or perfection. So these arguments might better be called arguments from perfection. This article deals with the main arguments from perfection and criticisms thereof in historical order. It first explicates Anselm's key phrase “something than which no greater can be thought” and then takes up his reasoning, then the question of whether its premises are true.

The term “ontological argument” was Kant's name for one member of a family of arguments that began with Anselm of Canterbury. These arguments all try to prove God's existence a priori, via reasoning about the entailments of a particular description of God. The description almost always involves God's greatness or perfection. Where it does not, the argument has a premise justified by God's greatness or perfection. 1 So these arguments might better be called arguments from perfection.

I deal with the main arguments from perfection and criticisms thereof in historical order.

Anselm: Proslogion 2

Anselm gave the first argument from perfection in his Proslogion (1078). The key passage (in ch. 2) is this:

We believe [God] to be something than which nothing greater can be thoughtThe Foolwhen he hears“something than which nothing greater can be thought,” understands what he hears, and what he understands is in his intellect. (But) it cannot exist in the intellect alone. For if it exists only in the intellect, it can be thought to exist also in reality, which is greater. If therefore itexists only in the intellect, this same thing than which a greater cannot be thought, is a thing than which a greater can be thought. But this surely cannot be. So something than which no greater can be thoughtexistsboth in the intellect and in reality. (Charlesworth 1965 , 116, my translation)

I first explicate Anselm's key phrase “something than which no greater can be thought” (henceforth “a G”). I then take up his reasoning, then the question of whether its premises are true.

“A G” is an indefinite description. Its form lets many things satisfy it (as with “something brown and red” and “something canine”). What the Fool understands is this description. A natural thought would be that what is “in his intellect,” if not just a token string of words, is the property the description expresses, being a G . But as the argument proceeds, it supposes that the Fool “has in mind” some particular thing that has the property, an “it” that cannot exist in the mind alone. Anselm seems to suppose, in short, that by understanding the description a G , one comes into some sort of direct cognitive relation with something that is a G: one thinks of or refers to a particular G. For Anselm, then, being such that no greater can be thought means being such that no one nondivine can refer to a greater possible object, under any description. 2 A G is a greatest possible being to which we can refer. If there is hierarchy of greatness with a topmost level to which we can refer, then, “a G” automatically picks out only something(s) on the topmost level. If we can refer to an unending progression of ever greater possible beings, “a G” does not refer.

“A G” has a modal element: it speaks of items to which we can refer. To make sense of this “can,” I now introduce a bit of technical terminology that will be repeatedly useful. The sentence “Possibly there are ostriches” asserts that in at least one history the universe could have, ostriches would exist. In fact, one such history has taken place. “Possibly Churchill runs a three-minute mile” asserts that in at least one history the universe could have, Churchill pulls off this surprising feat. Churchill has not yet done this, and barring reincarnation or resurrection, he will not. So it appears that actual history is not any of those in which Churchill does this: no such history has taken place. But still, it's in some sense possible that he do so. Every sentence instancing the form possibly P asserts the existence of at least one history the universe could have in which P. Every sentence instancing the form necessarily P asserts that there is no history the universe could have in which ¬P. The sentence “necessarily 2+2=4” asserts that there is no history the universe could have in which this is false; that is that in every possible history, 2+2=4. Every sentence using “can,” of course, is equivalent to one using “possibly” (e.g., “There can be ostriches”).

Philosophers call histories the universe could have possible worlds . So we can now explicate Anselm this way: something x is a G only if no nondivine being in any possible world can refer to any being greater than x actually is. Now surely, for every possible being, possibly someone or other nondivine refers to it. If that's so, then possibly something is greater than x only if possibly someone refers to that greater thing. If so, we can simplify our account of a G , for being a G is equivalent to being something than which there can be no greater. From now on, let's take Anselm to be talking of this property.

In Proslogion 5, Anselm reasons that unless it is to be less than we can think it to be, a G must be “whatever it is better to be than not to be” (Charlesworth 1965 , 120), that is, have every attribute F such that having F is better than lacking F. Now if something had every such attribute, it would be a G (a G being one thing it is better to be than not to be). So if something is not a G, it lacks some F a G has, such that having this F is better than lacking F. Thus, Proslogion 5 implies that a G is greater than any possible non-G in at least one respect. Further, there is no respect in which a non-G surpasses a G: if a non-G has some attribute it is better to have than to lack, any G has this too, and only such attributes are respects in which something might surpass a G. 3 So overall, any G is greater than any non-G. As it's obvious that nothing in the material world is a G, we can infer that a G must at least be greater than any actual material object—including the universe. Here is a particularly impressive attribute: being greater than every other possible being in some respect and equaled by no other possible being in any respect. Such a G would be a most perfect possible being. Anselm would almost certainly hold that a G must be a most perfect possible being: if a G were not so, we could apparently think of a greater, namely one that was so. But his argument doesn't make use of this description.

Talk of Gs naturally raises questions like What is greatness? or Greater in what way? Anselm doesn't answer. But he clearly means greatness or being greater to be or involve some sort of value-property the God of Western theism has supremely. So Findlay's ( 1955 ) suggestion that we take these in terms of worthiness of worship can't be too far off the mark: let's say that greatness is either desert of worship or some combination of attributes on which this supervenes. 4 As it turns out, we needn't be more specific than this.

In Proslogion 4, Anselm asserts that

Df. God = that than which no greater can be thought,

the definite description implying that there is just one G. Anselm nowhere argues that there is just one. And this is not obvious. Something without a greater might nonetheless have an equal. If Anselm cannot rule it out that there could be two or more equal Gs, he faces a problem. For his argument will apply to as many possible Gs as there are, prima facie, and so if it works will prove that there are many Gs. If there are, the definite description “ that than which no greater” will not refer—in which case, Anselm's argument will prove that God does not exist, given (Df). Why just one possible G? One can only speculate:

i. Anselm argues that being a G entails being intrinsically simple, that is, not having distinct purely intrinsic attributes ( Proslogion 12; see Monologion 16–17). Suppose that this is so. For any x, being x is intrinsic to x: it is a matter settled entirely within x's boundaries, so to speak. Being simple is also intrinsic. So for any x, if x is simple, being simple and being x must be the same attribute. But then any simple being will be identical to x. So there can be at most one simple being. So if being a G entails being simple, there can be at most one G—and if attribute-identities are necessary, at most one possible G. Thus, there is at least a good argument from premises Anselm clearly accepted to back his belief that at most one possible being is a G.

ii. As the doctrine of divine simplicity is controversial, perhaps a better answer lies with what Anselm means by “greatness.” It's axiomatic in Western theism that whatever precisely worship is , at most one thing deserves it, and this thing coexists with no rivals for worship (see, e.g., Isaiah 40:25, 44:6–7, 46:5, 9). Anselm argues that any G must as such exist necessarily and necessarily be a G. If he's right, and it's also the case that maximal greatness in a possible world W excludes having a rival in W, then in no possible world does a G coexist with another G, and there is at most one possible G.

I now turn to Anselm's reasoning.

The Reasoning

On one reading, Anselm's premises are

1. Someone thinks of a possible object which is a G, and 2. If any possible G is thought of but not actual, it could have been greater than it actually is.

The reductio runs this way. By definition, if a possible object g is a G, no possible object in any possible state is greater than g actually is: g is in a state than which there is no greater. Let g be the G someone thinks of. Then, as a G, g is in a state than which there is no greater. Per (2), if g is not actual, g could have been greater than g actually is. So if g is not actual, g is not in a state than which there is no greater. So if g is not actual, g both is and is not in such a state. So g is actual. So a G exists.

The argument is valid. So let us ask if its premises are true.

Ontological Commitments?

(1) is not innocent. It asserts a relation between a thinker and a possible object that is actually a G, and so brings an object into our ontology. Anselm needs it to do so if (1) is to give him a G to which to apply (2). But then if he is not blatantly to beg the question of God's existence, Anselm must also assume that this possible object is there, and is a G, even if it does not exist. And odds are that Anselm did believe in nonexistent objects. 5 But this puts an unflattering gloss on his argument. For then it seems to amount to: grant that something actually is in a state with no greater. This thing either does or doesn't exist. But how could something that didn't so much as exist be as great as all that? And of course, if that's what the argument amounts to, it's hard to see why one should grant that something actually is in such a state. The step from this admission to the conclusion seems vanishingly small.

But Anselm's argument doesn't require his ontology. One could instead read (1) in light of non-Anselmian semantic assumptions. Suppose that one denied nonexistent objects, but held that one can use satisfiable descriptions as if they refer, whether or not they do, and can properly use claims like (2) to reason about satisfiers of descriptions, whether or not the descriptions are satisfied. This would amount to running Anselm's argument within a “free” logic. Such logics carry no ontological commitments. Taken in light of these new assumptions, (1) asserts only that someone tokens an indefinite description that is possibly satisfied. (1), then, turns out no more or less problematic than the claim that

1a. Possibly something is a G.

(2) assigns a degree of greatness to an object even if it does not actually exist; like (1), it must allow for nonexistent objects with greatness if it is not to beg the question. Even if the degree were automatically zero, this would still entail that nonexistents have properties. So we must replace (2) with a premise assigning greatness to nonexistents only in worlds in which they exist. The most straightforward replacement is probably

2a. If possibly something is a G, but actually nothing is a G, then in any possible world W in which something is a G, that G could be greater than it is in W.

If possibly something is a G, there is a world W in which something is a G. So (2a) immediately yields

2b. If possibly something is a G, but actually nothing is a G, then in some possible world W, something is a G but could be greater than it is in W.

Free logics let one use names or descriptions that do not refer as if they refer. So they reject the logical rules of universal instantiation (from “for all x, Φx,” infer Φs for any singular term s) and existential generalization (from any statement Fs, infer that there is something which is F; Lambert 1983 , 106–7). Thus, to show that Anselm's argument can go free-logical, one must state his reductio without using these rules. So here it is: given (1a) and (2b), if nothing is a G, then in some possible world W, something is a G but could be greater than it is in W. But it cannot be the case that in some world, a G could be greater than it is in that world: being a G is being in a state with no greater in any world. So it is not the case that nothing is a G. As far as I can see, then, given a free logic, Anselm's reductio goes through.

The Premises

If an argument is valid and its premises are true, its conclusion is true. I will not try to settle whether (1a) is true. But there is a case for (2a). For a G could be greater than it is in W just in case G lacks in W some great-making property compatible with the rest of its attributes in W. If no G exists, any G in any W lacks the property of existing in @, the actual world. But

3. For a G, for any W, existing in @ is great-making in W.

And if it is possible that a G exists, then for some G in some W, existing in @ is compatible with the rest of its attributes.

The controversial premise here is of course (3). There are two cases to consider here: W = @ and W ≠ @. For the first, I support (3) in two ways. One appeals to a general claim,

4. For any F and x, if x would be F were it to exist, then for x, existing in @ is F-making.

Suppose that Leftow would be human were he to exist. Then whoever gives Leftow existence ipso facto makes him be human. So for Leftow, existence is human-making: it makes him actually what he would be were he actual, and so human. But the properties a G would have if actual include being great. So for a G, existing in @ is great-making. Oppy ( 1995 ) suggests that (3) must rest on or be supplanted by some more general principle connecting greatness and existence, which atheists and agnostics would be reasonable to reject: “After all, there seems to be no good reason to suppose that existence in reality is a great-making property solely in the case of a [G]” (10, cf. 11). 6 But the only general principle needed is (4). (4) does not connect existence with greatness any more than with any other property, and I cannot see that atheists or agnostics have any particular reason to object to it.

The second line of argument begins that surely

5. Nothing that doesn't exist ought to be worshipped.

For worship is a kind of talking to, and it makes no sense to talk to something that isn't there. Atheists and agnostics will of course insist on (5). If (5) is true, then any G would be more deserving of worship if actual than if merely possible. For a merely possible G does not deserve worship at all, and an actual G does deserve worship. If greatness is worthiness of worship or whatever property(-ies) would subvene it, this implies that any G would be greater if actual than if merely possible, and because it is actual, not merely possible. So a G's being actual surely moves it at least a bit in the direction of maximal greatness. In fact, it moves it all the way, if (as it were) the G is all set to be great save for the little detail of actually existing. But then existing in @ is great-making for Gs.

Suppose, on the other hand, that W ≠ @. We then must ask why existing in some other world contributes to a G's greatness in W. One sort of reply appeals to arguments that necessary existence is great-making: if it is, then a fortiori existing in another world is. Now the claim that being a G entails existing necessarily leads to its own sort of argument from perfection. But it does so only given certain principles of modal logic. Pros. 2 does not commit itself to any such principles. So this sort of support would not make Pros. 2 depend on modal perfection-arguments. It would at most show that Pros . 2 has one root these other arguments do.

Another sort of response begins with two premises: that worship consists largely of giving thanks and praise, and that @, as it happens, contains concrete things whose maker might in some circumstance deserve thanks and praise for them, and for whose existence a G would account if it existed. A being that can have no greater is one than which none can be more worship-worthy. So it must deserve the greatest thanks and praise compatible with its nature. Those who worship, thank and praise God for their existence and for items in the world around them if they seem good. So if a G is to deserve maximal thanks and praise, it must be such as to deserve thanks and praise for whatever should inspire these in worlds it graces. All things in any way good in these worlds thus must owe it their very being; its contribution must suffice for their existence. The more complete this dependence, the greater the thanks and praise deserved. So another axis along which to magnify the thanks/praise a G is owed is depth of dependence: the deeper it is, the greater the thanks/praise deserved. One way dependence can be deeper is this: an item depending on the G could depend on it so thoroughly that it could not exist without the G's causal support. So via “perfect being” reasoning, we can conclude that whatever in any way ought to inspire thanks and praise and coexists with a G depends so completely on it for existence that it could not exist without the G.

Turning now to our G in W, @, again, contains many things warranting thanks and praise. Either some of these also exist in W, or none do. Suppose that some do. Then if the G does not exist in @, some things in W could have existed without depending on a G's contribution to their existence. But we've just ruled this out. And so if a G exists in W but not in @, nothing warranting thanks and praise in @ exists in W. If a G exists in W but not in @, nothing in @ could have depended on that G. For if it did, in any world, it would there depend on that G so completely that it could not exist without the G in any world—including @. So if the G does not exist in @, everything in @ is such that that G does not possibly account for its existence. If so, the G of W is not omnipotent: there are perfectly possible contingent beings for whose existence it cannot account. Surely omnipotence is great-making and exemplifiable; surely nothing can be a G without it. So existence in @ follows from a clearly great-making property. This may well make existing in @ great-making. In any case, on the present argument, nothing that does not exist in @ can be a G in any world. And so any G in any world, including W, exists in @.

I submit, then, that the amended, free-logical version of Proslogion 2's argument is valid, and one of its two premises has strong support.

Proslogion 3

In Proslogion 3, Anselm reasons that

something can be thought to be, which cannot be thought not to be. This is greater than what can be thought not to be. Whence if that than which no greater can be thought, can be thought not to be, itis not that than which no greater can be thoughtSo truly does something than which no greater can be thought exist, therefore, that it cannot be thought not to exist. (Charlesworth 1965 , 118)

Some claim that here Anselm gives a second argument for God's existence. They do so by reading Anselm this way:

6. Possibly something is a G, and 7. Being a G entails existing necessarily. So 8. Possibly a G exists necessarily. So 9. A G exists necessarily. So 10. A G exists.

I doubt on exegetical grounds that Anselm actually means to give this argument. But as Proslogion 3 has led some to this argument, we can discuss it here.

(6)–(10) is a valid argument in the S5 system of modal logic. Systems of modal logic—the logic of inferences involving “possibly” and “necessarily”—differ in the claims they make about the relations between possible worlds. The distinctive feature of the S5 system of modal logic is that in it, every world is possible relative to every other world: no matter which world were actual, the same set of worlds would be possible. To see how (6)–(10) works in such a set of worlds, let the boxes below represent all the worlds that are possible:

Let existing in at least one box represent being possible, and existing in all the boxes represent existing necessarily. (6) asserts that possibly a G exists. To represent this, we enter a G in one box:

Now (8) asserts not just that it's possible that a G exist, but that it's possible that a G exist necessarily . What this means, in terms of our boxes, is that a G is in one box, and in that box, it's true of the G that it exists in all the boxes (more precisely, all the boxes possible relative to it, which in S5 are all the boxes). So if (8) is true, G is in W1, and in W1 it's true that if G is in W1, it is also in W2–4, so that we have

Thus, given an S5 system of relations among the boxes, (8) does entail (9): G exists necessarily (in all boxes). Now if W1–4 are all the worlds there are, then one of them will turn out to be actual. G is in all of them, so no matter which one is actual, G will be actual with it. So (9) entails (10). In S5, this modal argument from perfection is valid.

Anselm's Real Argument

While Anselm probably did not intend (6)–(10), he did develop the first modal argument from perfection, in a slightly later work, the Reply to Gaunilo :

Whatever can be thought and does not exist, if it existed, would be ablenot to exist. (But) something than which no greater can be thoughtif it existed, would not be ablenot to exist—for which reason if it can be thought, it cannot not exist. (Charlesworth 1965 , 60)

Anselm's reasoning is this:

11. If it can be thought that a G exists and no G exists, any G would exist contingently if it did exist. 12. It is not possible that a G exist contingently. So 13. It is not the case that it can be thought that a G exists and no G exists.

14. If it can be thought that a G exists, some G exists. 15. It can be thought that a G exists. 16. Some G exists.

There are strong a priori arguments for (12). We can recast (11) as

17. If it is possible that a G exists and no G exists, any G would exist contingently if it did exist.

and alter the rest of the argument accordingly. The advantage of doing so is that (17) comes out true within the Brouwer system of modal logic, a weaker system S5 includes. The Brouwer system is weaker than S5 because it makes a weaker claim about possible worlds: rather than assert that every world is possible relative to every other, it asserts that relative possibility is symmetric: that if A is possible relative to B, B is possible relative to A. To see that (17) is true in Brouwer, suppose that these boxes represent all the possible worlds there are:

Let's say that W1 is actual, and relative to W1, W2 is possible. Our G, God, exists only in W2. So actually, God does not exist. But W2 is possible. So it's possible that God exist. Now suppose that W2 had been actual instead of W1. In that case, God would have been actual. But if relative possibility is symmetric, then because W2 is possible relative to W1, had W2 been actual, W1 would have been possible. So had W2 been actual, a world would have been possible in which God did not exist. So had W2 been actual, God would have existed contingently: which is to say that if our G possibly exists and does not, it would exist contingently if it did exist, assuming what the Brouwer system says about relations among possible worlds.

It's also worth noting that (6) and (12) suffice on their own to prove God's existence if the correct system of modal logic for metaphysical possibility includes Brouwer. To see this, suppose that these boxes represent all the possible worlds there are:

If W4 is actual, of course, God exists. Suppose instead that W3 is actual. Then if possibly God exists, God exists in at least one box possible relative to W3, and so God exists in W4. Per (12), God exists necessarily in W4. So if W4 were actual, God would exist necessarily, that is, in every world possible relative to W4. Per Brouwer, if W4 is possible relative to W3, W3 is also possible relative to W4. So God is necessary in W4 only if God also exists in W3. So if W3 is actual, God actually exists. So whether W3 or W4 is actual, God exists, and so given (6), (12), and Brouwer, God exists.

Modulo the change from (11) to (17), then, we can credit Anselm with the first valid modal argument from perfection.

Modal arguments from perfection face two difficulties. One lies in showing that the modal systems they invoke really are the correct logics for real metaphysical possibility. The other is epistemological. Consider Plantinga's (1974a) attribute of no-maximality, or being such that one does not coexist with a G. If this attribute is possibly exemplified, then given (12) and S5, being a G is not. A modal argument gives one reason to become a theist only if its proponent offers one not just the argument but some reason to believe the claim that being a G is possibly exemplified rather than the claim that no-maximality is. Many claim that modal arguments from perfection “beg the question” by asserting that being a G rather than no-maximality is possibly exemplified. They do not. Every argument asserts rather than justifies its own premises. If we need reason to believe in being a G rather than no-maximality, this shows not that a modal argument begs the question, but merely that another argument is needed, on behalf of one of its premises.

Gaunilo and Parody

Shortly after Anselm published the Proslogion , Gaunilo of Marmoutiers replied with a parody of the Proslogion 2 argument:

(An) island more excellent than all other lands truly exists somewhere in reality (if it exists) in your mind. For it is more excellent to exist not only in the mind but also in reality. So it must necessarily exist. For if it did not, any other land existing in reality would be more excellent. And so the island you conceived to be more excellent will not be more excellent. (Charlesworth 1965 , 164)

This parody isn't quite right, but we can construct the right sort on Gaunilo's behalf: let's take him to have meant that if we replace “a G” with “an island than which no greater can be thought,” the resulting argument works as well as Anselm's. There is no such island. So (says Gaunilo) we know the argument isn't sound, even if we can't pinpoint its flaw.

Unfortunately for Gaunilo, some sorts of parody are easily dismissed. There is no greatest possible island, for there can always be another island better at least for containing more of what makes any other island good (Plantinga 1974b, 91–92). 7 Oppy suggests that perhaps “the greatest possible island will have an infinite surface area andsupply of banana trees (etc.)Given (this) it will not be the case that it could have a greater supply of these things” ( 1995 , 165). Not so: for every order of infinity, there is a higher order. Oppy also suggests that traditional theists must concede the possibility of a greatest island, for their heaven is in effect an island than which no greater is possible, whose greatness lies inter alia in conferring “eternal life and infinite attributes on its inhabitants” (165). But on traditional theist belief, not heaven but God confers eternal life, and heaven is not surrounded by water. A physical heaven might be more like a new universe. But traditional theists don't hold that heaven is a best possible physical universe, only that being in heaven is the best possible state for us—and that it is so because heaven affords each of us our closest contact with God. Further, if greatness is (roughly) worship-worthiness, it's not true that a greatest possible island would be still greater if it existed. Nonexistent islands don't deserve worship, but neither do real ones, however lovely. Here, however, Oppy has a countersuggestion. Perhaps, he wonders, a greatest possible island would have “Godlike powers of providing for its inhabitants,” in which case, theists can rule out a greatest possible island only if they can rule out the possibility of “limited—localized—pantheism” (166). Oppy might have made this particularly pointed by asking Christians whether God could incarnate Himself in an island. But a divine island is great qua divine, not qua island. Despite Oppy, it remains the case that islands as such don't deserve worship. So Oppy has left the realm of Gaunilo's original parody, and moved into talk of what I call almost-Gods.

Deity is a kind. Most kinds can have more than one member: there are many cows. If deity is a kind, perhaps it can have many members, or could have had a different one. If it can or could have, parallel arguments from perfection will work for all possible Gods, yielding more Gods than monotheists want. So Anselm needs to show that

NO. There cannot in one possible world be two instances of deity .

One good argument for (NO) stems from a claim argued earlier, that a G must account for the existence of all good things with which it coexists. Gs are good things. So were there two Gs at once, each would have to account for the other's existence. Because —— accounts for ——'s existence is a transitive relation, this would entail that each accounts for its own existence. But this is impossible. Again, we saw earlier that a G's contribution must be both sufficient and necessary for the existence of all good things with which it coexists. If so, there cannot be two Gs at once. For suppose that A and B each suffice on their own for C's existence. Then without B's contribution, C could still exist, if A were still making its contribution. But then it's false that B's contribution is necessary for C's existence.

(NO) is true, and so multiple-G parodies are ruled out. So let's consider parodies via almost-Gods, deities whose only greater is God. Let's call one such being Zod, and say that Zod is just like God save for a slight difference in perfection we cannot conceive. Zod is to us indiscernible from God. But Zod cannot coexist with God. For God is uncreatable and has made everything other than Himself, and Zod would duplicate Him in these respects. And so we cannot accept arguments for both Zod and God. But we might read “a G” as “an almost-God than whom no greater can be thought”—describing a being whose only greater is God, who is not an almost-God. If Anselm can't explain why we should accept (1) and (2) on his reading of them but not on a parody-reading, we ought not assent to them on either reading. Further, if God is a necessary being, so is Zod. So given a modal logic including Brouwer, it's not the case both that Zod and that God possibly exist. 8 But if we can't tell Zod from God, how could we have reason to think one but not the other possible? Thus, parody yields reason to be agnostic about such claims as that being a G is possibly exemplified.

Almost-Gods threaten to multiply: perhaps for any particular degree of likeness to God, an almost-God like Him to that degree would be more worship-worthy if it existed than if it were merely possible. Whether it would, though, depends on what worship is. At least within Western monotheism, whose concept of worship Anselm presumably had in mind, worship is or includes praise without qualification or limit. What deserves only qualified or limited praise thus does not deserve worship. And anything that can have a superior can deserve only qualified or limited praise. It is great—but there can be a greater, and so its praise ought to be qualified accordingly. “O god, you are great—but there can be greater”: this does not sound like worship. If it isn't, and yet someone surpassable can deserve no more, nobody surpassable can deserve worship. Nothing can unless it has no possible greater simpliciter. And now here's the rub: an almost-God has no possible greater simpliciter only if it isn't possible that there be an Anselmian G. For as we've seen, a G is greater overall than any other possible being. If a G is possible, then, no almost-God can deserve worship, and so none can be more worship-worthy if actual. And so if a G is possible, one can dismiss this sort of parody—any reason to think a G possible gives one reason simply to ignore it. Perhaps, then, one can so tweak Anselm's property of greatness as to make parody difficult.

Here an objection arises. Polytheists worshipped; what they felt, did, and said is enough like what monotheists feel, do, and say to deserve the label. Some worshipped gods other gods outranked. So one can worship something surpassed. And so there is room for worship of almost-Gods. The tweaking move is at best trivial and at worst question-begging, for it so defines worship that only God can deserve it.

This objection is confused on at least two levels. For one thing, even if polytheists did worship, nothing follows about what deserved their worship: that something is worshipped implies nothing about whether it ought to be. And no polytheist god could deserve what monotheists call worship. In worship, monotheists give all their religious thanks and praise to God. So deserving worship in the Western-monotheist sense includes deserving all of one's religious thanks and praise. No polytheist god deserves all religious thanks and praise, for none is responsible for all of our blessings. So either polytheists misdirected monotheist worship at their gods or, more charitably, what polytheists did “in church” does not count as worship in the sense discussed above . Further, worship for Western monotheists includes the giving of thanks and praise without limit or qualification. Polytheists, just as such, cannot consistently do this for any single god. They must limit and qualify their praise for any god in light of what they must say to other gods: they should not praise Zeus for blessings Hera gave or praise Hera to a degree only Zeus deserves. In worship, monotheists give God all their religious loyalty. Polytheists, as such, cannot give all their religious loyalty in any act of worship. Polytheists' religious loyalties compete: time spent in Venus's temple is not spent in Mars's. Monotheists have only one temple to attend. If polytheists worship, then, their worship differs from monotheists'. There is a kind of worship only monotheists can give, for there are attitudes one can have only to a sole object of worship.

Next epicycle: perhaps one can define the almost-greatness of almost-Gods in terms of deserving almost-worship (or almost-sole-worship, etc.), and say that almost-Gods would be almost-greater if actual. What then? Well, the problem for a Pros . 2 parody comes in applying the parallel to (2a). There is no maximal degree of deserving almost-worship (as vs. worship). There is no state than which there is no almost-greater. So for every state an almost-God might be in, there is an almost-greater state something could be in, and so the parody-argument will fail. I now argue the no-maximal-degree claim.

God deserves worship. Maximal likeness to God would be duplication, and so would yield something deserving worship, not almost-worship. If likeness to God is graded on a dense or continuous scale, then there is no maximum likeness to God short of duplication: for every nonduplicate of God, something can be more like God than it is. If God deserves worship, becoming more like God is coming closer to deserving worship. So plausibly, becoming more like God is also coming closer to deserving almost-worship, or (once over the threshold for this) deserving ever more almost-worship. If likeness to God has no maximum short of deserving worship (by duplication), there is no maximum state of almost deserving worship (almost duplicating God). This doesn't entail that there's no maximum state of deserving almost-worship, but it surely suggests it.

Still, it's not implausible that in some cases likeness to God is a granular matter, that is, comes in discrete degrees, with a maximum just shy of duplication. For we can describe such a scale: just like God save for knowing four public truths God knows, or three, or twoOn such scales, if there are maximal states, they are along the lines of being just like God save for not knowing one public truth an omniscient being would know, or being unable to do one task omnipotence, could accomplish, or being able to commit one sin. I doubt that beings like this really are possible—what could keep someone who has all eternity to figure things out, is omnipotent, and knows all the other public truths from learning the last? Be that as it may, someone with just one of these defects would be more like God than someone with all three. But which defect leaves one closest to God? Would someone not quite omnipotent be more like God than someone not quite omniscient? Someone is most like a perfect being if he or she is unlike it only in the least important (“perfecting”) respect, and so this amounts to the question Which is least important: omniscience, omnipotence, or moral perfection? Given the shakiness of all intuitions here, the best reply may be that each one-defect being is more like God in his or her nondefective respects than anything defective in these respects is, but there's no answer to the question Which is most like God overall? This sparks a suggestion: perhaps each one-defect being is in a state with no greater short of being God, and so is maximally Godlike short of duplication. But this suggestion is correct only if there are no relevant gradations within each one-defect state, and that's questionable.

Consider possible beings just one truth short of public-truth omniscience. Some don't know this truth, some that. Which truth they don't know can affect their Godlikeness. Some truths are more important than others. So the lack of some truths is more important than the lack of others: it seems less important that God know the weight of a particular gnat in early Mesopotamia than that God know that floods kill. It's more Godlike (“perfecting”) to get important things right. So beings are less Godlike the more important the truths they lack. Again, lacking some truths entails greater cognitive defect than lacking others: not knowing about the gnat is minor, while not knowing that modus ponens is valid is major. But it would take some doing to show that there are least important truths or lacks or defects. If some truths or lacks are more important than others, none are least important, and a being is the more Godlike in knowledge the less important the truth it lacks (or the less important the lack of this truth, or the defect it entails), then not all not-quite-omniscient beings are equally Godlike and there probably is no such thing as a most-Godlike not-quite-omniscient being. Like comments apply to lacks of power and abilities to sin.

The more like God in greatness-relevant ways, the closer to deserving worship. So if there is no greatest nonduplicative likeness to God, for every possible being deserving almost-worship, there is a state something can be in that would put it closer to deserving worship, and so make it deserve more or greater almost-worship. If possibly God exists, then, there is no state than which there is no greater for almost-Gods. Of course, if God is impossible, then again no possible being can duplicate Him, and the points just made about greater likeness to God remain, for they did not turn on the claim that God possibly exists. Possible items can be graded for likeness with impossible ones; the more nearly circular a thing, the more it is like a circular square.

So the last-epicycle parodic argument doesn't go through. On the other hand, almost-Gods make harder the epistemic problem modal arguments face: it's hard to see how to back belief that possibly God exists over belief that possibly Zod exists. And with the modal arguments there in the background, one wonders how well one can argue for (1a). For (it seems) any reason to accept (1a) would have also to be a reason to favor God over Zod. But in fact, the dialectical situation is this. To take a modal argument as reason to believe in God, one must have reason to believe that God rather than Zod is possible. For modal arguments from perfection will work as well for Zod as for God. But to take the Pros . 2 argument as a reason, one need only have reason to believe that God is possible, rather than more reason to believe this than to believe that Zod is.

Considering parodies for the modal argument shows that the existence of God (or Zod) would have modal consequences. If God exists, then given Brouwer, it is not so much as possible that Zod does: it's necessarily false that Zod exists. So the existence of God would have consequences for modal truths not involving the concept of God: God would have a modal footprint. And Anselm in fact held that what necessary truths there are depends on God ( Cur Deus Homo II, 17).

The Fifth of Descartes' Meditations on First Philosophy ([ 1641 ] 1993 ) offers the last fully original argument from perfection. It begins from a general attempt to show that some conceptual truths are not just conceptual truths, but rather reveal facts about natures independent of the mind:

I find within meideas of certain things that, even if perhaps they do not exist anywhere outside me, still cannot be said to be nothing. And althoughI think them at will, nevertheless they are not something I have fabricated; rather they have their own true and immutable natures. For example, when I imagine a triangle, even if perhaps no such figure exists outside my thought anywhere in the world and never has, the triangle still has a certain determinate nature, essence or form which is unchangeable and eternal, which I did not fabricate, and which does not depend on my mind. This is evident from the fact that various properties can be demonstrated regarding this triangle (which) Iclearly acknowledge, whether I want to or not. For this reason they were not fabricated by meAll these properties are patently trueand thus they are something and not nothing. (42–43)

Descartes then suggests that the nature of God is akin to the nature of a triangle in being something mind-independent which the mind grasps:

The idea of God, that isof a supremely perfect being, is one I discover to be no less within me than the idea of any figurethat it belongs to God's nature that he always existsI understand no less clearly and distinctly thanwhen I demonstrate in regard to some figurethat somethingbelongs to the nature of that figureThusthe existence of God ought to have for me at least thecertainty that truths of mathematics (have). (43–44)

This promises a quasi-mathematical demonstration. Descartes' attempt to keep the promise runs this way:

Existence can no more be separated from the essence of God than the fact that its three angles equal two right angles can be separated from the essence of a triangleit isa contradiction to think of God (that is, a supremely perfect being) lacking existence (that is, lacking a perfection)it isnecessary for me to suppose God exists, once I have made the supposition that he has all perfections (since existence is one of the perfections)Not that my thought brings this about or imposes any necessity on anything, but rather the necessity of the thing itselfforces me to think this. (44)

Descartes then adds further reasons to believe that his idea of God is “an image of a true and immutable nature” (45). The broad outline of Descartes' argument, then, is this: he grasps what he claims are mind-independent truths about the kind of thing God would be if there were one. And uniquely, in the case of God, the mind-independent truths about the kind require that the kind has an instance. To try to show why, Descartes tries to show that “God does not exist” entails a contradiction.

It is surprisingly hard to say exactly what this last phase of Descartes' argument is up to. I offer three readings of it, one of which subdivides.

Meditation V: One Reading

On one reading, Descartes' premises are that

18. If God does not exist, a being with all perfections lacks a perfection, and 19. A being with all perfections lacks a perfection entails a contradiction.

If both are true, Descartes may think, then if God does not exist, a contradiction is true. But (18) is ambiguous, between

18a. If God does not exist, then if anything has all perfections, it lacks a perfection, and 18b. If God does not exist, there is something with all perfections which lacks a perfection. (Van Inwagen 1993 , 80–81)

To get a valid argument with (18a), we must read (19) as

19a. If anything has all perfections, it lacks a perfection entails a contradiction.

But (19a) is false. That conditional does not by itself entail a contradiction. It entails only that nothing has all perfections, which is what one would expect if a perfect being does not exist. So if the argument including (18a) is valid, it is unsound.

For Descartes, God is the sole possible being with all perfections, and so (18b) amounts to

20. If God does not exist, God exists and lacks a perfection.

(20) is false unless God actually does exist necessarily, in which case “God does not exist” is impossible and so implies anything. But then why should an atheist or agnostic accept (20)? It is on its face quite unintuitive. On another reading, (18b) asserts that if God does not exist, He “is” there, in some sense of “is” compatible with nonexistence, and has contradictory properties. This reading clearly commits us to a Meinongian ontology of nonexistent impossible objects, for it asserts that if God does not exist, He is one. On such views, “there is” in “there is something with all perfections which lacks a perfection” does not express existence. It is instead a “wide” quantifier ranging over existent and nonexistent objects. To get a valid argument with (18b), we must read (19) as

19b. There is something with all perfections which lacks a perfection entails a contradiction.

But with the quantifier read “widely,” (19b) is false. On a Meinongian ontology, it is no contradiction for there to “be” contradictory nonexistent objects. Such objects are perfectly normal features of reality. What would be contradictory would be for one of them to exist . So the (18)–(19) argument is unsound on two readings, and on a third has a counterintuitive premise supporting which would require another, independent argument for God's (necessary) existence. Let's therefore consider a different analysis.

Meditation V: Second Try

Med . V speaks of what we do and must suppose, that is, of what our idea of God includes. Descartes later offered a “synthetic” presentation of material from his Meditations, and as an argument to what he seems to claim is to the same effect as Meditation V gave:

To say that something is contained in the nature or concept of anything is the same as to say that it is true of that thing. But necessary existence is contained in the concept of God. Hence it is true to affirm that necessary existence exists in Him, or God Himself exists. (HR II 57)

Here the argument is in terms of concepts. There is also a reference to necessary existence, which suggests a modal argument. But by “necessary existence” Descartes means only actual existence the nature of the thing guarantees: that “actual existence is necessarilylinked to God's other attributes” (HR II 20). So Descartes may here suggest that the Med . V argument is really this:

21. For all x, if being F is part of the concept of x, then Fx. 22. It is part of the concept of God that if God's nature is what it is, God exists. So 23. If God's nature is what it is, God exists. 24. God's nature is what it is. So 25. God exists.

The problem here is that (21) is false. It's part of the concept of Santa that he has a beard, but it's false that Santa has a beard, for it's false that anything really both is Santa and is bearded. “Santa is bearded” doesn't say anything true. It is just the right thing to say if you're telling Santa stories.

But perhaps (21) is dispensable. All Descartes really needs is

21a. For all x, if being F is part of the concept of God, then Fgod.

One can read Descartes' Meditation III argument about the concept of God as an attempt to warrant (21a). It is, in effect, an argument that the concept of God has contents such that nobody has this concept unless it has an instance—that the causal story behind anyone's having that concept must include a God. If recent externalists are right, there are many such concepts, for example, water . And if the concept of a sort of item is externally determined in the right way, then something like (21a) will hold for it. Suppose that an appropriate externalist story about natural kind concepts is correct, and that water is a natural kind. Then because the concept of water is determined by the real external nature of water, if being H 2 O is part of that concept, it follows that water is H 2 O. It's not clear a priori why God or perfect being could not be an externally determined concept. And that Descartes was in general the patron saint of anti-externalism hardly precludes his claiming that there is one exception to it, which the argument from perfection reveals. On the other hand, any argument that externalism holds for the concept of God is ipso facto one that God really exists. If to back a premise in an argument for God, one needs a second, discrete argument for God, then the first argument cannot be stronger than the second and is not independent of it. So if it took such an argument to back (21a), an argument resting on (21a) would be useless.

Meditation V: Third Try

Our third reading of Meditation V begins by noting again its talk of God's essence and what it includes. Descartes later claimed that the Meditation V argument is:

That which we clearly and distinctly understand to belong to the true and immutable nature of anything, its essence, can be truly affirmed of that thingto exist belongs to [God]'s true and immutable nature; thereforeHe exists. (HR II 19)

In accord with this, we might render the Med . V argument as

26. If the “true and immutable nature” of x includes being F, then Fx. 27. The “true and immutable nature” of God includes existence. So 28. God exists.

To respect Descartes' claim that this somehow encapsulates Med . V, we might expand the argument by deriving (27) from

29. The “true and immutable nature” of God includes having all perfections, and 30. Existence is a perfection.

Perhaps Descartes did not see (21)–(25) and (26)–(30) as distinct. He distinguishes ideas that grasp “true and immutable natures” from ideas that are just “fictitiousdue to a mental synthesis” (HR II 20). If an idea does not have its content simply due to a mental operation, it grasps a mind-independent truth. That is, it has its content by grasping something that is somehow also extramentally the case. Descartes' thought, then, seems to be that some ideas grasp “natures” that have some status beyond them, the idea of God being one; for these ideas, the “nature” is just the idea's content, and so we can switch indifferently between nature-talk and talk of concepts (ideas' contents).

Descartes' talk of “true and immutable natures” has two functions in (26)–(30). One is trying to lend credibility to (29). If it's part of a thing's nature that it is F, says Descartes, we did not simply dream this up, and so we can trust our impression that such a thing would be F. But apart from this, it also sets up the claim that (27) and (29) concern some entity or truth independent of the mind. If there really is some entity or truth that logically requires that God exist, then there would be a contradiction in objective reality (not just in our ideas about it) if God did not.

Like (21), (26) is dubious but dispensable. All Descartes needs is (27), which we can recast as

27a. There is a “true and immutable nature” P which includes all perfections and is (uniquely) such that if it exists, it has an instance,

whence he can reason that

31. P exists. (27a, simplification) 32. If P exists, it has an instance. (27a, simplification) 33. P has an instance. (31, 32, MP)

Traits of our idea of God are supposed to assure us that it captures a “true and immutable nature.” Why is (27a)'s second conjunct supposed to be true? One story Descartes tells is the (18)–(19) argument. But in at least one place, he tells another story about why existence is uniquely inseparable from the divine essence:

It is not true that essence and existence can be thought the one apart from the other in Godbecause God is His existence. (HR II 228)

That God = God's existence explains the inseparability of God's essence and God's existence only if God = God's essence—a standard part of the doctrine of divine simplicity Descartes inherited from his Jesuit education. So what Descartes is really saying here is that the divine essence = the divine existence. The reason (27a) is true, then, could be that if there is a divine nature, it is identical with the existence of God. If this is so, then if there is in extramental reality such a nature, there is also such an existence—and so God exists. Perhaps Descartes' doctrine of divine simplicity, asserted in Meditation III, can help his argument in Meditation V.

Descartes: Objections and Replies

Publication of the Meditations led to a series of exchanges between Descartes and prominent intellectuals. The best criticisms of Descartes' argument from perfection came from Pierre Gassendi and Johannes Caterus. Caterus wrote:

Though it be conceded that an entity of the highest perfection implies existence by its very name, yet it does not follow that that very existence is anything actual in the real world, but merely that the concept of existence is insepatably united with the concept of highest being. (The) complex “existing lion” includes both lion andexistence, and includes them essentially, for if you take away either it will not be the same complexdoes not its existence flow from the essence of this composite “existent lion”? Yet (this) does not constrain either part of the complex to existTherefore, also, even thougha being of supreme perfection includes existence in the concept of its essence, yet it does not follow that its existence is anything actual. (HR II, 7–8)

One can put Caterus's thought this way: from premises about the content of a concept, only conclusions about the content of a concept can validly follow.

Descartes' reply in a nutshell is that his premises deal in “what belongs to the true and immutable essence of a thing,” not “what is attributed to it merely by a fiction of the intellect” (HR II 19)—that is, are not merely about concepts' contents, but about extramental facts. His criterion for this seems to be that elements of a “merely fictitious” nature can rightly be separated conceptually: winged horse is “fictitious” because we can rightly conceive of horses without wings (HR II 20). On the other hand, if elements FG belong together as part of a “true and immutable nature,” we cannot rightly conceive them apart: being F entails being G, or conversely (HR II 21). Thus, Descartes goes on to try to show that existence really does belong to God's “true and immutable nature” without merely reiterating his Med . V argument, by arguing that the nature of God's power itself entails His existence (HR II 21). But if one must show that some divine attribute entails God's existence to show that existence is of God's nature, Descartes has a problem. For if the Med . V argument really does include a premise about God's true, immutable nature including existence, it is then an argument for God the defense of whose premises requires another, independent argument for God's existence. If it is, it is dialectically useless. For if one can demonstrate God's existence a priori in another way, the Med . V argument is unneeded: it can't yield any further, independent warrant for belief in God. If one can't, it has an indefensible premise.

Gassendi wrote:

Existence is a perfection neither in God nor in anything else; it is rather that in the absence of which there is no perfectionthat which does not exist has neither perfection nor imperfection, and that which exists (has) its existenceas that by means of which the thing itself equally with its perfections is in existencenor if the thing lacks existence is it said to be imperfect, (but rather) to be nothing. (HR II 186)

Descartes' reply is that possible existence is a perfection in the case of a triangle, making “the idea of a triangle superior to the ideas of chimeras,” and similarly necessary existence is a perfection in God's case, making the idea of God superior to other ideas (HR II 228–29). This does not immediately address Gassendi's point about mere existence; perhaps Descartes means to add that any property a perfection entails is itself a perfection. This claim would not be implausible, as we see below in discussing Gödel.

Gassendi's second major argument was this:

Although you say that existence quite as much as other perfections is included in the idea of a being of the highest perfection, you (just) affirm what has to be proved, and assume your conclusion as a premise. For I might alsosay that in the idea of a perfect Pegasus (is) contained not only the perfection of having wings but also that of existing. For just as God is thought to be perfect in every kind of perfection, so is Pegasus thought to be perfect in its own kind. (HR II 187)

Descartes offers no reply to the parody. Perhaps he would treat “existing Pegasus” as he did Caterus's “existing lion”: the “complex” captures no “true, immutable nature”—since it's not the case that the attribute of being Pegasus is such that necessarily, if it exists, it has an instance—and so here we do not escape the conceptual order. The Pegasus argument from perfection, Descartes might say, falls to the Caterus objection. But if Descartes cannot support his claim that God's nature includes existence without independent a priori proof that God exists, Gassendi is right that it begs the question.

Leibniz worked intensely on arguments from perfection in the 1670s. He held that Descartes' argument was valid but incomplete, needing the addition of a proof that it is at least possible that God exists. His own preferred argument was modal:

If a being from whose essence existence follows is possibleit existsGod is a being from whose essence existence followsTherefore if God is possible, He exists. (Adams 1994 , 137, n.9)

“A being from whose essence existence follows” is just a necessary being. So Leibniz's argument is really that

If possibly a necessary being exists, it exists. God is by nature a necessary being. So If possibly God exists, God exists.

The first premise is just an instance of the characteristic axiom of the Brouwer system of modal logic; the argument is sound in Brouwer. The conclusion leaves Leibniz's case for God incomplete, needing, as Leibniz said of Descartes, a proof that possibly God exists. Leibniz tries to provide one.

Leibniz's possibility-argument (Plantinga 1965 , 54–56) treats God as the being whose nature is a conjunction of all and only perfections, perfections being properties that are “simple,” “positive,” and “absolute.” Simple properties do not consist of other properties. They are primitive. Positive properties are those whose natures do not include the negation of other properties. If the property F is a constituent of the property ¬F, every simple property is positive. Positive properties needn't be simple, though. F • G is a positive property if F and G are positive. A property is absolute if and only if its nature involves no limitations of any sort. Leibniz's argument, then, is in essence this: it's possible that God exist just in case all properties in the nature He'd have if actual are compatible. But if properties are simple, they cannot be incompatible because properties of which they consist are incompatible. If properties are positive, their natures do not include the negations of other properties. That is, for all FG, if F and G are positive, F's nature is not and does not include not having G, and G's is not and does not include not having F. But properties F and G are incompatible, thinks Leibniz, only if F includes ¬G, G includes ¬F, some property F includes includes ¬G, or some property G includes includes ¬F. Thus, if any absolute properties are simple and positive, they are compatible.

Leibniz's argument raises a number of questions: Are there simple, absolute, positive qualities? Do they include necessary existence? Do they include colors, and do colors pose a problem for the argument? Can the argument be parodied? And what about the gap between consistency and metaphysical possibility?

Simple, Positive Properties

Leibniz wanted this to come out a proof that God possibly exists, and so presumably took perfections to include such properties as omnipotence, omniscience, and perfect benevolence. These involve no limits of quantity or degree. Presumably they need not be instanced by an imperfect subject—they are compatible with “infinity” and “perfection.” So their natures involve no limitations in that respect. It is a limitation to be something with knowledge and will only if there is something better to be, and this is not at all clear. But these are not obviously unanalyzable; plausible accounts of each abound. Leibniz's likely reply would be to say that perfect power, knowledge, and goodness are primitive properties—that although we offer accounts of them in terms of (say) generic power, knowledge, and goodness, in metaphysical fact power (for instance) in general consists in a likeness to the perfect exemplar of power, which thus figures as a primitive constituent in the general, shareable attribute of power. This amounts to applying a resemblance-nominalist account of attributes to the divine case, letting God serve as the paradigm instance: and Leibniz was indeed a nominalist, and speaks of created attributes as imperfect imitations of divine attributes in his Monadology (#48). If the standard divine attributes come out primitive, then they are also positive, and we've already seen that they're “absolute.” Perhaps Leibniz can claim that necessary existence is the paradigm of which nonnecessary existence is an imperfect imitation. This claim is at least standard in theological tradition; one finds it, for example, in Anselm.

Colors are a problem for Leibniz. Phenomenal redness and greenness seem unanalyzable. They are also positive qualities of experience. They also seem absolute. For what limits are involved in seeming red? Not materiality: a discarnate soul could hallucinate in color, and plausibly in a hallucination something appears red. But no spot in any visual field can have both properties: they are incompatible. Now here Leibniz could perhaps reply that just for this reason, colors are not positive in his sense. Each is, after all, a determinate of a determinable, phenomenal color. And the nature of determinables may come to Leibniz's aid. For a plausible view of determinables would see them as simply disjunctions of their determinates, such that each n-tuple of the properties of which a determinable consists is internally inconsistent—in which case, each determinate implies the negation of each other determinate. If this is correct, the phenomenal colors are not Leibniz-positive. Each's nature in some manner contains the negation of the rest: certainly it entails these. So perhaps Leibniz's cause is not utterly hopeless here.

Parody and Possibility

Leibniz's argument does seem vulnerable to parody (Adams 1994 , 150–51). Nothing he says indicates that his simple perfections entail one another. And it's hard to see how he could allow this. If omniscience did entail omnipotence, say, it would not be in virtue of “containing” the negation of nonomnipotence (since it doesn't contain the negation of any property). If the perfections do not entail each other, it seems possible to conjoin all save omniscience with almost-omniscience. For as none contain the negation of any other property, none contain the negation of almost-omniscience. But then the other perfections are consistent with almost-omniscience—or at least Leibniz's argument gives us as much reason to think this as to think that the perfections are all consistent. And so the argument gives us as much reason to grant the possibility of a necessarily existing almost-omniscient almost-God as we do the existence of God. But they can't both be possible. Just because we do see that it is vulnerable to parody, it's clear that Leibniz has a problem with the gap between consistency and real metaphysical possibility. The concepts of God and almost-God are equally consistent, on his showing. But it cannot be that both are possible, for at most one of these beings really exists. So we can't take Leibniz to have shown that it is possible that God or an absolutely perfect being exists.

Kant's Critique of Pure Reason ([ 1781 ] 1956 ) is often treated as the death knell of arguments from perfection. Kant claimed against Descartes that “ ‘being’ isnota predicatewhich could be added to the concept of a thingIt is merely the positing of a thing” (A598/B626). This denies (30), at least if we assume that every perfection is expressed by a “predicate,” something that describes or characterizes an object. On this assumption, it is very nearly one of Gassendi's moves. Kant also argued this way:

34. All necessary truths are really conditional in form (“The absolute necessity of the judgment is only a conditioned necessity ofthe predicate in the judgment” [A703–4/B621–22]). 35. Any conditional expansion of a purported necessary existential truth would be analytic as well as existential. 36. There are no analytic existential propositions (A708/B626). 9 37. So no necessary proposition asserts the existence of anything.

(36) and (37) follow Hume. But Kant's way of supporting them is, for better or worse, his own. If (36) or (37) is true, then Descartes' argument cannot be sound, if its contention is in effect that “God exists” is analytic. If an argument is unsound, it either has a false premise or makes an invalid inference, and one who asserts that an argument is unsound must back the claim by showing one or the other. Kant's denial of (30) does this.

Kant supports (34) with only an example, that “necessarily a triangle has three sides” is really “necessarily, for all x, if x is a triangle, x has three sides” (A704/B622). His case for (35) is left implicit. In parallel to the triangle example, “necessarily, God exists” would on Kant's account really assert “necessarily, for all x, if x is a God, x exists.” This is an “identical proposition” (A704/B622), since “x is a God” includes the note that x exists, at least on the plausible assumption that only existing things have any attributes at all. If this is an “identical proposition,” it is also an analytic proposition, because its consequent merely makes explicit something its antecedent clearly includes. So if Kant's conditional account of necessity-claims is correct, then any necessary existential proposition is analytic. Kant's denial that existence is a “predicate”—by which he means something that describes or characterizes an object—helps back (36). Analytic propositions unfold the contents of a concept of some item. Concepts characterize their objects, that is, ascribe to them conjunctions of characterizing properties. So analytic propositions can only ascribe characterizing properties. So if existence is not a characterizing property, there can be no analytic existentials.

How much did Kant actually achieve? As to the claim that existence is not a predicate, Anselm's backing for (2), as explained above, does not involve any particular doctrine about the logical status of existence, nor even the claim that existence has some general great-making or perfective aspect. The point about existence doesn't even really cut against Descartes. One version of his argument uses the premise that existence is a perfection, but the having of a perfection could be expressed other than by what Kant would call a “real predicate.” Another version claims that necessary existence is a perfection—but to claim that necessary existence is a property is not to claim that any existential proposition is necessary. Propositions predicating such a property need not be quantified at all. In any case, the claims that existence is not a predicate or a characterizing predicate are quite likely false. We can well understand a woman who concedes that her hus band, Harvey, is not as brave as Batman or as brilliant as Lex Luthor, then adds “But at least Harvey exists!” This claim predicates existence of Harvey, telling us something substantive about him that “enlarges our concept” of Harvey, namely, that he is not a fictional character.

As to Kant's other line of attack, mathematics features numerous apparent necessary and nonconditional existential truths, for example, that there is a prime number between one and ten. (Kant's friends might dig their heels in and insist that this is really something like a claim that if anything is a series of natural numbers, it includesBut this would pretty plainly be stretching things.) Note that worries about the ontological status of numbers aren't really to the point here: the truths involved are of this form, whatever precisely it is that makes them true, and even if one assigns some unusual interpretation to the existential quantifier in mathematical contexts. So Kant's (34) seems frail indeed, and without it, (35) is at best irrelevant. If the logicists are right, these necessary truths are all analytic. If they are not, these are synthetic propositions which ( pace Kant) do not concern how things must appear to us. Either way, Kant's theory of necessity is in serious trouble.

Kant actually said little that earlier writers had not already said, and Kant's objections (I've claimed) were duds. But they were not thought so, and so arguments from perfection found few friends for the next two centuries. In 1970, mathematician Kurt Gödel developed an argument related to Leibniz's. The reasoning keys on a concept of a “positive” property that Gödel did not explain well. C. Anthony Anderson suggests that we take being positive as being “necessary for and compatible with perfection,” or such that “its absence in an entity entails that the entity is imperfect and its presence does not entail (this)” ( 1990 , 297). The two descriptions are equivalent. If a property is necessary for perfection, its absence in A entails that A is imperfect, and conversely. If a property is compatible with perfection, its presence in A does not entail that A is imperfect, and conversely. Gödel's proof (as Anderson emends it) makes these assumptions:

Definition 1. X is divine if and only if x has as essential properties all and only positive properties. Definition 2. A is an essence of x if and only if for every property B, x has B necessarily just in case x's having A entails x's having B. Definition 3. X necessarily exists if and only if every essence of x is necessarily exemplified. Axiom 1. If a property is positive, its negation is not positive. Axiom 2. Any property a positive property entails is positive. Axiom 3. The property of being divine is positive. Axiom 4. If a property is positive, it is necessarily positive. Axiom 5. Necessary existence is positive.

Since being perfect is necessary for and compatible with perfection, on Anderson's reading, Definition 1 yields the claim that anything divine is by nature a perfect being. Again, on D. 1, a divine being has essentially every property necessary for perfection. Presumably having every property necessary for perfection suffices for perfection. (If it did not, something more would be necessary to attain perfection.) So D. 1 licenses the use of “perfect being theology” to fill out the concept of a divine being. If entailment is strict implication, Definition 2 encapsulates one standard account of what an essence is. Given D. 2, Definition 3 follows at once.

I now present the argument. Axiom 3 has it that the property of being divine is positive. D. 1 has it that every positive property is essential to a divine being. So being divine is essential to a divine being. D. 2 entails that any being has each of its essential properties in every world in which it exists, for if x has B necessarily, x's having A entails x's having B only if x has A necessarily. So per D. 2, any divine being is necessarily divine—divine in all possible worlds in which it exists. Per D. 1 and A. 5, any divine being is essentially a necessary existent. So any divine being is by nature divine and necessary in every possible world.

Axioms 1 and 2 jointly entail that any positive property is consistent. For a property is inconsistent just in case it entails its own negation. Per Axiom 1, if a property is positive, its negation is not positive. But per Axiom 2, if a property is positive, it entails only positive properties. So no positive property entails its own negation.

If every positive property is consistent, and being divine is positive, being divine is consistent. It is necessarily so per A. 4. We can confirm this another way: being divine is having all and only positive properties essentially. But if positive properties entail only positive properties (A. 2), and no negation of any positive property is positive (A. 1), no positive property entails the negation of any positive property. But then the set of all positive properties is consistent; none of its members entails the negation of any of its members. 10 Suppose now that if being divine is consistent, it is instanced in some possible world. Then given what we've argued so far, there is in some possible world a necessarily existent necessarily divine being: that is, it is possibly necessary that “a divine being exists” is true. Given this and the Brouwer axiom, it follows that a divine being exists.

Gödel's argument faces two basic questions. One is whether there is a con‐ tentful, theologically appropriate gloss of “positive” on which the axioms are true. The other is whether there is a sort of possibility such that (a) a concept's being syntactically consistent entails that it is possible in that sense that it be instanced, and (b) the Brouwer axiom is true for that sort of possibility and necessity.

The answer to the first question is yes. Talk of God as a perfect being is certainly appropriate theologically, and perfect being theology has been the main tool to give content to the concept of God philosophically almost as long as there has been philosophical theology. And on Anderson's gloss, the axioms come out true.

Anderson's gloss validates Axiom 1. Suppose that a property F is positive. Then by Anderson's gloss, if A lacks F, A is imperfect. If A has not-F, A lacks F. So if A has not-F, A is imperfect, and so not-F is not compatible with perfection, and so not positive. Anderson's gloss validates Axiom 2. On Anderson's gloss, if a property is not positive, either it is not necessary for or it is not compatible with perfection. If having a property F entails having some property that is not compatible with perfection, having F is not compatible with perfection—and so any property that entails something for this reason nonpositive is itself nonpositive. If a property entails a property not necessary for perfection, it entails a property a divine being can lack. Any property a divine being can lack is not part of its essence. A divine being's essence includes or entails whatever properties it has necessarily (D. 2); so any property a divine being can lack is contingent. But only properties had contingently entail the having of contingent properties. So any property that entails a property not necessary for perfection is itself contingent and not part of a divine being's essence. But a divine being's essence includes all positive properties (D. 1). So any property entailing a property that is not positive in this second way is itself not positive. Axiom 3 seems patent, for given D. 1, being divine amounts to a conjunction of all positive properties, and it's hard to see how such a conjunction could fail to be positive. As to Axiom 4, on Anderson's gloss, a property's being positive consists in two facts about property-entailment. It's plausible that properties entail what they do necessarily. As to Axiom 5, necessary existence is certainly compatible with perfection, and perfect being reasoning suggests that it is necessary for it.

There remains the modal question, of whether a concept of possibility and necessity such that being syntactically consistent (entailing no explicit contradiction) entails being in this way possible also conforms to the Brouwer axiom. Syntactic consistency amounts to “logical possibility,” in one sense of the term. But not all that is possible in this narrow logical sense is really or metaphysically possible: there is no formal, explicit contradiction in the claim that something is red and green all over at once, and yet this claim is not metaphysically possible. So there is a gap between what Gödel establishes and its being metaphysically possible that a divine being exist. And it's a substantive question whether the Brouwer axiom governs real metaphysical possibility. We can describe coherently a set of possible worlds in which the Brouwer axiom doesn't hold, and in which, while it's possibly necessary that God exists, God does not exist. We need only two worlds to do so, in fact:

Suppose that W2 is actual, and W1 is possible relative to W2 but not vice versa. Then were W2 actual, W1 would be possible. As we're supposing that there are only these two worlds, a God who exists in W1 exists in every world possible relative to W1, if W2 is not possible relative to W1. So in W1, God exists necessarily (and W2 is impossible). Thus, since W1 is possible relative to W2, in this setup, God is possibly necessary and yet does not exist.

Gödel's argument (as emended) shows us that the concepts of a perfect being and of divinity are consistent, given a reasonable concept of perfection. But the gap between consistency and metaphysical possibility and the need to establish that the logic of metaphysical possibility includes the Brouwer axiom stand between it and the Holy Grail of proving God's existence. As well, as a modal argument, Gödel's faces the epistemic problems we've observed: the portion of the argument that contends that possibly a divine being exists may admit of significant parody. On the other hand, consistency is evidence for possibility, though defeasibly so, and if I've assessed Proslogion 2 correctly, that argument is promising and does not require us to deal with the epistemic problems the modal argument faces. There is (I think) little good to be said for Descartes' argument. But the Pros . 2 argument appears to survive objections; to accept its premise (1a) we needn't have more reason to believe in God's possibility than in Zod's; and we do have evidence that possibly God exists. So while there is of course much more to be said here, perhaps Anselm's argument has a future.

Leibniz's argument, for instance, reasons simply from the claim that God is a necessary being (see below). But the latter rests on the claims that necessary existence is a perfection and that God is a perfect being.

Nobody nondivine is clumsy but necessary. Proslogion 15 asserts that God is greater than can be thought, using the same language involved in a G . Anselm could not mean to say that God is too great to be thought of or described simpliciter, since he surely thinks that God thinks of Himself. So he must mean a G in terms of thinkers other than God. But Anselm wouldn't want to read a G simply in terms of what we can describe or refer to, for he believes in angels, and surely he'd hold that God is too great for angels as well as humans to describe adequately. Still, since “nobody nondivine” is clumsy, I henceforth replace it with “we.”

If it is better to lack than to have F—that is, if F is an imperfection—then it is better to have than to lack ¬F, and so a G has ¬F. So a G has no imperfections. So nothing could surpass a G by surpassing one of its imperfections. If an attribute is neither a perfection nor an imperfection—neither raises nor lowers greatness—it's hard to see how it could be a respect in which one being could surpass another. For if being F makes A greater than G, presumably being F raises A's greatness past B's.

Oppy ( 1995 ) suggests that we need reason to think that a G, if actual, would be “a being of religious significance” since there may well be numbers too great (large) for us to “form a positive conception of” (16). Agreed. The only nonlogical vocabulary in “a G” is “thought of” and “greater.” Since no religious significance attaches to the first, the second must provide some. The Findlay suggestion in effect stipulates that it does. And why not?

Anselm's argument requires that understanding “the G” puts one in cognitive relation to an entity, the G, which then “exists in intellectu .” On this general approach, understanding “Santa Claus” puts one in cognitive relation with Santa Claus. Santa Claus then is the object of one's thought. But Santa Claus does not exist.

But see also p. 68, where Oppy ( 1995 ) seems to waver.

Can there also always be another being a bit better than any being we pick (Oppy 1995 , 19)? We have the concept of God, which has a number of notes and is supposed in virtue of them to be a concept of the greatest possible being. And we find this connection intuitive: it's pretty hard to think of something better than being necessary, omnipotent, omniscient, morally perfect, and so on. So if one can show it possible that God exist, one can answer the question no. Those who offer arguments from perfection must show that this is possible anyway. So “Is it the case that for any possible being, there is always a greater?” adds nothing to their argumentative task. Moreover, —— is a greatest possible island wears its unsatisfiability on its sleeve. —— is a greatest possible being does not, if only because we're less clear on what makes beings as such “great,” or what greatness is in beings. Further, on the reading of greatness I've suggested, it turns out trivially true that God is the greatest being possible, if God possibly exists.

To see the need for Brouwer, suppose (contra Brouwer) that relative possibility is not symmetric. Then there could be worlds like these:

For simplicity, suppose that W1–3 are all the worlds there are, that only adjacent boxes bear links of direct relative possibility, and that W2 is actual. Say that W1 and W3 are possible relative to W2, but not vice versa. Then both God and Zod exist necessarily (each exists in the only world possible relative to the world in which it exists). And they do not possibly coexist. But both possibly exist, as W1 and W3 are both possible relative to the actual world.

Kant also believed in synthetic necessities. (He discussed these under the rubric of “synthetic a priori” truths. But he also held that whatever is knowable a priori is necessarily true.) But these, he held, all concern how things must appear to our senses, and God, he held, cannot appear to our senses.

Which probably entails that not every prima facie member of the set is actually a member. Being omniscient seems to many a prima facie perfection/positive property. So does being atemporal. Nobody is omniscient who does not know what time it is now. But many think that no atemporal being can know this (e.g., Kretzmann 1966 ). One conclusion from this might be that there are at least two incompatible sets of perfections, differing at least in that one includes atemporality but not omniscience and the other includes omniscience but not atemporality. But if we accept the Gödel/Anderson reasoning, no genuine perfections are incompatible. So on their account, what follows is instead that at most one of atemporality and omniscience is actually a perfection.

Works Cited

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Charlesworth, M.  J. 1965 . St. Anselm's Proslogion with a Reply on Behalf of the Fool by Gaunilo and the Author's Reply to Gaunilo. Trans. M. J. Charlesworth. Oxford: Clarendon Press.

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Kant, Immanuel . [1781] 1956 . Critique of Pure Reason. Trans. Norman Kemp Smith. London: Macmillan.

Kretzmann, Norman . 1966 . “ Omniscience and Immutability. ” Journal of Philosophy 63: 409–21. 10.2307/2023849

Lambert, Karel . 1983 . Meinong and the Principle of Independence. New York: Cambridge University Press.

Oppy, Graham . 1995 . Ontological Arguments and Belief in God. New York: Cambridge University Press.

Plantinga, Alvin , ed. 1965 . The Ontological Argument. Garden City, N.Y.: Doubleday.

Van Inwagen, Peter . 1993 . Metaphysics. Boulder, Colo.: Westview Press.

For Further Reading

Adams, Robert . 1971 . “ The Logical Structure of Anselm's Arguments. ” Philosophical Review 80: 647–84.

Alston, William . 1960 . “ The Ontological Argument Revisited. ” Philosophical Review 69: 452–74. 10.2307/2183480

Barnes, Jonathan . 1972 . The Ontological Argument. New York: St. Martin's Press.

Chandler, Hugh . 1993 . “ Some Ontological Arguments. ” Faith and Philosophy 10: 18–32.

Clarke, Bowman , 1971 . “ Modal Disproofs and Proofs for God. ” Southern Journal of Philosophy 9: 247–58.

Coburn, Robert . 1963 . “ Professor Malcolm on God. ” Australasian Journal of Philosophy 41: 143–62. 10.1080/00048406312341151

Davis, Steven . 1976 . “ Does the Ontological Argument Beg the Question ?” International Journal for Philosophy of Religion 7: 433–42. 10.1007/BF00136308

Devine, Philip . 1975 . “ Does St. Anselm Beg the Question ?” Philosophy 50: 271–81. 10.1017/S0031819100025110

Dore, Clement . 1984 . Theism. Dordrecht: D. Reidel .

Dore, Clement . 1984 . “ The Possibility of God. ” Faith and Philosophy 1: 303–15.

Forgie, William , 1972 . “ Frege's Objection to the Ontological Argument. ” Nous 6: 251–65.

Forgie, William . 1976 . “ Is the Cartesian Ontological Argument Defensible ?” New Scholasticism 50: 108–21.

Forgie, William . 1990 . “ The Caterus Objection. ” International Journal for Philosophy of Religion 28: 81–104. 10.1007/BF00131743

Forgie, William . 1991 . “ The Modal Ontological Argument and the Necessary A Posteriori. ” International Journal for Philosophy of Religion 29: 129–41. 10.1007/BF00141327

Gale, Richard , 1986 . “ A Priori Arguments from God's Abstractness. ” Nous 20: 531–43.

Gale, Richard . 1988 . “ Freedom vs. Unsurpassable Greatness. ” International Journal for Philosophy of Religion 23: 65–75. 10.1007/BF00138712

Gale, Richard . 1991 . On the Nature and Existence of God. Cambridge, England: Cambridge University Press.

Gotterbarn, Dale . 1976 . “ Leibniz' Completion of Descartes' Proof. ” Studia Leibnitiana 8: 105–12.

Grim, Patrick . 1979 . “ Plantinga's God. ” Sophia 18: 35–42. 10.1007/BF02790688

Grim, Patrick . 1979 . “ Plantinga's God and Other Monstrosities. ” Religious Studies 15: 91–97. 10.1017/S0034412500011112

Grim, Patrick . 1981 . “ Plantinga, Hartshorne and the Ontological Argument. ” Sophia 20: 12–16. 10.1007/BF02789922

Grim, Patrick   1982 . “In Behalf of ‘In Behalf of the Fool.’ ” International Journal for Philosophy of Religion 13: 33–42. 10.1007/BF00148937

Hartshorne, Charles . 1962 . The Logic of Perfection. LaSalle, Ill.: Open Court Press.

Hartshorne, Charles . 1965 . Anselm's Discovery. LaSalle, Ill.: Open Court Press.

Hazen, Alan . 1998 . “ On Gödel's Ontological Proof. ” Australasian Journal of Philosophy 76: 361–77. 10.1080/00048409812348501

Hopkins, Jasper . 1972 . A Companion to the Study of St. Anselm. Minneapolis: University of Minnesota Press.

Hopkins, Jasper . 1976 . “Anselm's Debate with Gaunilo.” In Analecta Anselmiana V, ed. H. Kohlenberger , 25–53. Frankfurt: Minerva GmbH.

Kane, Robert . 1990 . “ The Modal Ontological Argument. ” Mind 93: 336–50.

Kenny, Anthony . 1968 . “Descartes' Ontological Argument.” In Fact and Existence , ed. Joseph Margolis , 18–36. New York: Oxford University Press.

Leftow, Brian . 1988 . “ Anselmian Polytheism. ” International Journal for Philosophy of Religion 23: 77–104. 10.1007/BF00138713

Leftow, Brian . 1990 . “ Individual and Attribute in the Ontological Argument. ” Faith and Philosophy 7: 235–42.

Lewis, David. 1970 . “ Anselm and Actuality. ” Nous 4: 175–88.

Mackie, John . 1982 . The Miracle of Theism. New York: Oxford University Press.

Malcolm, Norman . 1960 . “ Anselm's Ontological Arguments. ” Philosophical Review 69: 41–62. 10.2307/2182266

Mann, William . 1976 . “ The Perfect Island. ” Mind 85: 417–21. 10.1093/mind/LXXXV.339.417

Mann, William . 1991 . “Definite Descriptions and the Ontological Argument.” In Philosophical Applications of Free Logic , ed. Karel Lambert. New York: Oxford University Press.

Mason, P.   1978 . “ The Devil and St. Anselm. ” International Journal for Philosophy of Religion 9: 1–15. 10.1007/BF00138726

Oppenheimer, Paul , and Edward Zalta . 1991 . “On the Logic of the Ontological Argument.” In Philosophical Perspectives V , ed. James Tomberlin , 509–29. Atascadero, Calif.: Ridgeview Press.

Plantinga, Alvin . 1967 . God and Other Minds. Ithaca, N.Y.: Cornell University Press.

Plantinga, Alvin . 1986 . “ Is Theism Really a Miracle ?” Faith and Philosophy 3: 109–34.

Rowe, William . 1976 . “ The Ontological Argument and Question-Begging. ” International Journal for Philosophy of Religion 7: 425–32. 10.1007/BF00136307

Shaffer, Jerome . 1962 . “ Existence, Predication and the Ontological Argument. ” Mind 71: 307–25. 10.1093/mind/LXXI.283.307

Sobel, Jordan . 1987 . “Gödel's Ontological Proof.” In On Being and Saying , ed. Judith Thomson , 241–261. Cambridge, Mass.: MIT Press.

Stone, Jim . 1989 . “ Anselm's Proof. ” Philosophical Studies 57: 79–94. 10.1007/BF00355663

Tooley, Michael . 1981 . “ Plantinga's Defense of the Ontological Argument. ” Mind 90: 422–27. 10.1093/mind/XC.359.422

Van Inwagen, Peter . 1977 . “ Ontological Arguments. ” Nous 11: 375–95.

Wainwright, William . 1978 . “ Unihorses and the Ontological Argument. ” Sophia 17: 27–32. 10.1007/BF02798116

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A Level Philosophy & Religious Studies

The Ontological Argument

Introduction.

“it is easier to feel convinced that [the Ontological Argument] must be fallacious than it is to find out precisely where the fallacy lies.” – Bertrand Russell

Ontological arguments are a priori. They are based on an analysis of the concept of God. They essentially argue that if you think carefully about what God is, you’ll understand that God must exist.

Ontological arguments are deductive. The truth of the premises logically entails the truth of the conclusion.

St Anselm’s Ontological argument

P1. God is the greatest conceivable being (by definition) P2. It is greater to exist in reality than the mind alone P3. God exists in the mind C1. Therefore, God exists in reality

Anselm uses the illustration of a painter who has an idea of what they will paint in their mind before painting it in reality. This shows that ideas can exist in the mind.

Anselm points to Psalm 14:1 “the fool says in his heart, ‘there is no God’.”

An atheist says they do not believe in God, but they must therefore have an idea of God in their mind.

The force of Anselm’s argument is then that it would be incoherent to think that God exists in the mind alone, since then we could conceive of something greater, i.e., God also existing in reality. Yet, God is the greatest being, so conceiving of anything greater is incoherent. So, our idea of God must therefore be of a being that exists in reality. To say that God does not exist in reality is to say that the greatest being is not the greatest being. It is self-contradictory.

“that, than which nothing greater can be conceived, cannot exist 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 reality.” – Anselm.

Whether God is within our understanding

A strength of the ontological argument its definition of God

Anselm uses a theologically and philosophically convincing definition of God, carefully designed to avoid the problem of defining something that is beyond our understanding. Anselm presents an analogy. We can’t fully look at the sun but can still see daylight. Similarly, we can’t fully know God, but can at least understand that he is the greatest conceivable being.

“If you say that what is not entirely understood is not understood and is not in the understanding: say, then, that since someone is not able to gaze upon the purest light of the sun does not see light that is nothing but sunlight.”  – Anselm

Weakness: God is not ‘in’ the mind/understanding

Gaunilo raises an objection to P3; the premise that the greatest conceivable being exists in the mind/understanding. Gaunilo draws on the traditional Christian premise that God is beyond our understanding to argue that God therefore cannot be in the understanding.

Anselm cannot then proceed to reason about whether it would be greater also in reality. The ontological argument seems to fail because it relies on our ability to understand and reason about things that are beyond our ability to understand or reason about.

Aquinas also made this argument against Anselm – that God’s nature, such as the ‘eternal law’ is beyond our understanding and that people have different understandings of God.

“Perhaps not everyone who hears this word “God” understands it to signify something than which nothing greater can be thought” – Aquinas.

Gaunilo even doubts that we can understand this idea of the greatest conceivable being:  

“of God, or a being greater than all others, I could not conceive at all” – Gaunilo.

“So much for the assertion that this supreme nature already is in my understanding.” – Gaunilo.

Evaluation defending the ontological argument

However, Gaunilo’s argument is unsuccessful because a full understanding of the greatest conceivable being or of God’s nature is not required for the ontological argument to work.

Peter van Inwagen explains that Anselm would not accept that we either understand God fully or not at all. Our limited understanding of God is enough to justify attributing the name “that than which nothing greater can be conceived” to God.

God has traits but infinitely, i.e., omnipotence, omniscience etc. It is impossible to conceive of anything greater. So, we can understand enough of that idea. We may not be able to conceive of the ‘being’ itself, as Gaunilo says, but that seems to commit a straw man fallacy. Anselm doesn’t rely on conceiving the being itself. We can grasp the concept of a being greater than which none may be conceived. We can then follow Anselm’s reasoning that since it is greater to exist, that being must exist.

Evaluation criticizing the ontological argument

Gaunilo has a point. When we think about the concept of a being greater than anything we could possibly imagine, the idea of that actual being is not in our understanding.

Furthermore, the insights of Apophatic theology show that reasoning about God is impossible. Pseudo-Dionysius argues that if we are true to God’s transcendent unknowability, we would recognize that God is simply beyond any human concepts that we can understand. God therefore cannot be grasped by the understanding and so cannot be ‘in’ the understanding.

Pseudo-Dionysius explicitly says that God is ‘beyond assertion and denial’. So although the atheist is indeed wrong to deny God, proponents of the ontological argument are also wrong to assert God. God is beyond all these philosophical terms, even beyond truth and falsity itself.

Gaunilo’s ‘lost island’ response to Anselm

Deductive arguments are strong because the only way to attack them is to deny the truth of the premises (soundness). This is stronger than inductive arguments because they can also be attacked by arguing that the conclusion is false even if the premises are true.

Weakness: Gaunilo denies that the ontological argument is actually a valid deductive argument, attacking the inference from the premises to the conclusion of God existing in reality

“I have in my understanding all manner of unreal objects” – Gaunilo.

“I should not admit that this being is in my understanding and concept even in the way in which many objects whose real existence is uncertain and doubtful, are in my understanding and concept. For it should be proved first that this being itself really exists somewhere; and then, from the fact that it is greater than all, we shall not hesitate to infer that it also subsists in itself.” – Gaunilo.

Anselm’s argument could succeed in showing that if God exists, then God is the greatest being and even that it subsists in itself, i.e., has necessary existence. However, this is not enough to show that God does exist necessarily.

“he who says that this being exists, because otherwise the being which is greater than all will not be greater than all, does not attend strictly enough to what he is saying. – Gaunilo.

Gaunilo then illustrates this with the case of a perfect lost island, which is an illustration of a thing whose real existence is ‘uncertain and doubtful’ yet is in his understanding as a concept.

Applying the logic of Anselm’s argument to this island has an absurd result (reductio ad absurdum). It is greater for this island to exist in reality, so it must exist. This would work not just for an island. The greatest or supremely perfect member of every category must exist. This is sometimes called the ‘overload’ objection because it suggests that reality would be overloaded with greatest/perfect things.

In response to Gaunilo, Anselm strengthened his argument into a 2 nd form.

Something is greater if it doesn’t depend on anything for its existence. An Island by definition is land enclosed by water, so part of the concept of an Island involves a dependence on things such as an ocean or a planet to exist. So, the greatest possible Island will still be contingent, which means by definition it could either exist or not.

This is why a priori analysis of its definition cannot prove its existence. The existence of contingent beings cannot be proven a priori since their existence is not a matter of definition. Their existence is a matter of whether what they depend on exists.

There is nothing in the concept of the greatest being that involves dependence, making it a necessary being. So, Anselm can now argue that this is why the argument works for God but is absurd when applied to anything contingent.

Anselm’s 2 nd form of his argument successfully refutes the relevance of the perfect island. A priori arguments cannot prove the existence of contingent things like islands since their existence is not a matter of definition. However, the greatest being is necessary, so its existence can be prove a priori.

However, Anselm arguably failed to respond to Gaunilo’s central contention.

Even if Anselm is right that we cannot conceive of God without existence, that only proves that God is a necessary being, such that if God existed it would be in a special way where God could not cease to exist. This is not the same as proving that this necessary being actually does exist. Anselm doesn’t deal with this point.

Descartes & Anselm vs Kant’s development of Gaunilo

Descartes’ Ontological argument

Descartes aims to strengthen the ontological argument with his rationalist epistemology. This claims that we can gain certain knowledge of some truths a priori, through rational intuition. This involves our mind’s ability to simply know certain truths. We can simply think about the concept of God as the supremely perfect being. We then rationally appreciate that God contains the perfection of existence. This is similar to how a rational understanding of a triangle reveals that it contains three sides.

“t he idea of God, or a supremely perfect being, is one that I find within me just as surely as the idea of any shape or number. And my understanding that it belongs to his nature that he always exists is no less clear and distinct than is the case when I prove of any shape or number that some property belongs to its nature” – Descartes

Descartes’ ontological argument functions through intuition rather than argument. Nonetheless he did put it into the form of a deductive argument:

P1 – I have an idea of a supremely perfect being which contains all perfections P2 – Existence is a perfection C3 – God exists

Descartes argument is notable for its simplicity. We know mathematical truths about triangles by simply thinking about our clear and distinct concept of a triangle. Similarly, we can know God exists by thinking about our clear and distinct concept of a supremely perfect being.

The argument is deliberately short, highlighting that its main point is that God’s existence can be known intuitively, not requiring a process of reasoning.

Weakness: Kant’s 1st objection: A priori reasoning cannot establish existence

Gaunilo tried to show that Anselm only succeeds in showing that if God exists, then God exists necessarily. The ontological argument has not shown that God-the-necessary-being does exist. Kant developed this type of objection.

Kant argues that Anselm and Descartes treat ‘existence’ as a predicate, as a description of God. Descartes implies that perfection is an ‘attribute’ of God. Anselm argues God must exist in order to be God. They try to show that you cannot think of God without existence, because it is a defining quality of God. The idea of ‘God’ and the idea of ‘existence’ are necessarily connected.

Kant objects that existence being a predicate of God does not establish God’s existence in reality. He Descartes example of a triangle. It is necessary that ‘having three sides’ is part of the concept of a triangle. This shows that if a triangle exists, it must have three sides.

Similarly, we could grant that the ontological argument shows that ‘necessary existence’ is part of the concept of God. Kant’s objection is that this only shows that if God exists, then God exists necessarily. It doesn’t show that God-the-necessary-being does exist.

Existence necessarily being part of the definition of God only shows that God is the idea of a necessary being. We can still deny that this necessary being or being greater than which cannot be conceived or maximally great or unlimited being actually exists.

Like Gaunilo, Kant is drawing a distinction between judgement and reality. A priori reasoning showing that existence is necessary to the definition of God in our minds is not the same as showing that necessary being actually exists in reality.

“The unconditioned necessity of judgements is not the same as an absolute necessity of things” – Kant.

“the illusion of this logical necessity has proved so powerful that when one has made a concept a priori of a thing that was set up so that its existence was comprehended within the range of its meaning, one believed one could infer with certainty that because existence necessarily pertains to the object of this concept, i.e., under the condition that I posit this thing as given (existing), its existence can also be posited necessarily” – Kant.

Kant’s first critique is unsuccessful because it is self-contradictory.

“ I think that Caterus, Kant, and numerous other philosophers have been mistaken in supposing that the proposition ‘God is a necessary being’ (or “God necessarily exists”) is equivalent to the conditional proposition ‘If God exists then He necessarily exists’ … Can anything be clearer than that the conjunction ‘God necessarily exists but it is possible that He does not exist’ is self-contradictory?” – Malcolm

Kant’s 1st objection seems to accept that the ontological argument shows that God is necessary. So, Kant must then accept that God is a being which contains its own reason for existence and is thus defined by the impossibility of non-existence.

It’s incoherent of Kant to grant God’s necessity while maintaining the possibility of God’s non-existence. So, the Ontological argument does show that God-the-necessary-being actually exists.

Malcolm tries to object that it’s incoherent to say God necessarily exists, but possibly doesn’t exist. However, that misunderstands Kant’s argument.

The issue is, Malcolm has only shown that God is a non-dependent being. In his ontological argument, Malcolm argued that if God exists, God exists necessarily because nothing could cause God to cease existing, as God is unlimited and non-dependent. This is what Malcolm established as God’s necessity. But this only establishes that God is necessary in the sense of being non-dependent, not in the sense of must exist. A being could be non-dependent and yet not exist. If it existed, then it would be necessary.

So, the necessity of God’s existence established by the ontological argument only relates to the manner of God’s existence if God exists. 

Ontological arguments cannot show that God actually exists, then.

The most famous modern defender on the ontological argument is Plantinga. Even he admits that this critique from Kant cannot really be solved and that at most the ontological argument can make religious belief rational – it cannot prove that God actually does exist, however. Very few people defend the ontological argument these days except for Plantinga, and even he doesn’t defend it as actually proving that God actually exists necessarily.

“reformulated versions of St. Anselm’s argument … cannot, perhaps, be said to prove or establish their conclusion. But since it is rational to accept their central premise, they do show that it is rational to accept that conclusion” – Plantinga.

Whether existence is a predicate

Strength: Anselm strengthens his argument in Proslogion chapter 3 to include necessary existence.

In chapter 2 Anselm spoke of existence being greater than non-existence. In chapter 3 he better justifies that premise. Existence is greater than non-existence because a being is greater if it cannot cease existing. A being whose nonexistence is impossible is greater than a being whose non-existence is possible.

If a being can cease to exist, that is because it depends on something else for its existence. Malcolm points out that dependence is a kind of limitation and in common language these concepts are linked to inferiority. A being which doesn’t depend on anything else (is necessary) is therefore unlimited and so is the greatest conceivable being.

This is Anselm’s argument in its strongest form. A being greater than which cannot be conceived must be one whose nonexistence is impossible.

Weakness: Kant’s 2 nd objection: existence is not a predicate.  

Anselm argues that if God didn’t exist, God wouldn’t be what God is; the greatest conceivable being. Descartes says that existence is part of what God is. Kant thinks they both assume that existence is a predicate, a description of a quality that God possesses.

Kant objects that existence is not a quality or attribute that defines a thing. To say something exists is not to describe that thing. If I say my cat exists, I do not describe a feature of the cat. I may be describing reality in a general sense, but I am not describing a defining quality of the cat. So existence isn’t a predicate.

Kant’s illustration is 100 thalers (coins). Imagine you have 100 thalers in your mind as a mere concept. Then imagine you also have 100 thalers in existence, not only in the mind. You have two cases of 100 thalers, one which exists in reality and the other which only exists in your mind.

If Anselm was correct that existence is part of the definition of the concept of a thing, then the thalers which exist should be conceptually different to the thalers that do not.

However, the concept 100 thalers is no different whether a mere concept in your mind or instantiated in reality. 100 thalers is just 100 thalers. It has the predicates of shininess, roundness and 100 etc. Being only in the mind doesn’t make the concept somehow less of a complete description of what 100 thalers is. So, existence is not part of the definition of a thing. It is not a predicate or property of the definition of a thing.

So, Anselm and Descartes are wrong when they claim it’s incoherent to think of God without existence.

Malcolm criticised Kant, arguing that Anselm’s second form had been right all along. Kant’s argument worked regarding contingent but not necessary existence. He made the same mistake Gaunilo’s lost island argument made, which was to think we could test the logic of the ontological argument through application to contingent things, such as islands or thalers.

Something is contingent if it is dependent on something else for its existence. The reason for the existence of a contingent thing is therefore external to it and so does not describe or define it. However, a necessary being doesn’t depend on anything else for its existence, so it contains the reason for its existence within itself. Since the reason for its existence is contained within itself, necessary existence must be a defining part of a thing in a way that contingent existence is not. So, necessary existence is a predicate. The ontological argument, which relies on necessary existence, is therefore defended from Kant’s critique.

Anselm and Malcolm seem correct that necessary existence is a predicate of God. Contingent existence of things like cats and coins is not a predicate, since their reason for existence is something else. However, necessary existence does define and describe a thing.

However, Kant’s first criticism might still succeed:

Even if necessary existence were a predicate of God, that only shows that if God exists, then God necessarily exists. It doesn’t show that God does exist.

Malcolm’s ontological argument

N. Malcolm’s created his own version of the ontological argument, referring to God as an unlimited being.

P1. God either exists or does not exist. P2. If God exists, God cannot go out of existence as that would require dependence on something else. So, if God exists, God’s existence is necessarily P3. If God does not exist, God cannot come into existence as that would make God dependent on whatever brought God into existence. So, if God does not exist, God’s existence is impossible. C1. So, God’s existence is either necessary or impossible P4. The concept of God is not self-contradictory (like a four-sided triangle), therefore God’s existence is not impossible. C2. Therefore, God exists necessarily.

In P4, Malcolm’s ontological argument most clearly shows how all versions of the argument rely on the concept of God being coherent. Nonetheless, there are numerous philosophical debates about that, including:

• The paradox of the stone
• The Euthyphro dilemma
• The incompatibility of free will and omniscience
• The logical problem of evil

Empiricist response: Hume on the impossibility of a necessary being

“there is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by any arguments a priori” – Hume.

A necessary being must exist – it cannot be the case that it does not exist. This means that we shouldn’t even be able to conceive of a necessary being not existing without contradiction. However, Hume claims that whatever we conceive of as existing, we can conceive of as not existing. It follows that there is no being that we cannot conceive to not exist, so no being can exist necessarily. Hume concludes:

“The words, therefore, necessary existence, have no meaning.”  

This argument references “Hume’s fork”:

A priori reasoning can only tell us about the relations between ideas , i.e. analytic knowledge (true by definition). E.g. “a bachelor is an unmarried man”.

A posteriori reasoning can only tell us about matters of fact , i.e. synthetic knowledge (true by the way the world is). E.g. “The sun will rise tomorrow”.

Matters of fact, such as whether a being exists, cannot be established a priori, according to this argument. Hume’s basis for the fork is that if a particular truth is a matter of logic/definition, then it will be true or false no matter the factual state of the universe. E.g., one plus one will always equal two, regardless of what happens to be factually true of the universe. This suggests there is a disconnect between logical truth and factual truth. The term “necessary existence” seem to ignore this disconnect. A being having necessary existence does not seem to be a matter of fact since it must exist, regardless of the factual state of the universe. Yet, what exists is a matter of fact, not relations of ideas. It’s invalid to claim that a being’s existence is logically necessary since a being’s existence cannot be established through logic. Since Hume’s fork has shown that logical truth is disconnected from factual truth, the idea that something could necessarily exist is incoherent.

The ontological argument therefore fails because it attempts to establish a matter of fact (God’s existence) through a priori reasoning.

University of Notre Dame

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The Ontological Argument from Descartes to Hegel

Kevin J. Harrelson, The Ontological Argument from Descartes to Hegel , Humanity Books, 2009, $39.98 (hbk), ISBN 9781591026396.

Reviewed by Charles Taliaferro, St. Olaf College

In 1945, Bertrand Russell announced in his famous The History of Western Philosophy (a brilliant but sometimes eccentric and flawed book) that the ontological argument has been proved to be invalid, despite the fact that the soundness of the argument would be very good news indeed for philosophy:

The real question is: Is there anything we can think of which, by the mere fact that we can think of it, is shown to exist outside our thoughts. Every philosopher would like to say yes, because a philosopher’s job is to find out things about the world by thinking rather than observing. 1

While Russell pronounced the argument dead (albeit with regret), perhaps Anthony Kenny was wiser in his four volume New History of Western Philosophy when he ended the fourth volume with a warning to those who think the argument has been refuted:

Plantinga’s reinstatement of the [ontological] argument, using logical techniques more modern than any available to Russell, serves as a salutary warning of the danger that awaits any historian of logic who declares a philosophical issue definitively closed. 2

Kevin J. Harrelson has written a welcome historical and critical analysis of the ontological argument in early modern European philosophy. In the Introduction, he writes:

In the following chapters I argue that the strategy for proving a priori the existence of God that remains in place during this period, from Descartes’ initial argument in the Discourse on the Method (1637) to Hegel’s final lectures in Berlin (1831), is both internally consistent and free of any easily identifiable error. More importantly, I try to show that the most common objections to the modern ontological proof, raised by the likes of Gassendi, Hobbes, Hume, and Kant, fail to identify any conclusive and universal fallacy. (p. 18)

His book is not, however, “an outright defense of the ontological argument”, for Harrelson is convinced most versions of the ontological argument face serious obstacles and are not persuasive to those not already committed to what he finds philosophically problematic. The book is rich with historical references and nuanced readings of canonical texts, and is packed with arguments and counter-arguments.

The book opens with a compact overview of the ontological arguments found in Anselm, the scholastics, Descartes, and Leibniz. Some of the arguments’ exposition is a bit hard to follow. In discussing the relationship between perfection and necessary existence (which Anselmians usually seek to secure on the grounds that existing necessarily is a perfection or great-making attribute), Harrelson writes: “If God is indeed identical to his own existence, then it could only represent a shortcoming of human reason to distinguish the notion of a ‘perfect being’ from that of ‘necessary existence’” (p. 25). Why is this a problem? Can’t a case for the ontological argument begin with a consideration of great-making properties and an inquirer come to reason that necessary existence plus theistic attributes would be (or is) more excellent than theistic attributes and contingent existence? If one does not realize this prior to entertaining the argument, perhaps that is a “shortcoming”, but no worse than if someone did not realize 6 is the smallest perfect number before she reasoned that 6 is equivalent to 3 + 2+ 1.

In the same chapter, and on the same page as the claim just considered, Harrelson writes, “the peculiar identification of ‘God’ and ‘necessary existence’ renders misleading all theological statements about the existence of the deity” (p. 25). It is not clear, however, which philosopher (if any) claims that what we mean by “perfection” is “necessary existence” (as in “grandmother” is “a female whose child has a child”). Harrelson writes:

In early modern philosophy we find rather that theological propositions are understood to be akin to identical statements, and the philosophers in question fall just short of claiming that “perfect being” and “necessary existence” have the same meaning. “Necessary existence,” like God’s other predicates, is identical with God’s whole nature. This identity of subject and predicate would seem to exempt theological statements from the rules governing normal attributive statements. (p. 25)

Why, however, would a defender of the ontological argument claim that “necessary existence” means the same as “perfect being”, or claim that necessary existence “is identical with God’s whole nature”, rather than claim that necessary existence (or existing necessarily) is a mode of being distinct from being contingent (or having the property being contingent) ? Presumably, for an Anselmian theist, claiming that God exists necessarily involves claiming that there necessarily exists a being of unsurpassable excellence or perfection. I do not yet see how linking necessity and perfection is a theological disaster. At the least, some clarification of how the thesis of divine simplicity comes into play on this issue would have been desirable.

In the same chapter, Harrelson has an interesting treatment of Descartes’ analogy about the idea of a triangle in discussing the idea of God. The format Harrelson employs in clarifying the points at issue is complex.

The following is a short list of those objections, other than the possibility and Thomistic, that are prevalent in the modern period. After each objection I give a caricature of the kind of reply that is frequently found among proponents of the modern argument. I also give a brief explanation of the debate, in which I try to indicate, very roughly, the historical contexts in which the respective objections and replies appear and reappear. (p. 29)

The deliberate use of caricature made the reasoning less easy to follow (for me, anyway).

Thus, the problem with the argument is that it involves the existence of God (!), experience and/or intuition (perhaps especially theological intuition), and insight. One difficulty readers will have so far is that it is not easy to see “the downfall” of the argument without seeing more of “the rise”.

First, from the fact that our perception is incorporated in the premise of the argument it follows that the conclusion is not true for everyone. In other words, whoever does not actually perceive the connection between “a supremely perfect being” and “necessary existence” cannot assent to the claim in the minor premise, in which case the conclusion remains undemonstrated. It is not the case that these individuals fail to grasp a premise that is objectively true; rather, their perceiving a certain “truth” is itself part of the premise. The premise is in fact false in any instance in which the perception is lacking. The ontological argument is thus unsound in those cases. Regardless of whether the ontological argument is ever sound, then, it will sometimes be unsound. The objections will always be, in some sense, in the right, despite their inability to discover an internal flaw in the argument. (p. 67)

This strikes me as odd. Any argument in philosophy might well be considered unsound if not everyone grasps its entailment relations. Even a simple entailment like “if all humans are mortal, no immortal being is a human” might sometimes be unsound because someone, somewhere does not accept the entailment.

In “Refutation of Atheism”, there is a welcome discussion of Cambridge Platonist treatments of the ontological argument. Harrelson has some sympathy with Henry More and Ralph Cudworth, even if he thinks both present arguments with fatal flaws or fail to persuade. As before, I find Harrelson’s autopsy of the argument neither obvious or clear. Here is an analysis of More:

Like Descartes, [More] assents to the following maxim: "we are first to have a settled notion of what God is , before we go about to demonstrate that he is." The various subsidiary arguments to the minor premise (the proof of innateness, the deduction of necessary existence from the idea of God, etc.) serve this end, comprising a preliminary examination of the essence or notion of God. The inference to God’s actual existence appears only at the end of this discussion. This last fact, however, represents the fatal consequence of the systematic presentation of the ontological argument: in order to clarify the various steps in the argument, it was necessary to distinguish the essence of God (i.e., “what God is”) from his existence (“that he is”). The systematic presentation of the ontological argument thereby contradicts the basic presupposition of that same argument, viz., that the essence and existence of God are inseparable. (pp. 87-88)

I do not quite see the problem. More does not think God’s essence and existence can be metaphysically separated, but he thinks one can epistemically consider God’s essence and then come to see that it (together with the thesis that God exists either necessarily or God’s existence is impossible , plus the premise that God’s existence is possible ) entails that God exists.

Harrelson offers a helpful exposition of the work of Ralph Cudworth and Samuel Clark. He is probably correct that Locke’s attack on innate ideas undermined the popularity of the ontological argument, though there are many versions of the argument that do not require or presuppose the existence of innate ideas.

In the chapter “Being and Intuition”, Harrelson takes up the work of Malebranche. There is a useful examination of how Malebranche advances the ontological argument beyond Descartes. At least one of Harrelson’s objections to Malebranche seems strained: “the revised form of the argument is indefensible against the nominalist’s objection that ‘being’ is a mere concept” (p. 115). It is indefensible, unless of course nominalism turns out to be deeply problematic and then the objection carries no weight.

Chapter four contains a helpful analysis of Spinoza’s work, showing how his version of the ontological argument is closely tied in with the whole of Spinoza’s philosophy: "No one can accept [Spinoza’s] argument without accepting his other doctrines in toto , or at least without offering alternative versions of them." (p. 135)

Chapter five offers a detailed exploration of the ontological argument in pre-Kantian German philosophy. Arguments by Leibniz, Wolff, Baumgarten, and Crusius are addressed.

Chapter six on Kant is excellent. Harrelson places Kant’s famous criticism of the ontological argument in perspective and shows why it is not decisive. Harrelson thinks Kant was effective in challenging the authority of the ontological argument largely because of Kant’s general case about the limits of human thought:

The ontological argument, in 1785, is still not the object of any directly successful critique. Its temporary disappearance is a product only of the belief that humans are incapable of obtaining any genuine cognition beyond the field of “experience,” as this term is defined in the opening chapters of the Critique of Pure Reason . (p. 191)

The final chapter on Hegel provides a good context for Harrelson’s thesis that the ontological argument might work for some people. If one can (in Hegel’s terms) “elevate” one’s mind to God, the argument succeeds:

Whoever “grasps” or comprehends that “being is the concept,” i.e., whoever gazes from the summit of absolute knowledge and thereby understands the inferences of Hegelian logic, also perceives the existence of God via participation in God’s self-knowledge. (p. 220)

In Harrelson’s view, while (to echo Russell) every philosopher would like to have such elevation, few of us succeed and so Hegel’s ontological argument (like Descartes’) fails in its ambition as a demonstration or proof.

The correctness and relevance of the modal ontological argument

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• Volume 199 , pages 2727–2743, ( 2021 )

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This paper deals with some metaphilosophical aspects of the modal ontological argument originating from Charles Hartshorne. One of the specific premises of the argument expresses the idea that the existence of God is not contingent. Several well-known versions of the argument have been formulated that appeal to different ways of clarifying the latter. A question arises: which of the formally correct and relevant versions is proper or basic? The paper points to some criteria of formal correctness, and distinguishes two types of relevance for these versions: strong and weak. Its aim is to furnish a strictly worked out answer to the question, taking into account each of these types. As a result, a very simple, formally correct and (weakly) relevant version of the modal ontological argument is formulated. The results obtained are also used to criticize a popular belief about the relations in which the main versions of the modal ontological argument stand to one another.

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1 Introduction

Ontological arguments amount to a priori arguments for philosophical theism : i.e. the thesis that God, in a philosophical sense of the word, exists. There are many (at least seven) types of such arguments (Oppy 2019 ). One of them is the modal ontological argument (hereinafter MOA), an argument formalizable in a simple zero-order language of (applied) modal logic or an (appropriately enriched) standard first-order language of the theory of possible worlds. Footnote 1

More particularly, in the form in which it originated in the work of Charles Hartshorne, the MOA is formulated within a zero-order modal theory, in which the only extra logical constant is the sentence “God exists”. This argument is based on two specific metaphysical premises. The first of these states that the existence of God is possible. This premise is nowadays most often considered to be the core of the MOA. Footnote 2 The second premise is an explication of the idea that the existence of God is not contingent (Hartshorne 1944 ). Depending on the way in which this idea is explicated, different versions of the MOA come to be formulated.

More complex versions, coming from Alvin Plantinga, are formulated within a standard first-order theory of possible worlds extended by the concept of God (Plantinga 1974 ). Footnote 3 In this paper we will confine ourselves to just the Hartshorne-style zero-order MOA-versions. (For convenience, we will continue to use the phrases “Hartshorne-style argument” and “MOA” interchangeably.) Footnote 4

One of the simpler (zero-order) MOA-versions (though not necessarily the simplest—see the next footnote and Sect.  4 below) is presented in the Stanford Encyclopedia of Philosophy as follows:

It is possible that God exists. God is not a contingent being, i.e., either it is not possible that God exists, or it is necessary that God exists. Hence, it is necessary that God exists. Hence, God exists. Footnote 5

Despite its formal simplicity, arguments of this type are still the subject of numerous analyses. In particular, their persuasive power is studied, and often also questioned (see Oppy 1996 , 2019 ). We do not, however, intend to address that issue here (except for the analysis of the persuasive function of one of the MOA versions). The main subject of our present study is, rather, the general structure of Hartshorne-style MOAs, the basic properties of their most important versions, and some relations between them. Its main purpose is to distinguish the basic versions from all other MOAs that meet predetermined criteria of formal correctness and relevance.

The following view seems prevalent in the philosophical literature: certain general premises generated by stronger systems of modal logic are logically essential, or at least philosophically the most adequate ones, where the MOA is concerned. Footnote 6 This line of thinking was clearly expressed by philosophers quite early on. For example, Kane ( 1984 ) lists the general premises (taken from B -system) as the third important element—after two specific premises—in the construction of the MOA. Footnote 7 In recent years, a similar view has been presented by van Inwagen ( 2012 , 2018 ) in a more cautious way, clearly suggesting two theses: (1) each formally correct and relevant MOA-version has either strong specific premises and weak general premises (generated by modal logic) or, equally, weak specific premises and strong general premises; (2) the second part of this equivalence is the philosophically preferable one (on account of its clarity and elegance). Footnote 8

In this paper, I will try to show that if we assume some quite natural criteria for the philosophical evaluation and selection of MOA-versions, both of the above theses turn out to be wrong.

The main points of reference here are those normal systems of modal logic that are the most widely known ones (at least in the context of MOA): i.e. systems in which Modus Ponens and the Rule of Necessitation make up the only primitive rules. The purpose of the study is not to analyze possible logical systems underlying various MOA-versions, but rather to analyze these versions on the basis of a predetermined system of modal logic: one that is—in a way—the “internal” logic of every MOA-version. Thus, the determination of a given MOA-version will not be done by changing this “internal” logic (e.g. by adding new logical rules), but by changing or adding new premises, including both substantive (metaphysical) and general premises taken from various “external” systems of modal logic. Footnote 9

The findings proposed in the next section provide fairly strict answers to two questions: namely, what the overall structure of the (Hartshorne-style) MOA and the criterion for the latter’s relevance look like. There are also two other closely related and logically basic questions, which concern what the correctness and simplicity criteria for MOA-versions are, and which of the known correct and relevant versions is the simplest. In the philosophical literature, we do not find precise answers to these questions. Answers to the above questions can then help with solving the key metaphilosophical issue of which MOA-version is proper or basic.

The structure of the ensuing discussion is as follows: the next two sections set out the structure of the MOA and criteria for its formal correctness and simplicity, while Sect.  4 furnishes proofs of the correctness of the T-version, S5-version, and B-version; Sect.  5 then presents a correct, simplified MOA-version, and Sect. 6 distinguishes two types of philosophical relevance for MOA-versions; finally, Sect.  7 uses these findings to criticize a popular view related to van Inwagen’s theses (1) and (2).

2 The structure and relevance of the modal ontological argument

For the purposes of the present analysis, we shall retain the principle of the possibility of the existence of God (expressed in the first sentence of the quotation presented in the previous section), expressing this with the symbolization “Mg” and assuming it to be obligatory for every MOA-version:

Taking the symbols “Lg” to represent the sentence “It is necessary that God exists”, the second sentence of the above quotation can then be rewritten using the formula:

The sentence g (“God exists”) is derived from the premises (1) and (2), with the use of the law of double negation, disjunctive syllogism ( modus tollendo ponens ), and the following general (non-specific) premise resulting from the application of the axiom ‘L α ⇒ α ’ of the T -system of modal logic Footnote 11 :

Considering the above, we may conclude that this argument for (philosophical) theism is a valid (i.e. logically correct) MOA-version that is based on two specific premises, (1) and (2), and one general premise T. Footnote 12

In other MOA-versions, the choice of specific and general premises changes. In typical versions, (2) is weakened and the T -system is strengthened by adding new general premises. Footnote 13 These additional premises are usually taken from the S5 -system or, less often, from the weaker B -system.

The T -system is the weakest system of normal modal logic in which the uncontroversial formulae of the form ‘L α ⇒ α ’ are theses. Regardless of whether or not these formulae are used in a particular MOA-version, it is the basic system for explicating the meaning of the modal operators L and M. Therefore, we will take it to be a logical basis for MOA. Footnote 14

Now, the MOA - structure can be presented as an arrangement < T , (1), α , X , g> , where α is a sentence of the language of the (applied) modal logic including the sentence g, and X is a set of sentences resulting from the substitution of propositional variables by the sentence g in the specific theses of modal propositional systems. The sentence α represents here a specific premise expressing the idea that the existence of God is not contingent, and the set X represents the set of all general premises of the given MOA-version.

Since the elements T , (1) and g are fixed in every MOA-version, we will write it in a shortened form as “[ α , X ]”:

By virtue of this convention, any version of the argument for philosophical theism applying the theses of the T -system and having specific premises that include {(1)} (non-modifiable) and { α } ∪  X (modifiable) may be represented by the MOA-structure [ α , X ]. It is easy to see that the arrangement [(2), {T}] is an example of such MOA-structures, and that it is one that represents the T-version of the MOA.

The MOA-version (represented by the structure) [ α , X ] will be relevant if and only if α is a sentence clarifying the idea that the existence of God is not contingent, and α is logically equivalent (based on the T -system) to a specific (i.e. non-general) sentence or conjunction of sentences with the following forms: ‘ β ⇒ g’, ‘L( β ⇒ g)’, ‘g ⇒ β ’, or ‘L(g ⇒ β )’. The latter requirement’s being fulfilled serves to reassure us that any clarification of the idea of God furnishes logically nontrivial, necessary and/or sufficient conditions for the existence of God. We embrace the idea that a formulation of such conditions will be methodologically fundamental where any philosophical theory of God is concerned.

The T-version [(2), {T}] is an example of the relevant MOA-versions. The following metatheorem testifies to this:

Metatheorem 1

Sentence (2) is logically equivalent (based on the T-system) to each of the following sentences:

(g ⇒ Lg) ∧ (Mg ⇒ g),

 ~ (Mg ∧ M ~ g).

Ad (i): For the implication from left to right, assume that ~ Mg ∨ Lg. This sentence is equivalent to the sentence ‘Mg ⇒ Lg’. Hence, and from the two specific theses of the T -system ‘g ⇒ Mg’ and ‘Lg ⇒ g’, we get, by hypothetical syllogism, g ⇒ Lg and Mg ⇒ g. For the right-to-left implication, assume that (g ⇒ Lg) ∧ (Mg ⇒ g). Again, by hypothetical syllogism, we get Mg ⇒ Lg. Ad (ii): based on de Morgan’s law and the standard logical relations between modal operators. Q.E.D.

Sentence (i) of Metatheorem 1 specifies the necessary (g ⇒ Lg) and sufficient (Mg ⇒ g) conditions for God to exist. Sentence (ii), in turn, expresses the non-contingency of God’s existence in the Aristotelian sense of bilateral possibility (cf. Łukasiewicz 1957 ).

Let us now consider the argument having the structure [‘MLg’, {B}], where B is the sentence that results when we substitute g for p in the thesis ‘ML p ⇒ p ’ of the B -system:

This argument is an example of an MOA-version for which we do not see any possibility of demonstrating that it meets the criterion of relevance adopted. (We will leave open the matter of proving that this version does not really meet the relevance criterion). Footnote 15

3 Criteria of formal correctness and simplicity for modal ontological arguments

In order to properly formulate a criterion of correctness for the MOA, we shall introduce the following definition:

Definition 1

If { α }, X are sets of sentences of the language of the T -system with the constant g, then X ⊢ α if and only if α is an element of the smallest set containing all substitutions of theses of the T -system and elements of X and closed under two inference rules – modus ponendo ponens and Gödel’s Rule of Necessitation.

As usual, for short, we will write ‘ α 1 , …, α n , X ⊢ β ’ instead of ‘{ α 1 , …, α n ,} ∪  X ⊢ β ’, and ‘⊢ β ’ instead of ‘∅ ⊢ β ’. We introduce the abbreviation:

We will, also for short, write ‘ TG ( α )’ instead of ‘ TG ( α , X )’, if TG ( α , X ) =  TG ( α , ∅).

Thus, TG ( α , X ) is the smallest set of sentences containing all substitutions of theses of the T -system and elements of the set {(1), α } ∪  X , and closed under modus ponendo ponens and the Rule of Necessitation. In other words, TG ( α , X ) is the theory resulting from the strengthening of T by adding axioms (1), α and all elements of X . Each such theory will be called a mini-theory of God . Footnote 16

Intuitively, all elements of the set {(1), α } ∪  X are a priori sentences; so if they are true, they are necessarily true. This assumption seems to be fully justified in the context of the considerations pertaining to MOA. It also justifies the above finding to the effect that the Rule of Necessitation applies not only to the laws of logic, but also to all of the elements of the set {(1), α } ∪  X . (This finding will allow us to simplify some of the proofs below.)

A MOA-version [ α , X ] will be non-circular if and only if the sentence g is not derived from { α } ∪  X on the basis of the T -system alone (and therefore is not derived from { α } ∪  X independently of (1)); formally: α , X ⊬ g. Footnote 17

The argument [‘Lg’, {T}] is an example of a valid but circular MOA-version. The fact that this argument can be treated as a relevant MOA-version at all is evidenced by the fact that the sentence ‘Lg’ explicates the idea of the existence of God as a necessary existence. (More precisely, it states that God necessarily exists). It is valid and circular, because sentence g is directly derivable from sentence ‘Lg’ and premise T—and thus is so regardless of (1).

As we can see, the concept of “validity” (used for a given MOA-version) has a broader extension than the concept of “formal correctness”. Let us adopt the following definition:

Definition 2

The MOA-version [ α , X ] will be formally correct if and only if the following three conditions are met:

(1), α , X ⊢ g (VALIDITY),

TG ( α , X ) is consistent (CONSISTENCY), and

α , X ⊬ g (NON-CIRCULARITY).

Valid MOA-versions can be compared with respect to the number of general premises they possess and the deductive strengths of their own mini-theories. Let us adopt the following definition:

Definition 3

[ α , X ] will be a simpler MOA-version than [ β , Y ] if and only if at least one of the following conditions is met (where | X | is the cardinality of the set X ):

| X | <| Y | and TG ( α , X ) ⊆  TG ( β , Y ), or

| X | ≤| Y | and TG ( α , X ) ⊊  TG ( β , Y ).

Definitions 1 – 3 will be deployed in due course – in the remainder of this paper.

4 Formal correctness of three standard versions of the argument

The T-version [(2), {T}] is an example of a relevant and formally correct MOA-version. Indeed, and especially given Metatheorem 1 , the relevance of this version is obvious, and the following METATHEOREM holds true:

Metatheorem 2

The T-version of the MOA, represented by the structure [(2), {T}] , is formally correct.

The following three conditions must be shown to have been met:

(1), (2), {T} ⊢ g.

TG (2) is consistent, and.

(2), {T} ⊬ g.

The sentence g was derived from the axioms of TG (2) in Sect.  1 .

The language of the mini-theory TG (2) can be formally interpreted as follows: the sentence g is interpreted as the constant 1 (true sentence), the operators M and L are interpreted as the assertion operator A (in the sense defined by the axiom ‘A p ⇔ p ’), and the truth-connectives are left unchanged. As a result, the set TG (2) becomes the theory TG (2)*, which contains only the constant 1 and the set of sentences resulting from the substitution of variables by this constant in tautologies of classical sentential logic (with the operator A). As we know, the set TG (2)* is consistent; consequently, the set TG (2) is also consistent.

If g were derived from TG (2), then g would be true in any model in which TG (2) is true. To see that the opposite is the case, let us consider a TG (2) # -theory that will be the result of the following interpretation: g is interpreted as the constant 0 (false sentence), the M and L operators are interpreted as the assertion operator A, and the truth-connectives are left unchanged. The TG (2) # -theory will therefore consist only of the relevant substitutions in respect of the tautologies. (More particularly, Axiom (2) # will be such a substitution, equivalent to the sentence ‘ ~ 0 ∨ 0’). At the same time, the sentence g # (the constant 0) is false. Q.E.D.

In the literature on the MOA, what is known as Anselm’s Principle has tended to receive more frequent consideration than Principle (2):

Historically, the first formalized version of the MOA (originating from Hartshorne) should be represented by the structure [(3), {T, N, S5}], where N is the result of substitution of g for p , and Lg for q , in the thesis ‘L( p ⇒ q ) ⇒ (M p ⇒ M q )’ of T :

and S5 is the formula being a result of the similar substitution in the specific axiom of the modal system S5 :

Metatheorem 3

The S5-version of the MOA, represented by the structure [(3), {T, N, S5}] , is formally correct.

The parts of the proof relating to the conditions of consistency and non-circularity are analogous to the relevant parts of the proof of Metatheorem 2 . It is therefore sufficient to show that the condition.

is satisfied. By using the law of detachment ( modus ponens ) twice, we obtain the thesis ‘M g ⇒ ML g ’ (from (3) and N), and then ‘ML g ’ (by using the axiom (1)). Hence, from S5, we get the sentence ‘Lg’. Hence, from T, we get the sentence g. Q.E.D.

The deductive basis of the S5-version can be simplified so as to arrive at the weaker mini-theory TG (3, {N, B}). This shows a key fragment of the proof of the following metatheorem:

Metatheorem 4

The B-version of the MOA, represented by the structure [(3), {N, B}] , is formally correct.

is satisfied. By using N and the law of detachment, we obtain the thesis ‘M g ⇒ ML g ’, and then ‘ML g ’. Hence, from B, we get the sentence g . Q.E.D. Footnote 18

Using Definition 3 , we can compare all of the above MOA-versions. The following metatheorem will be the result of such a comparison:

Metatheorem 5

[(2), {T}] is a simpler MOA-version than [(3), {T, N, S5}] and [(3), {N, B}].

It is enough to note that two conditions are met: (i) |{T}| <|{T, N, S5}| & TG (2) ⊆  TG (3, {S5}), and (ii) |{T}| <|{N, B}| & TG (2) ⊆  TG (3, {B}). Ad (i): Given that T is an S5 -thesis, it is sufficient to show that (2) is a thesis of the mini-theory TG (3, {S5}). This becomes apparent when we consider that ‘Lg’ is a thesis of this theory (see the penultimate step of Metatheorem 2 ). Hence, we may conclude that (2) is also a thesis of it. Ad (ii): Given that T is a B -thesis, it is sufficient to show that (2) is a thesis of the mini-theory TG (3, {B}). If we consider Metatheorem 4 , then we realize that the sentence g is a thesis of this theory. Eliminating operator L in formula (3) (according to the T-schema ‘L α ⇒ α ’), we get ‘g ⇒ Lg’. Hence, by modus ponens , we get Lg. Thus, we may conclude that (2) is a thesis of the mini-theory TG (3, {B}). Q.E.D.

5 The simplest relevant version of the modal ontological argument

It will now be shown that there is a correct version of the MOA that is simpler than all the versions considered so far. Footnote 19 The essence of this argument, which in logical terms borders on triviality, is the sentence.

We shall demonstrate in what follows that this premise correctly (though not necessarily completely) clarifies the idea of the existence of God as a necessary existence.

Indeed, such a claim is indirectly evidenced by the fact that such a postulate has appeared in the works of Hartshorne and Plantinga. It was also clearly accepted by Malcolm:

What Anselm has proved is that the notion of contingent existence or of contingent nonexistence cannot have any application to God. His existence must either be logically necessary or logically impossible.[…] If God, a being a greater than which cannot be conceived, does not exist then He cannot come into existence.[…] Since He cannot come into existence, if He does not exist His existence is impossible. (Malcolm 1960 , p. 49)

The direct justification for the thesis that (4) is the correct explication for the existence of God as a necessary existence runs in essence as follows: the idea is fully expressed (taking into account the object language of modal logic) in the form of Principle (2), and each of Postulates (3) and (4) can be treated as a part of the explication of the idea expressed by (2). Thus, if we assume that Principle (3) appropriately elucidates the aforementioned idea, then we should also assume that Principle (4) correctly explicates it. Footnote 20

This explicative dependence of both postulates on (2) is quite clearly visible in the context of possible-worlds semantics. Both postulates are similarly derived from the ontological principle, relating to (2), that God exists either in every possible world (accessible from the actual world) or not in any of them. According to this principle, if God exists in the actual world, then God exists in every possible world – which is the content of (3). Footnote 21 Similarly, if God does not exist in the actual world, then God does not exist in any possible world—which in turn, is the content of (4).

The following metatheorem shows the logical connection of both postulates with (2):

Metatheorem 6

⊢(2) if and only if ⊢(3) and ⊢(4).

The proof here is similar to that of Metatheorem 1 . For the implication from left to right, assume ⊢‘ ~ Mg ∨ Lg’. Hence, and from ⊢ ‘g ⇒ Mg’ and ⊢ ‘Lg ⇒ g’, we get ⊢ ‘g ⇒ Lg’ and (taking into consideration the Rule of Necessitation) ⊢ ‘L(g ⇒ Lg) ∧ (~ g ⇒ ~ Mg)’. For the implication from right to left, assume ⊢ ‘L(g ⇒ Lg) ∧ (~ g ⇒ ~ Mg)’. We therefore get ⊢ ‘(g ⇒ Lg) ∧ (Mg ⇒ g)’ and, by hypothetical syllogism: ⊢ ‘Mg ⇒ Lg’. Q.E.D.

It turns out that Premise (2) is logically equivalent to the conjunction of Premises (3) and (4).

Moreover, (4) is deductively weaker than both (2) and (3), where this stated by another metatheorem:

Metatheorem 7

The following relations hold:

(3), {N, B} ⊢(4),

Ad (i): obvious. Ad (ii): as with the proof of Metatheorem 4. Ad (iii) and (iv): it is enough to consider the possible-worlds model in which g is true in the actual world and there are possible worlds accessible from the actual world in which g is not true; we see that in this model (4) is true and (2) and (3) are not true. Q.E.D.

Consequently, since (3) means that the existence of God is either impossible or necessary, (2) and (3) only partially clarify this meaning. In fact, (3) only means that if God exists, he exists by necessity, and (4) only means that if God does not exist, he does not exist by necessity. Only both sentences taken together fully express the idea of God as a non-contingent being. Footnote 22

Thus, from an ontological and a logical point of view, Premise (4) is not less obvious, more controversial or in any other sense stronger than (3). On the contrary, taking into account conditions (ii) and (iv) of Metatheorem 7, Premise (4) is deductively weaker than (3). If we accept (3) as an intuitively acceptable premise in one or other of the MOA-versions, we must surely proceed likewise with (4). Premise (3) is commonly treated as being the result of a typical explication of the basic idea of God, so there is no reason for Premise (4) to be treated any differently. Footnote 23

We sometimes encounter sentence (4), or its logical equivalents, being treated as premises for a complex MOA-version, in which they are further formally justified (see, e.g., Spencer 2018 , p. 214). However, it appears that the persuasive power of (4) is no less than that of similar premises such as are normally accepted without any formal justification. Thus, (4) does not call for such justification more than in the case of other MOA-versions.

Since Postulate (4) explicates the idea of the existence of God as a necessary existence, the mini-theory TG ((4), ∅) represents one of the MOA-versions meeting the relevance condition.

Metatheorem 8

The “empty” structure [(4), ∅] represents a formally correct MOA-version which is simpler than T-version.

The parts of the proof relating to the conditions of consistency and non-circularity are analogous to the relevant parts of the proof of Metatheorem 2 . It is therefore enough to note that |∅| <|{T}|, and the sentence g is a thesis of the mini-theory TG (2) (as we know from Metatheorem 2 ). Therefore, it is all the more the case that (4) is one of its theses. Q.E.D.

6 Some stronger correctness and relevance criteria for the modal ontological argument

The logical triviality of the “empty” MOA-version suggests that the range of formally correct and philosophically relevant MOA-versions should be limited in such a way that this version can be considered incorrect or irrelevant. Consider the two options outlined below.

It seems that the only correspondingly appropriate way to limit the concept of formal correctness is to introduce an additional condition into Definition 2 :

This condition states (in simple terms) that the specific premise that defines the concept of God as a non-contingent being cannot be a consequence of the thesis that God exists. At first glance, the following metatheorem would seem to support a possible decision to introduce this condition:

Metatheorem 9

If [ α , X ] is valid and g, X ⊢ α , then X ⊢ ‘g ⇔ (Mg ∧ α )’.

The second part of the antecedent of the metatheorem is equivalent (in virtue of the deduction theorem) to the meta-formula X ⊢ ‘g ⇒ α ’. Hence, taking into consideration that X ⊢ ‘g ⇒ Mg’, we get: X ⊢ ‘g ⇒ (Mg ∧ α )’. Since [ α , X ] is valid, the reverse implication is also derivable: X ⊢ ‘(Mg ∧ α ) ⇒ g’. Q.E.D.

Metatheorem 9 shows that any valid MOA-version that does not meet (C) is “circular” in the sense that the conjunction of its premises is logically equivalent to its conclusion. (We are, at the very least, using the term “logically” here just as it pertains to modal logic).

The following metatheorem shows that the “empty” MOA-version does not meet Condition (C):

Metatheorem 10

If we take into account the Duns Scotus Law, we realize that ⊢ ‘g ⇒ (~ g ⇒ ~ Mg)’. Keeping in mind the deductive theorem, we thus get: g ⊢‘ ~ g ⇒ ~ Mg’. Q.E.D.

Unlike (4), Premises (2) and (3) cannot be derived from Sentence g alone, together with the laws of modal logic. (Let us leave this observation without proof). This shows that Condition (C) could be used to eliminate the “empty” MOA-version.

The question is whether Condition (C) should be accepted. Let us recall its general content: the premise defining the concept of God cannot be a consequence of the thesis that God exists. But why not? There is no logical, methodological or philosophical reason to accept such a restriction. Apparently, an attempt to introduce it would be an ad hoc solution, only aimed at eliminating one of the MOA-versions.

Let us therefore consider the second option (referred to at the beginning of this section). A quite natural way of strengthening the relevance condition was already indicated in Metatheorems 1 and 6 , and in the analysis of the previous section. Its philosophical basis runs as follows: an MOA-version will be strongly relevant if the premise defining the philosophical concept of God (as a non-contingent being) is a complete explication of that concept. According to Metatheorems 1 and 6 , and the analyses carried out in the previous section, Postulate (2) is such a complete explication, as opposed to Postulates (3) and (4). Consequently, the T-version is, in contrast to other versions, a strongly relevant MOA-version.

Let us recall the general distinction between the two types of MOA-relevance, and try to find a good philosophical basis for it. Postulate (2) is a complete explication of the idea of God as a non-contingent being in the language of the mini-theory TG (2). Postulate (3) is its partial explication, because (3) only expresses a necessary condition for the existence of God. Postulate (4) is also its partial explication, but for another reason: because (4) only expresses a sufficient condition for this existence. Unlike mini-theories TG (3, {S5}), TG (3, {B}) and TG (4), the mini-theory TG (2) generates both a necessary and a sufficient condition for God to exist. So, from a philosophical and theoretical point of view, TG (2) is a better mini-theory than the others and, consequently, the T-version is a better MOA-version than the others.

This does not mean that the MOA-version represented by the structure [(2), {T}] is absolutely preferable to the structures [(3), {T, N, S5}], [(3), {N, B}] and [(4), ∅]. On the contrary, according to the general theory of argumentation, there are many types of relevance depending on kinds of arguments and their conversational contexts (see, e.g., Walton 1998 ). In the case of a philosophical argument, its context can be determined equally by its persuasive and explicative (or, more precisely, theoretical-explicative) purpose. If the argument is formulated in a persuasive context, the requirement of full explication of the notions used in it, and therefore the requirement of strong relevance, does not apply.

Consequently, we should use two criteria in assessing MOA-versions: the weak criterion and the strong criterion of MOA-relevance. If the MOA-version is formulated to convince someone that the God of philosophers exists, the strong criterion is unnecessary. This criterion, on the other hand, is essential for the evaluation of each MOA-version formulated in order to examine the consequences of the explicatively complete concept of God.

Given the persuasive function of the argument, the “empty” MOA-version would seem to be the optimal one. There are at least two reasons for this assessment. Firstly, the “empty” version is the simplest of the formally correct and (weakly) relevant MOA-versions (cf. Metatheorems 5 – 7 ). Secondly, the “empty” version mounts an effective defence against a typical counter-argument that purports to show the persuasive weakness of the standard MOA-versions. Let us replace Premise (1) with the sentence ‘M ~ g’ (“It is possible that God does not exist”); hence, from (2) we derive (in the T -system) ‘ ~ g’ (“God does not exist”), and from (3) we also therefore derive (in the S5 -system) ‘ ~ g’ (cf. Oppy 1996 , 2019 ). It is easy to see that there is no analogous counter-argument to the “empty” MOA-version.

7 Conclusions

The questions posed at the end of the first section can now be answered quite precisely. Each relevant (zero-order, Hartshorne-style) MOA-version has the structure < T , (1), α , X , g>, where α is a specific premise clarifying the idea that the existence of God is not contingent and X is a set of general premises resulting from modal logic. The formal correctness criterion for such versions consists of conditions of VALIDITY, CONSISTENCY, and NON-CIRCULARITY.

It turns out that the simplest known MOA-version fulfilling these conditions has the structure [(4), ∅]. In contrast to the previously presented versions (T, S5 and B), this “empty” MOA-version is devoid of general premises taken from modal logic. Thus, its entire strength lies in its specific philosophical premises, not in its logic. (These premises state that the existence of God is possible, and that if God does not exist, the existence of God is impossible.)

Given the persuasive function of the argument, the “empty” version seems to be the basic MOA-version on account of its simplicity (consisting in its formal simplicity and the deductive weakness of its mini-theory of God) and its resistance to a well-known counter-argument from the possibility of the non-existence of God.

Of all the MOA-versions considered here, only the T-version meets the strong relevance condition of explicative completeness, because only this version expresses precisely the idea (from Aristotle) of God’s non-contingency in the form of a necessary and a sufficient condition for the existence of God. For this reason, from a theoretical point of view (although not necessarily from a persuasive point of view), the T-version should be treated as the basic MOA-version.

These conclusions undermine the view (mentioned in Sect.  1 ) that certain general premises generated by stronger systems of modal logic are logically essential, or at least highly adequate philosophically, where the MOA is concerned. Let us recall both of the theses suggested by van Inwagen:

Each formally correct and relevant MOA-version has either strong specific premises and weak general premises (van Inwagen 2012 , p. 158, is referring here to the T-version) or, equally, weak specific premises and strong general premises (he is referring here to the S5-version).

The option indicated in the second part of this equivalence (contained in Thesis 1) is philosophically better.

Both theses turn out to be false, assuming the criteria adopted here for the evaluation and selection of MOA-versions. Footnote 24

Firstly, there is a formally correct and persuasively relevant MOA-version (namely, the “empty” version) that has relatively weak specific (metaphysical) premises and no general premises. Thus, van Inwagen’s specification omits the type of MOA-version that plays a key role in our analysis. Moreover, the equivalence contained in Thesis 1 is incorrect in one important respect: the T-version (indicated on the left) is not philosophically equivalent to the S5-version (indicated on the right). The S5-version, unlike the T-version, is based on an explicatively incomplete mini-theory of God. (More precisely, the mini-theory of God underlying the S5-version generates a necessary condition for the existence of God, but does not generate a sufficient condition for it.)

Secondly, both because of this explicative incompleteness and on account of its complexity, the S5-version is philosophically inferior to the T-version. Moreover, despite the lesser complexity of the B-version (relative to the S5-version), that same conclusion applies to it, too.

These versions basically have their origins in Hartshorne ( 1944 ), Malcolm ( 1960 ) and Plantinga ( 1974 ) (although their intuitive formulation already appears in St. Anselm’s Proslogion III ; see Eder and Ramharter 2015 , p. 2815; Oppy 2019 ). They are clearly distinguished here from arguments of the Gödelian kind formulated within second- or higher-order modal theories of positive properties (see Sobel 1987 ; Benzmüller 2020 ). The MOAs and Gödelian-type arguments are also specified in Oppy’s taxonomy as two different kinds of ontological argument (Oppy 2019 ). The abbreviation “MOA” is most commonly used in the philosophical literature to designate the former. (Even so, the term “modal ontological argument” could also be used in the broader sense of “ontological argument using modal concepts”, and in this sense, Gödelian-type arguments could also be called “modal”.).

G.W. Leibniz is commonly regarded as the philosopher who was the first to point out the key role of this premise in the ontological argument (see, e.g., Perzanowski 1991 ; Antognazza 2018 ). The issue of the reliability of this premise will not be discussed in the present paper.

The argument presented in Lewis ( 1970 ) is an example of another MOA-version formulated within an extensional first-order theory of possible worlds.

These versions belong to one of the four kinds of MOA distinguished in Oppy ( 1996 , pp. 70–78). This kind includes arguments of the sort Oppy calls “ontological arguments involving necessity”.

Oppy ( 2019 ). This version was already considered by Hartshorne (see Goodwin 1978 ). An even simpler MOA-version is analyzed in Oppy ( 1996 ): “It is possible that it is necessarily the case that God exists. Hence, God exists.” This version is valid on the basis of systems B and S5 of modal logic. Other B- and S5-versions of the MOA will be presented in Sect.  4 . (The T , B and S5 systems are deductive extensions of classical sentential logic, and belong to what is known as normal modal logic. Chellas ( 1980 ) offers an accessible characterization of what they involve.).

It is worth noting that in recent years an opposite trend has appeared in studies on Gödelian-type of ontological argument (which was originally formulated within the S5 -system, cf. Sobel 1987 ). For example, Świętorzecka and Łyczak ( 2018 ) reconstruct a version of this argument within the S4 -system, and Benzmüller ( 2020 ) within the T -system and even in the weaker K -system. The analyses undertaken in this paper go in a similar direction, that is, towards the search for possibly simple and adequate MOA-versions.

This view was also adopted in a well-known philosophical dictionary, in which one of the S5-versions was commented on as follows: “The correct response to this argument is to disallow the apparently reasonable concession that it is possible that such a being exist. This concession is much more dangerous than it looks, since in the modal logic involved, from possibly necessarily p , we can derive necessarily p ” (Blackburn 1994 , p. 269). Cf. also the footnote 18 .

“Here is the version I think is the clearest and most elegant” (van Inwagen 2012 , p. 157). “[…] the modal logic of the argument is S5 , the strongest modal system. This is not the case with every version of the modal argument. Some are valid in weaker modal systems, but those arguments require additional premises” ( ibid. , p. 158). “One could regard the first premise of each of Hartshorne’s arguments [equivalent to formula (2)] as substitutes for an appeal to the strong modal system S5 ” (van Inwagen 2018 , p. 242).

Taking into account this assumption, we will omit systems possessing a different set of primitive rules, such as D+  and T+  (with the so called MacIntosh rule), which are sometimes also considered in MOA-related literature (see Chellas and Segerberg 1994 ).

The operators M and L can also be understood as formal counterparts of Anselm’s “it is conceivable that” and “it is not conceivable that not”, respectively (see Eder and Ramharter 2015 , p. 2814). In turn, for the constant g, the following interpretation is sometimes proposed: “There is a necessarily existent being that has all perfections essentially” (van Inwagen 2012 , p. 158, 2018 , p. 242).

Throughout the article, the bold symbols “ T ”, “ B ” and “ S5 ” stand for systems of modal logic, while the ordinary symbols “T”, “B” and “S5” stand for sentences falling under axiom schemes appropriate to these systems.

Apart from specific and general premises, the laws of classical sentential logic are also applied in each version of the MOA (as in any other argument). This fact, which we take to be quite obvious, is not one of which we intend to make any special use.

The issue of the proper selection of general premises in the MOA was already raised in Kane ( 1984 ).

Similar assumptions can sometimes be found in literature on the MOA. Cf., for example, van Inwagen’s remark that the formula ‘L p ⇒ p ’ “must be valid in every modal system in which the sentential operators represent possibility and necessity in any intuitive sense” (van Inwagen 2018 , p. 242). Cf. also Eder and Ramharter ( 2015 ), where the authors state that the T -system would “seem to be mandatory on any modal conception of conceivability which can claim to be faithful to Anselm’s reasoning” (p. 2814).

This MOA-version corresponds to the version considered in Oppy ( 1996 ) (cf. footnote 5 ).

The difference between objects [α, X ] and TG (α, X ) is worth emphasizing. The former is the abstract structure of a given MOA-version, while the latter is a set of propositions underlying that version.

The formal concept of “circularity” used above differs from the intuitive concept of “begging the question” used by van Inwagen ( 2018 ).

See Kane ( 1984 , p. 339). According to Kane, this fact proves that the B-version is the right version of the MOA. Even so, the view that the S5-system is essential for a proper analysis of the MOA has been quite popular in the philosophical literature—cf., for example, this statement: “[…] all modal ontological arguments are valid in S5 (and they are valid in no weaker modal system […])” (van Inwagen 2009 , pp. 219–20).

I shall make use of a similar result here to that arrived at in Biłat ( 2012a , b ). (The latter being a slightly shortened translation of the former.).

We omit here the fact that (3) is additionally preceded by a necessity operator. This is possible thanks to the assumption made here that Rule of Necessitation can also be applied to extra-logical a priori statements such as (4). (The modal status of (3) and (4), as a priori statements, is exactly the same).

This principle can also be expressed in Plantinga’s style: if God exists in the actual world, then God is maximally great (cf. Plantinga 1974 ).

This topic of the explicative dependencies, respectively, of (3) and (4) on (2), will be developed in the next section.

Postulate (4) is logically equivalent to the thesis ‘Mg ⇒ g’. This MOA-version is even closer to Leibniz’s conception of God (defined as the necessarily existing being), according to which if God is possible, then God exists (see Griffin 2012 , pp. 42, 43; Antognazza 2018 , pp. 75, 78–89).

It is worth noting that Theses 1 and 2 do not perform any key role as regards the main theses of van Inwagen’s article quoted above.

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What Is the Ontological Argument?

There are many arguments for the existence of God. Perhaps one of the most famous — and most often misunderstood — is the ontological argument. Dr. J.R. Gilhooly, Assistant Professor of Theology at Cedarville University, explains this argument in the video below.

Unlike most arguments that start with an observation about the world and work back to a Creator, the ontological argument starts with the idea that based on the meaning of the word “God,” there has to be a God. There are many ways to make this argument but the simplest way is this: If it’s possible that God exists, then God exists.

As Christians, we should know and understand that there are good, sound arguments for God’s existence and be prepared to explain them to our skeptical friends.

Watch the video below for more with Dr. Gilhooly.

Cedarville University's  Master of Divinity  degree and online  Master of Ministry  degree are equipping men and women who desire to serve God in vocational ministry. With multiple accelerated options to choose from, Cedarville’s M.Div. provides a comprehensive program that is committed to biblical authority, the Great Commission, and equipping the next generation of leaders in the local church.

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The Ontological Argument and the Existence of God

An undergraduate essay on the Ontological Proof, drawing the conclusion that while it is fatally flawed as a logical proof, it nevertheless remains valuable as a demonstration of the coherence of the concept of God

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The Downside Review

Martin Benson

Anselm of Canterbury’s Proslogion is a Benedictine prayer-exercise that contains a famous argument for the existence of God. This article highlights how the argument is intertwined with the prayer. The article argues that since the understanding of God leads to a joyous affect, the logic of the argument must be causally connected with joy. While much of the secondary literature applies a division between ‘prayer’ and ‘proof’, this article suggests a reading of the Proslogion proof as a prayer-practice, and the prayer-practice is in turn analyzed through the logic of the proof. The result is a description of how contemplation of the argument drives affect, leading to the conclusion that the affect of joy achieves the intended result of the proof: the joy leads the mind to God. The article thus shows that the Proslogion is an intellectual affective prayer-practice.

John D. Wilsey

Luke Arredondo

Edgar Ter Danielyan

I propose that reading 2 and 3 Proslogion as advancing exclusively philosophical arguments and virtually ignoring the context of both the book and its author is untenable. I agree with Smart who identifies a radical misunderstanding of the Sitz im Leben of Proslogion in exclusively philosophical readings of Anselm: 'Anselm's attitude is radically misunderstood if it is thought of as being purely philosophical and theoretical. His real aim is practical, polemical, apologetic. He is engaged in combatting scepticism - the scepticism of the fool who says in his heart, "There is no God”. Against this position Saint Anselm advances the view that not only does God exist, He exists with an absolute necessity which excludes the possibility of His being even conceived correctly as possibly not existing. In pursuit of such a purpose Anselm might legitimately use any order of cognition he thought sound and he might therefore be expected to depend on faith at least in part; only had his end been pure philosophic speculation would such an appeal to the supra-rational have been out of court.' Smart’s view is supported in particular by Anselm’s reply to Gaunilo, where he writes ‘Now my strongest argument that this is false is to appeal to your faith and to your conscience’ - such an appeal being reasonable only if the arguments advanced are not exclusively philosophical.

Scottish Journal of Theology

Sigurd Baark

Shaun Smith

James Cleary

This paper seeks to analyse the metaphysical presuppositions of the word maius in Anselm’s argument. In order to do so, the paper first interprets Anselm and his philosophy and then the Proslogion. Since maius is seldom used in the Proslogion outside the phrase referring to God, the paper recurs to the Monologion for a further interpretation. This then permits the author to analyse maius in the context in which Anselm used the word, focussing both on what it refers to as well as what metaphysical presuppositions the word entails in the phrase id quo maius cogitari non potest. A modified version of this paper was named runner-up to the Marco Arosio Award 2015"

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Ontological Arguments

Ontological arguments are arguments, for the conclusion that God exists, from premises which are supposed to derive from some source other than observation of the world—e.g., from reason alone. In other words, ontological arguments are arguments from nothing but analytic, a priori and necessary premises to the conclusion that God exists.

The first, and best-known, ontological argument was proposed by St. Anselm of Canterbury in the 11th. century C.E. In his Proslogion , St. Anselm claims to derive the existence of God from the concept of a being than which no greater can be conceived . St. Anselm reasoned that, if such a being fails to exist, then a greater being—namely, a being than which no greater can be conceived, and which exists —can be conceived. But this would be absurd: nothing can be greater than a being than which no greater can be conceived. So a being than which no greater can be conceived—i.e., God—exists.

In the seventeenth century, René Descartes defended a family of similar arguments. For instance, in the Fifth Meditation , Descartes claims to provide a proof demonstrating the existence of God from the idea of a supremely perfect being. Descartes argues that there is no less contradiction in conceiving a supremely perfect being who lacks existence than there is in conceiving a triangle whose interior angles do not sum to 180 degrees. Hence, he supposes, since we do conceive a supremely perfect being—we do have the idea of a supremely perfect being—we must conclude that a supremely perfect being exists.

In the early eighteenth century, Gottfried Leibniz attempted to fill what he took to be a shortcoming in Descartes’ view. According to Leibniz, Descartes’ arguments fail unless one first shows that the idea of a supremely perfect being is coherent, or that it is possible for there to be a supremely perfect being. Leibniz argued that, since perfections are unanalysable, it is impossible to demonstrate that perfections are incompatible—and he concluded from this that all perfections can co-exist together in a single entity.

In more recent times, Kurt Gödel, Charles Hartshorne, Norman Malcolm and Alvin Plantinga have all presented much-discussed ontological arguments which bear interesting connections to the earlier arguments of St. Anselm, Descartes and Leibniz. Of these, the most interesting are those of Gödel and Plantinga; in these cases, however, it is unclear whether we should really say that these authors claim that the arguments are proofs of the existence of God.

Critiques of ontological arguments begin with Gaunilo, a contemporary of St. Anselm. Perhaps the best known criticisms of ontological arguments are due to Immanuel Kant, in his Critique of Pure Reason . Most famously, Kant claims that ontological arguments are vitiated by their reliance upon the implicit assumption that “existence” is a predicate. However, as Bertrand Russell observed, it is much easier to be persuaded that ontological arguments are no good than it is to say exactly what is wrong with them. This helps to explain why ontological arguments have fascinated philosophers for almost a thousand years.

In various ways, the account provided to this point is rough, and susceptible of improvement. Sections 1–5 in what follows provide some of the requisite embellishments, though—as is usually the case in philosophy—there are many issues taken up here which could be pursued at much greater length. Sections 6–8 take up some of the central questions at a slightly more sophisticated level of discussion. Section 9 is a quick overview of very recent work on ontological arguments:

1. History of Ontological Arguments

2. taxonomy of ontological arguments, 3. characterisation of ontological arguments, 4. objections to ontological arguments, 5. parodies of ontological arguments, 6. gödel’s ontological argument, 7. a victorious ontological argument, 8. st. anselm’s ontological argument, 9. ontological arguments in the 21st century, primary texts, other texts, other internet resources, related entries.

For a useful discussion of the history of ontological arguments in the modern period, see Harrelson 2009.

According to a modification of the taxonomy of Oppy 1995, there are eight major kinds of ontological arguments, viz:

• definitional ontological arguments;
• conceptual (or hyperintensional) ontological arguments;
• modal ontological arguments;
• Meinongian ontological arguments;
• experiential ontological arguments;
• mereological ontological arguments;
• higher-order ontological arguments; and
• ‘Hegelian’ ontological arguments;

Examples of all but the last follow. These are mostly toy examples. But they serve to highlight the deficiencies which more complex examples also share.

Note: I provide no example of a ‘Hegelian’ ontological argument because I know of no formulation of such an argument. Many people assert that Hegel provided an ontological argument; but, when pressed for a list of the premises of the argument, Hegel’s friends fail to deliver. (For a defense of Hegel against these charges—but not for a supply of the premises of ‘the Hegelian ontological argument’—see Redding and Bubbio 2014.)

God is a being which has every perfection. (This is true as a matter of definition.) Existence is a perfection. Hence God exists.

I conceive of a being than which no greater can be conceived. If a being than which no greater can be conceived does not exist, then I can conceive of a being greater than a being than which no greater can be conceived—namely, a being than which no greater can be conceived that exists. I cannot conceive of a being greater than a being than which no greater can be conceived. Hence, a being than which no greater can be conceived exists.

It is possible that that God exists. God is not a contingent being, i.e., either it is not possible that God exists, or it is necessary that God exists. Hence, it is necessary that God exists. Hence, God exists. (See Malcolm 1960, Hartshorne 1965, and Plantinga 1974 for closely related arguments.)

[It is analytic, necessary and a priori that] Each instance of the schema “The F G is F ” expresses a truth. Hence the sentence “The existent perfect being is existent” expresses a truth. Hence, the existent perfect being is existent. Hence, God is existent, i.e. God exists. (The last step is justified by the observation that, as a matter of definition, if there is exactly one existent perfect being, then that being is God.)

The word ‘God’ has a meaning that is revealed in religious experience. The word ‘God’ has a meaning only if God exists. Hence, God exists. (See Rescher 1959 for a live version of this argument.)

I exist. Therefore something exists. Whenever a bunch of things exist, their mereological sum also exists. Therefore the sum of all things exists. Therefore God—the sum of all things—exists.

Say that a God-property is a property that is possessed by God in all and only those worlds in which God exists. Not all properties are God properties. Any property entailed by a collection of God-properties is itself a God-property. The God-properties include necessary existence, necessary omnipotence, necessary omniscience, and necessary perfect goodness. Hence, there is a necessarily existent, necessarily omnipotent, necessarily omniscient, and necessarily perfectly good being (namely, God).

Of course, this taxonomy is not exclusive: an argument can belong to several categories at once. Moreover, an argument can be ambiguous between a range of readings, each of which belongs to different categories. This latter fact may help to explain part of the curious fascination of ontological arguments. Finally, the taxonomy can be further specialised: there are, for example, at least four importantly different kinds of modal ontological arguments which should be distinguished. (See, e.g., Ross 1969 for a rather different kind of modal ontological argument.)

It is not easy to give a good characterisation of ontological arguments. The traditional characterisation involves the use of problematic notions—analyticity, necessity, and a priority —and also fails to apply to many arguments to which defenders have affixed the label “ontological”. (Consider, for example, the claim that I conceive of a being than which no greater can be conceived. This claim is clearly not analytic (its truth doesn’t follow immediately from the meanings of the words used to express it), nor necessary (I might never have entertained the concept), nor a priori (except perhaps in my own case, though even this is unclear—perhaps even I don’t know independently of experience that I have this concept.)) However, it is unclear how that traditional characterisation should be improved upon.

Perhaps one might resolve to use the label “ontological argument” for any argument which gets classified as “an ontological argument” by its proponent(s). This procedure would make good sense if one thought that there is a natural kind—ontological arguments—which our practice carves out, but for which is hard to specify defining conditions. Moreover, this procedure can be adapted as a pro tem stop gap: when there is a better definition to hand, that definition will be adopted instead. On the other hand, it seems worthwhile to attempt a more informative definition.

Focus on the case of ontological arguments for the conclusion that God exists. One characteristic feature of these arguments is the use which they make of “referential vocabulary”—names, definite descriptions, indefinite descriptions, quantified noun phrases, etc.—whose ontological commitments—for occurrences of this vocabulary in “referential position”—non-theists do not accept.

Theists and non-theists alike (can) agree that there is spatio-temporal, or causal, or nomic, or modal structure to the world (the basis for cosmological arguments); and that there are certain kinds of complexity of organisation, structure and function in the world (the basis for teleological arguments); and so on. But theists and non-theists are in dispute about whether there are perfect beings, or beings than which no greater can be conceived, or … ; thus, theists and non-theists are in dispute about the indirect subject matter of the premises of ontological arguments.

Of course, the premises of ontological arguments often do not deal directly with perfect beings, beings than which no greater can be conceived, etc.; rather, they deal with descriptions of, or ideas of, or concepts of, or the possibility of the existence of, these things. However, the basic point remains: ontological arguments require the use of vocabulary which non-theists should certainly find problematic when it is used in ontologically committing contexts (i.e not inside the scope of prophylactic operators—such as “according to the story” or “by the lights of theists” or “by the definition”—which can be taken to afford protection against unwanted commitments).

Note that this characterisation does not beg the question against the possibility of the construction of a successful ontological argument—i.e., it does not lead immediately to the conclusion that all ontological arguments are question-begging (in virtue of the ontologically committing vocabulary which they employ). For it may be that the vocabulary in question only gets used in premises under the protection of prophylactic operators (which ward off the unwanted commitments.) Of course, there will then be questions about whether the resulting arguments can possibly be valid—how could the commitments turn up in the conclusion if they are not there in the premises?—but those are further questions, which would remain to be addressed.

Objections to ontological arguments take many forms. Some objections are intended to apply only to particular ontological arguments, or particular forms of ontological arguments; other objections are intended to apply to all ontological arguments. It is a controversial question whether there are any successful general objections to ontological arguments.

One general criticism of ontological arguments which have appeared hitherto is this: none of them is persuasive , i.e., none of them provides those who do not already accept the conclusion that God exists—and who are reasonable, reflective, well-informed, etc.—with either a pro tanto reason or an all-things-considered reason to accept that conclusion. Any reading of any ontological argument which has been produced so far which is sufficiently clearly stated to admit of evaluation yields a result which is invalid, or possesses a set of premises which it is clear in advance that no reasonable, reflective, well-informed, etc. non-theists will accept, or has a benign conclusion which has no religious significance, or else falls prey to more than one of the above failings.

For each of the families of arguments introduced in the earlier taxonomy, we can give general reasons why arguments of that family fall under the general criticism. In what follows, we shall apply these general considerations to the exemplar arguments introduced in section 2.

(1) Definitional arguments: These are arguments in which ontologically committing vocabulary is introduced solely via a definition. An obvious problem is that claims involving that vocabulary cannot then be non-question-beggingly detached from the scope of that definition. (The inference from ‘By definition, God is an existent being’ to ‘God exists’ is patently invalid; while the inference to ‘By definition, God exists’ is valid, but uninteresting. In the example given earlier, the premises licence the claim that, as a matter of definition, God possesses the perfection of existence. But, as just noted, there is no valid inference from this claim to the further claim that God exists.)

(2) Conceptual arguments: These are arguments in which ontologically committing vocabulary is introduced solely within the scope of hyperintensional operators (e.g. ‘believes that’, ‘conceives of’, etc.). Often, these operators have two readings, one of which can cancel ontological commitment, and the other of which cannot. On the reading which can give cancellation (as in the most likely reading of ‘John believes in Santa Claus’), the inference to a conclusion in which the ontological commitment is not cancelled will be invalid. On the reading which cannot cancel ontological commitment (as in that reading of ‘John thinks about God’ which can only be true if there is a God to think about), the premises are question-begging: they incur ontological commitments which non-theists reject. In our sample argument, the claim, that I conceive of an existent being than which no greater being can be conceived, admits of the two kinds of readings just distinguished. On the one hand, on the reading which gives cancellation, the inference to the conclusion that there is a being than which no greater can be conceived is plainly invalid. On the other hand, on the reading in which there is no cancellation, it is clear that this claim is one which no reasonable, etc. non-theist will accept: if you doubt that there is a being than which no greater can be conceived, then, of course, you doubt whether you can have thoughts about such a being.

(3) Modal arguments: These are arguments with premises which concern modal claims about God, i.e., claims about the possibility or necessity of God’s attributes and existence. Suppose that we agree to think about possibility and necessity in terms of possible worlds: a claim is possibly true just in case it is true in at least one possible world; a claim is necessarily true just in case it is true in every possible world; and a claim is contingent just in case it is true in some possible worlds and false in others. Some theists hold that God is a necessarily existent being, i.e., that God exists in every possible world; all non-theists reject the claim that God exists in the actual world. The sample argument consists, in effect, of two premises:

• God exists in at least one possible world.
• God exists in all possible worlds if God exists in any.

A minimally rational non-theist would not accept both of these premises – they entail that God exists in every possible world whereas a minimally rational non-theists would insist that there is at least one possible world in which God does not exist. Given that that a minimally rational non-theist accepts that there is at least one possible world in which God does not exist, such a non-theist could offer the following counterargument:

• God fails to exist in at least one possible world.

These premises entail that God exists in no possible world, and hence that God does not exist in the actual world. Considered together, the argument and the counterargument just mentioned plainly do not give anyone a reason to prefer theism to non-theism, and nor do they give anyone a reason to prefer non-theism to theism. So the sample argument is unsuccessful: it doesn’t supply an all-things-considered reason to prefer theism to non-theism (just as the counterargument doesn’t supply an all-things-considered reason to prefer non-theism to theism).

(4) Meinongian arguments: These are arguments which depend somehow or other on Meinongian theories of objects. Consider the schema ‘The F G is F ’. Naive Meinongians will suppose that if F is instantiated with any property, then the result is true (and, quite likely, necessary, analytic and a priori). So, for example, the round square is round; the bald current King of France is bald; and so on. However, more sophisticiated Meinongians will insist that there must be some restriction on the substitution instances for F, in order to allow one to draw the obvious and important ontological distinction between the following two groups: {Bill Clinton, the sun, the Eiffel Tower} and {Santa Claus, Mickey Mouse, the round square}. Choice of vocabulary here is controversial: Let us suppose (for the sake of example) that the right thing to say is that the former things exist and the latter do not. Under this supposition, ‘existent’ will not be a suitable substitution instance for F—obviously, since we all agree that there is no existent round square. Of course, nothing hangs on the choice of ‘existent’ as the crucial piece of vocabulary. The point is that non-theists are not prepared to include god(s) in the former group of objects—and hence will be unpersuaded by any argument which tries to use whatever vocabulary is used to discriminate between the two classes as the basis for an argument that god(s) belong to the former group. (Cognoscenti will recognise that the crucial point is that Meinongian ontological arguments fail to respect the distinction between nuclear (assumptible, characterising) properties and non-nuclear (non-assumptible, non-characterising) properties. It should, of course, be noted that neither Meinong, nor any of his well-known modern supporters—e.g. Terence Parsons, Richard Sylvan—ever endorses a Meinongian ontological argument; and it should also be noted that most motivate the distinction between nuclear and non-nuclear properties in part by a need to avoid Meinongian ontological arguments. The reason for calling these arguments “Meinongian” is that they rely on quantification over—or reference to—non-existent objects; there is no perjorative intent in the use of this label.)

(5) Experiential arguments: These are arguments which try to make use of ‘externalist’ or ‘object-involving’ accounts of content. It should not be surprising that they fail. After all, those accounts of content need to have something to say about expressions which fail to refer (‘Santa Claus’, ‘phlogiston’, etc.). But, however the account goes, non-theists will insist that expressions which purport to refer to god(s) should be given exactly the same kind of treatment.

(6) Mereological arguments: Those who dislike mereology will not be impressed by these arguments. However, even those who accept principles of unrestricted composition—i.e., who accept principles which claim, e.g., that, whenever there are some things, there is something which is the sum or fusion of all of those things—need not be perturbed by them: for it is plausible to think that the conclusions of these arguments have no religious significance whatsoever—they are merely arguments for, e.g., the existence of the physical universe.

(7) Higher-Order arguments: The key to these arguments is the observation that any collection of properties, that (a) does not include all properties and (b) is closed under entailment, is possibly jointly instantiated. If it is impossible that God exists — as all who deny that God exists suppose, on the further assumption that, were God to exist, God would exist of necessity — then it cannot be true both that the God-properties are closed under entailment and that there are properties that are not God-properties. Those who take themselves to have good independent reason to deny that there are any gods will take themselves to have good independent reason to deny that there are God-properties that form a non-trivial collection that is closed under entailment.

Even if the forgoing analyses are correct, it is important to note that no argument has been given for the conclusion that no ontological argument can be successful. Even if all of the kinds of arguments produced to date are pretty clearly unsuccessful—i.e., not such as ought to give non-theists reason to accept the conclusion that God exists—it remains an open question whether there is some other kind of hitherto undiscovered ontological argument which does succeed. (Perhaps it is worth adding here that there is fairly widespread consensus, even amongst theists, that no known ontological arguments for the existence of God are persuasive. Most categories of ontological argument have some actual defenders; but none has a large following.)

Many other objections to (some) ontological arguments have been proposed. All of the following have been alleged to be the key to the explanation of the failure of (at least some) ontological arguments: (1) existence is not a predicate (see, e.g., Kant, Smart 1955, Alston 1960); (2) the concept of god is meaningless/incoherent/ inconsistent (see, e.g., Findlay 1949); (3) ontological arguments are ruled out by “the missing explanation argument” (see Johnston 1992; (4) ontological arguments all trade on mistaken uses of singular terms (see, e.g., Barnes 1972; (5) existence is not a perfection (see almost any textbook in philosophy of religion); (6) ontological arguments presuppose a Meinongian approach to ontology (see, e.g., Dummett 1993); and (7) ontological arguments are question-begging, i.e., presuppose what they set out to prove (see, e.g., Rowe 1989). There are many things to say about these objections: the most important point is that almost all of them require far more controversial assumptions than non-theists require in order to be able to reject ontological arguments with good conscience. Trying to support most of these claims merely in order to beat up on ontological arguments is like using a steamroller to crack a nut (in circumstances in which one is unsure that one can get the steamroller to move!).

Of course, all of the above discussion is directed merely to the claim that ontological arguments are not dialectically efficacious—i.e., they give reasonable non-theists no reason to change their views. It might be wondered whether there is some other use which ontological arguments have—e.g., as Plantinga claims, in establishing the reasonableness of theism. This seems unlikely. After all, at best these arguments show that certain sets of sentences (beliefs, etc.) are incompatible—one cannot reject the conclusions of these arguments while accepting their premises. But the arguments themselves say nothing about the reasonableness of accepting the premisses. So the arguments themselves say nothing about the (unconditional) reasonableness of accepting the conclusions of these arguments. Those who are disposed to think that theism is irrational need find nothing in ontological arguments to make them change their minds (and those who are disposed to think that theism is true should take no comfort from them either).

Positive ontological arguments—i.e., arguments FOR the existence of god(s)—invariably admit of various kinds of parodies, i.e., parallel arguments which seem at least equally acceptable to non-theists, but which establish absurd or contradictory conclusions. For many positive ontological arguments, there are parodies which purport to establish the non-existence of god(s); and for many positive ontological arguments there are lots (usually a large infinity!) of similar arguments which purport to establish the existence of lots (usally a large infinity) of distinct god-like beings. Here are some modest examples:

(1) By definition, God is a non-existent being who has every (other) perfection. Hence God does not exist.

(2) I conceive of a being than which no greater can be conceived except that it only ever creates n universes. If such a being does not exist, then we can conceive of a greater being—namely, one exactly like it which does exist. But I cannot conceive of a being which is greater in this way. Hence, a being than which no greater can be conceived except that it only ever creates n universes exists.

(3) It is possible that God does not exist. God is not a contingent being, i.e., either it is not possible that God exists, or it is necessary that God exists. Hence it is not possible that God exists. Hence God does not exist.

(4) It is analytic, necessary, and a priori that the F G is F . Hence, the existent perfect being who creates exactly n universes is existent. Hence the perfect being who creates exactly n universes exists.

There are many kinds of parodies on Ontological Arguments. The aim is to construct arguments which non-theists can reasonably claim to have no more reason to accept than the original Ontological Arguments themselves. Of course, theists may well be able to hold that the originals are sound, and the parodies not—but that is an entirely unrelated issue. (All theists—and no non-theists—should grant that the following argument is sound, given that the connectives are to be interpretted classically: “Either 2+2=5, or God exists. Not 2+2=5. Hence God exists.” It should be completely obvious that this argument is useless.)

There are many parodic discussions of Ontological Arguments in the literature. A particularly pretty one is due to Raymond Smullyan (1984), in which the argument is attributed to “the unknown Dutch theologian van Dollard”. A relatively recent addition to the genre is described in Grey 2000, though the date of its construction is uncertain. It is the work of Douglas Gasking, one-time Professor of Philosophy at the University of Melbourne (with emendations by William Grey and Denis Robinson):

• The creation of the world is the most marvellous achievement imaginable.
• The merit of an achievement is the product of (a) its intrinsic quality, and (b) the ability of its creator.
• The greater the disability or handicap of the creator, the more impressive the achievement.
• The most formidable handicap for a creator would be non-existence.
• Therefore, if we suppose that the universe is the product of an existent creator, we can conceive a greater being—namely, one who created everything while not existing.
• An existing God, therefore, would not be a being than which a greater cannot be conceived, because an even more formidable and incredible creator would be a God which did not exist.
• (Hence) God does not exist.

This parody—at least in its current state—seems inferior to other parodies in the literature, including the early parodies of Gaunilo and Caterus. To mention but one difficulty, while we might suppose that it would be a greater achievement to create something if one did not exist than if one did exist, it doesn’t follow from this that a non-existent creator is greater ( qua being) than an existent creator. Perhaps it might be replied that this objection fails to take the first premise into account: if the creation of the world really is “the most marvellous achievement imaginable”, then surely there is some plausibility to the claim that the creator must have been non-existent (since that would make the achievement more marvellous than it would otherwise have been). But what reason is there to believe that the creation of the world is “the most marvellous achievement imaginable”, in the sense which is required for this argument? Surely it is quite easy to imagine even more marvellous achievements—e.g., the creation of many worlds at least as good as this one! (Of course, one might also want to say that, in fact, one cannot conceive of a non-existent being’s actually creating something: that is literally inconceivable. Etc.)

Chambers 2000 and Siegwart 2014 provide nice, recent discussions of Gaunilo’s parody of the Proslogion II argument.

There is a small, but steadily growing, literature on the ontological arguments which Gödel developed in his notebooks, but which did not appear in print until well after his death. These arguments have been discussed, annotated and amended by various leading logicians: the upshot is a family of arguments with impeccable logical credentials. (Interested readers are referred to Sobel 1987, Anderson 1990, Adams 1995b, and Hazen 1999 for the history of these arguments, and for the scholarly annotations and emendations.) Here, I shall give a brief presentation of the version of the argument which is developed by Anderson, and then make some comments on that version. This discussion follows the presentation and discussion in Oppy 1996, 2000.

Definition 1: x is God-like if and only if x has as essential properties those and only those properties which are positive Definition 2: A is an essence of x if and only if for every property B , x has B necessarily if and only if A entails B Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified Axiom 1: If a property is positive, then its negation is not positive. Axiom 2: Any property entailed by—i.e., strictly implied by—a positive property is positive Axiom 3: The property of being God-like is positive Axiom 4: If a property is positive, then it is necessarily positive Axiom 5: Necessary existence is positive Axiom 6: For any property P , if P is positive, then being necessarily P is positive. Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified. Corollary 1: The property of being God-like is consistent. Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing. Theorem 3: Necessarily, the property of being God-like is exemplified.

Given a sufficiently generous conception of properties, and granted the acceptability of the underlying modal logic, the listed theorems do follow from the axioms. (This point was argued in detail by Dana Scott, in lecture notes which circulated for many years and which were transcribed in Sobel 1987 and published in Sobel 2004. It is also made by Sobel, Anderson, and Adams.) So, criticisms of the argument are bound to focus on the axioms, or on the other assumptions which are required in order to construct the proof.

Some philosophers have denied the acceptability of the underlying modal logic. And some philosophers have rejected generous conceptions of properties in favour of sparse conceptions according to which only some predicates express properties. But suppose that we adopt neither of these avenues of potential criticism of the proof. What else might we say against it?

One important point to note is that no definition of the notion of “positive property” is supplied with the proof. At most, the various axioms which involve this concept can be taken to provide a partial implicit definition. If we suppose that the “positive properties” form a set, then the axioms provide us with the following information about this set:

• If a property belongs to the set, then its negation does not belong to the set.
• The set is closed under entailment.
• The property of having as essential properties just those properties which are in the set is itself a member of the set.
• The set has exactly the same members in all possible worlds.
• The property of necessary existence is in the set.
• If a property is in the set, then the property of having that property necessarily is also in the set.

On Gödel’s theoretical assumptions, we can show that any set which conforms to (1)–(6) is such that the property of having as essential properties just those properties which are in that set is exemplified. Gödel wants us to conclude that there is just one intuitive, theologically interesting set of properties which is such that the property of having as essential properties just the properties in that set is exemplified. But, on the one hand, what reason do we have to think that there is any theologically interesting set of properties which conforms to the Gödelian specification? And, on the other hand, what reason do we have to deny that, if there is one set of theologically interesting set of properties which conforms to the Gödelian specification, then there are many theologically threatening sets of properties which also conform to that specification?

In particular, there is some reason to think that the Gödelian ontological argument goes through just as well—or just as badly—with respect to other sets of properties (and in ways which are damaging to the original argument). Suppose that there is some set of independent properties { I , G 1 , G 2 , …} which can be used to generate the set of positive properties by closure under entailment and “necessitation”. (“Independence” means: no one of the properties in the set is entailed by all the rest. “Necessitation” means: if P is in the set, then so is necessarily having P . I is the property of having as essential properties just those properties which are in the set. G 1 , G 2 , … are further properties, of which we require at least two.) Consider any proper subset of the set { G 1 , G 2 , …}—{ H 1 , H 2 , …}, say—and define a new generating set { I *, H 1 , H 2 , …}, where I * is the property of having as essential properties just those properties which are in the newly generated set. A “proof” parallel to that offered by Gödel “establishes” that there is a being which has as essential properties just those properties in this new set. If there are as few as 7 independent properties in the original generating set, then we shall be able to establish the existence of 720 distinct“God-like” creatures by the kind of argument which Gödel offers. (The creatures are distinct because each has a different set of essential properties.)

Even if the above considerations are sufficient to cast doubt on the credentials of Gödel’s “proof”, they do not pinpoint where the “proof” goes wrong. If we accept that the role of Axioms 1, 2, 4, and 6 is really just to constrain the notion of “positive property” in the right way—or, in other words, if we suppose that Axioms 1, 2, 4, and 6 are “analytic truths” about “positive properties”—then there is good reason for opponents of the “proof” to be sceptical about Axioms 3 and 5. Kant would not have been happy with Axiom 5; and there is at least some reason to think that whether the property of being God-like is “positive” ought to depend upon whether or not there is a God-like being.

The “victorious” modal ontological argument of Plantinga 1974 goes roughly as follows: Say that an entity possesses “maximal excellence” if and only if it is omnipotent, omnscient, and morally perfect. Say, further, that an entity possesses “maximal greatness” if and only if it possesses maximal excellence in every possible world—that is, if and only if it is necessarily existent and necessarily maximally excellent. Then consider the following argument:

• There is a possible world in which there is an entity which possesses maximal greatness.
• (Hence) There is an entity which possesses maximal greatness.

Under suitable assumptions about the nature of accessibility relations between possible worlds, this argument is valid: from it is possible that it is necessary that p , one can infer that it is necessary that p . Setting aside the possibility that one might challenge this widely accepted modal principle, it seems that opponents of the argument are bound to challenge the acceptability of the premise.

And, of course, they do. Let’s just run the argument in reverse.

• There is no entity which possesses maximal greatness.
• (Hence) There is no possible world in which there is an entity which possesses maximal greatness.

Plainly enough, if you do not already accept the claim that there is an entity which possesses maximal greatness, then you won’t agree that the first of these arguments is more acceptable than the second. So, as a proof of the existence of a being which posseses maximal greatness, Plantinga’s argument seems to be a non-starter.

Perhaps somewhat surprisingly, Plantinga himself agrees: the “victorious” modal ontological argument is not a proof of the existence of a being which possesses maximal greatness. But how, then, is it “victorious”? Plantinga writes: “Our verdict on these reformulated versions of St. Anselm’s argument must be as follows. They cannot, perhaps, be said to prove or establish their conclusion. But since it is rational to accept their central premise, they do show that it is rational to accept that conclusion” (Plantinga 1974, 221).

It is pretty clear that Plantinga’s argument does not show what he claims that it shows. Consider, again, the argument: “Either God exists, or 2+2=5. It is not the case that 2+2=5. So God exists.” It is just a mistake for a theist to say: “Since the premise is true (and the argument is valid), this argument shows that the conclusion of the argument is true ”. No-one thinks that that argument shows any such thing. Similarly, it is just a mistake for a theist to say: “Since it is rational to accept the premise (and the argument is valid), this argument shows that it is rational to accept the conclusion of the argument”. Again, no one thinks that that argument shows any such thing. But why don’t these arguments show the things in question? There is room for argument about this. But it is at least plausible to claim that, in each case, any even minimally rational person who has doubts about the claimed status of the conclusion of the argument will have exactly the same doubts about the claimed status of the premise. If, for example, I doubt that it is rational to accept the claim that God exists, then you can be quite sure that I will doubt that it is rational to accept the claim that either 2+2=5 or God exists. But, of course, the very same point can be made about Plantinga’s argument: anyone with even minimal rationality who understands the premise and the conclusion of the argument, and who has doubts about the claim that it is rationally permissible to believe that there is an entity which possesses maximal greatness, will have exactly the same doubts about the claim that it is rationally permissible to believe that there is a possible world in which there is an entity which possesses maximal greatness.

For further discussion of Plantinga’s argument, see—for example—Adams 1988, Chandler 1993, Oppy 1995 (70–78, 248–259), Tooley 1981, and van Inwagen 1977).

There is an enormous literature on the material in Proslogion II-III . Some commentators deny that St. Anselm tried to put forward any proofs of the existence of God. Even among commentators who agree that St. Anselm intended to prove the existence of God, there is disagreement about where the proof is located. Some commentators claim that the main proof is in Proslogion II , and that the rest of the work draws out corollaries of that proof (see, e.g., Charlesworth 1965). Other commentators claim that the main proof is in Prologion III , and that the proof in Proslogion II is merely an inferior first attempt (see, e.g., Malcolm 1960). Yet other commentators claim that there is a single proof which spans at least Proslogion II-III —see, e.g., Campbell 1976 and, perhaps, the entire work—see, e.g., La Croix 1972. I shall ignore this aspect of the controversy about the Proslogion . Instead, I shall just focus on the question of the analysis of the material in Proslogion II on the assumption that there is an independent argument for the existence of God which is given therein.

Here is one translation of the crucial part of Proslogion II (due to William Mann (1972, 260–1); alternative translations can be found in Barnes 1972, Campbell 1976, Charlesworth 1965, and elsewhere):

Thus even the fool is convinced that something than which nothing greater can be conceived is in the understanding, since when he hears this, he understands it; and whatever is understood is in the understanding. And certainly that than which a greater cannot be conceived cannot be in the understanding alone. For if it is even in the understanding alone, it can be conceived to exist in reality also, which is greater. Thus if that than which a greater cannot be conceived is in the understanding alone, then that than which a greater cannot be conceived is itself that than which a greater can be conceived. But surely this cannot be. Thus without doubt something than which a greater cannot be conceived exists, both in the understanding and in reality.

There have been many ingenious attempts to find an argument which can be expressed in modern logical formalism, which is logically valid, and which might plausibly be claimed to be the argument which is expressed in this passage. To take a few prime examples, Adams 1971, Barnes 1972 and Oppenheimer and Zalta 1991 have all produced formally valid analyses of the argument in this passage. We begin with a brief presentation of each of these analyses, preceded by a presentation of the formulation of the argument given by Plantinga 1967, and including a presentation of some of the formulations of Lewis 1970. (Chambers 2000 works with the analysis of Adams 1971.)

8.1 Formulation 1

God exists in the understanding but not in reality. (Assumption for reductio )

Existence in reality is greater than existence in the understanding alone. (Premise)

A being having all of God’s properties plus existence in reality can be conceived. (Premise)

A being having all of God’s properties plus existence in reality is greater than God. (From (1) and (2).)

A being greater than God can be conceived. (From (3) and (4).)

It is false that a being greater than God can be conceived. (From definition of “God”.)

Hence, it is false that God exists in the understanding but not in reality. (From (1), (5), (6).)

God exists in the understanding. (Premise, to which even the Fool agrees.)

Hence God exists in reality. (From (7), (8).)

See Plantinga 1967.

8.2 Formulation 2

The Fool understands the expression “the being than which no greater can be conceived”. (Premise)

If a person understands an expression “ b ”, then b is in that person’s understanding. (Premise)

If a thing is in a person’s understanding, then the person can conceive of that thing’s existing in reality. (Premise)

Each thing which exists in reality is greater than any thing which exists only in the understanding. (Premise)

If a person can conceive of something, and that thing entails something else, then the person can also conceive of that other thing. (Premise)

If a person can conceive that a specified object has a given property, then that person can conceive that something or other has that property. (Premise)

Hence the being than which no greater can be conceived exists in reality. (From (1)-(6), by a complex series of steps here omitted.)

See Barnes 1972.

8.3 Formulation 3

There is a thing x , and a magnitude m , such that x exists in the understanding, m is the magnitude of x , and it it not possible that there is a thing y and a magnitude n such that n is the magnitude of y and n > m . (Premise)

For any thing x and magnitude m , if x exists in the understanding, m is the magnitude of x , and it is not possible that there is a thing y and magnitude n such that n is the magnitude of y and n > m , then it is possible that x exists in reality. (Premise)

For any thing x and magnitude m , if m is the magnitude of x , and it it not possible that there is a thing y and a magnitude n such that n is the magnitude of y and n > m , and x does not exist in reality, then it is not possible that if x exists in reality then there is a magnitude n such that n is greater than m and n is the magnitude of x . (Premise)

(Hence) There is a thing x and a magnitude m such that x exist in the understanding, and x exists in reality, and m is the magnitude of x , and it it not possible that there is a thing y and a magnitude n such that n is the magnitude of y and n > m . (From 1, 2, 3)

See Adams 1971.

8.4 Formulation 4

For any understandable being x , there is a world w such that x exists in w . (Premise)

For any understandable being x , and for any worlds w and v , if x exists in w , but x does not exist in v , then the greatness of x in w exceeds the greatness of x in v . (Premise)

There is an understandable being x such that for no world w and being y does the greatness of y in w exceed the greatness of x in the actual world. (Premise)

(Hence) There is a being x existing in the actual world such that for no world w and being y does the greatness of y in w exceed the greatness of x in the actual world. (From (1)-(3).)

See Lewis 1970.

Lewis also suggests an alternative to (3) which yields a valid argument:

(3′) There is an understandable being x such that for no worlds v and w and being y does the greatness of y in w exceed the greatness of x in v .

and two alternatives to (3)—not presented here—which yield invalid arguments. (Of course, there further two alternatives are crucial to Lewis’ overall analysis of the passage: essentially, Lewis suggests that Anselm equivocates between an invalid argument with plausible premises and a valid argument with question-begging premises. In this respect, Lewis’ analysis is quite different from the other analyses currently under discussion.)

8.5 Formulation 5

There is (in the understanding) something than which there is no greater. (Premise)

(Hence) There is (in the understanding) a unique thing than which there is no greater. (From (1), assuming that the “greater-than” relation is connected.)

(Hence) There is (in the understanding) something which is the thing than which there is no greater. (From (2), by a theorem about descriptions.)

(Hence) There is (in the understanding) nothing which is greater than the thing than which there is no greater. (From (3), by another theorem about descriptions.)

If that thing than which there is no greater does not exist (in reality), then there is (in the understanding) something which is greater than that thing than which there is no greater. (Premise)

(Hence) That thing than which there is no greater exists (in reality). (From (4) and (5).)

(Hence) God exists. (From (6).)

See Oppenheimer and Zalta 1991.

Oppenheimer and Zalta 2011 provides a “simplified” version of this argument, in which the number of controversial assumptions is reduced. Since they also provide a clear reason for thinking that this new version of the argument is not persuasive, I shall not consider it further here.

8.6 Critical Appraisal

Considered as interpretations of the argument presented in the Proslogion , these formulations are subject to various kinds of criticisms.

First , the modal interpretations of Lewis 1970 and Adams 1971 don’t square very well with the rest of the Proslogion : the claim that “being than which no greater can be conceived” should be read as “being than which no greater is possible” would have us render the claim of Proslogion 15 to be that God is a being greater than any which is possible. And that is surely a bad result.

Second , the Meinongian interpretations of Barnes 1972, Adams 1971 and Oppenheimer and Zalta 1991 produce arguments which, given the principles involved, could easily be much simplified, and which are obviously vulnerable to Gaunilo-type objections.

Consider, for example, the case of Oppenheimer and Zalta. They have Anselm committed to the claim that if anyone can understand the phrase “that than which F ”, then there is something in the understanding such that F (see their footnote 25); and they also have him committed to the claim that if there is something which is the F -thing, then it—i.e., the F -thing—has the property F (see page 7). Plainly though, if Anselm is really committed to these principles, then he could hardly fail to be committed to the more general principles: (1) if anyone can understand the phrase “an F ”, then there is at least one F -thing in the understanding; and (2) if there are some things which are the F -things, then they—i.e., the F -things—must have the property F . (It would surely be absurd to claim that Anselm is only committed to the less general principles: what could possibly have justified the restrictions to the special cases?)

But, then, mark the consequences. We all understand the expression “an existent perfect being”. So, by the first claim, there is at least one existent perfect being in the understanding. And, by the second claim, any existent perfect being is existent. So, from these two claims combined, there is—in reality—at least one existent perfect being.

This argument gives Anselm everything that he wants, and very much more briefly. (The Proslogion goes on and on, trying to establish the properties of that than which no greater can be conceived. How much easier if we can just explicitly build all of the properties which want to “derive” into the initial description.) So, if Anselm really were committed to the principles which Oppenheimer and Zalta appear to attribute to him, it is hard to understand why he didn’t give the simpler argument. And, of course, it is also hard to understand why he didn’t take Gaunilo’s criticism. After all, when it is set out in this way, it is obvious that the argument proves far too much.

Third , some of the arguments have Anselm committed to claims about greatness which do not seem to correspond with what he actually says. The natural reading of the text is that, if two beings are identical save that one exists only in the understanding and the other exists in reality as well, then the latter is greater than the former. But Barnes 1971, for example, has Anselm committed to the much stronger claim that any existing thing is greater than every non-existent thing.

Given these kinds of considerations, it is natural to wonder whether there are better interpretations of Proslogion II according to which the argument in question turns out NOT to be logically valid. Here is a modest attempt to provide such an analysis:

We start with the claim that the Fool understands the expression “being than which no greater can be conceived”, i.e., even the Fool can entertain the idea or possess the concept of a being than which no greater can be conceived. Now, entertaining this idea or possessing this concept requires the entertainer or possessor to recognise certain relationships which hold between given properties and the idea or concept in question. For example, given that you possess the concept of, or entertain the idea of, a smallest really existent Martian, it follows that you must recognise some kind of connection between the properties of being a Martian, really existing, and being smaller than other really existing Martians, and the concept or idea in question.

Following Anselm, we might say that, since you understand the expression “smallest really existent Martian”, there is, in your understanding, at least one smallest really existent Martian. (Or, apparently following Descartes, one might say that real existence is “part of”—or “contained in”—the idea of a smallest really existent Martian.) However, in saying this, it must be understood that we are not actually predicating properties of anything: we aren’t supposing that there is something which possesses the properties of being a Martian, really existing, and being no larger than any other Martian. (After all, we can safely suppose, we don’t think that any Martians really exist.) In other words, we must be able to have the concept of, or entertain the idea of, a smallest really existing Martian without believing that there really are any smallest Martians. Indeed, more strongly, we must be able to entertain the concept of a smallest really existent Martian—and to recognise that the property of “really existing” is part of this concept—while nonetheless maintaining that there are no smallest existent Martians.

It will be useful to introduce vocabulary to mark the point which is being made here. We could, for instance, distinguish between the properties which are encoded in an idea or concept, and the properties which are attributed in positive atomic beliefs which have that idea or concept as an ingredient. The idea “really existent Santa Claus” encodes the property of real existence; but it is perfectly possible to entertain this idea without attributing real existence to Santa Claus, i.e., without believing that Santa Claus really exists.

We can then apply this distinction to Anselm’s argument. On the one hand, the idea “being than which no greater can be conceived” encodes the property of real existence—this is what the reductio argument establishes (if it establishes anything at all). On the other hand, it is perfectly possible to entertain the idea of a being than which no greater can be conceived—and to recognise that this idea encodes the property of real existence—without attributing real existence to a being than which no greater can be conceived, i.e., without believing that a being than which no greater can be conceived really exists.

Of course, the argument which Anselm actually presents pays no attention to this distinction between encoding and attributing—i.e., between entertaining an idea and holding a belief—and nor does it pay attention to various other niceties. We begin from the point that the Fool entertains the idea of that than which no greater can be conceived (because the Fool understands the words “that than which no greater can be conceived”). From this, we move quickly to the claim that even the Fool is “convinced”—i.e., believes—that that than which no greater can be conceived possesses the property of existing in the understanding. And then the reductio argument is produced to establish that that than which no greater can be conceived cannot exist only in the understanding but must also possess the property of existing in reality as well (and all mention of the Fool, and what it is that the Fool believes, disappears).

As it stands, this is deeply problematic. How are we supposed to regiment the references to the Fool in the argument? Is the reductio argument supposed to tell us something about what even the Fool believes, or ought to believe? Are the earlier references to the Fool supposed to be inessential and eliminable? How are we so much as to understand the claim that even the Fool believes that that than which no greater can be conceived exists in the understanding? And how do we get from the Fool’s understanding the words “that than which no greater can be conceived” to his believing that that than which no greater can be conceived possesses the property of existing in the understanding?

Following the earlier line of thought, it seems that the argument might go something like this:

(Even) the Fool has the concept of that than which no greater can be conceived.

(Hence) (Even) the Fool believes that that than which no greater can be conceived exists in the understanding.

No one who believes that that than which no greater can be conceived exists in the understanding can reasonably believe that that than which no greater can be conceived exists only in the understanding.

(Hence) (Even) the Fool cannot reasonably deny that that than which no greater can be conceived exists in reality

(Hence) That than which no greater can be conceived exists in reality.

While this is not a good argument, it could appear compelling to one who failed to attend to the distinction between entertaining ideas and holding beliefs and who was a bit hazy on the distinction between the vehicles of belief and their contents. When the Fool entertains the concept of that than which no greater can be conceived he recognises that he is entertaining this concept (i.e., he believes that he is entertaining the concept of that than which no greater can be conceived—or, as we might say, that the concept is in his understanding). Conflating the concept with its object, this gives us the belief that than which no greater can be conceived possesses the property of existing in the understanding. Now, suppose as hypothesis for reductio , that we can reasonably believe that that than which no greater can be conceived possesses the property of existing only in the understanding. Ignoring the distinction between entertaining ideas and holding beliefs, this means that we when we entertain the idea of that than which no greater can be conceived, we entertain the idea of a being which exists only in the understanding. But that is absurd: when we entertain the idea of that than which no greater can be conceived, our idea encodes the property of existing in reality. So there is a contradiction, and we can conclude that, in order to be reasonable, we must believe that that than which no greater can be conceived exists in reality. But if any reasonable person must believe that that than which no greater can be conceived exists in reality, then surely it is the case that that than which no greater can be conceived exists in reality. And so we are done.

No doubt this suggestion about the interpretation of Anselm’s argument is deficient in various ways. However, the point of including it is illustrative rather than dogmatic. In the literature, there has been great resistance to the idea that the argument which Anselm gives is one which modern logicians would not hesitate to pronounce invalid. But it is very hard to see why there should be this resistance. (Certainly, it is not something for which there is much argument in the literature.) The text of the Proslogion is so rough, and so much in need of polishing, that we should not be too quick to dismiss the suggestion that Anselm’s argument is rather more like the argument most recently sketched than it is like the logically valid demonstrations provided by commentators such as Barnes, Adams, and Oppenheimer and Zalta. (For a more complex analysis of Proslogion II that has it yielding a valid argument, see Hinst 2014.)

Many recent discussions of ontological arguments are in compendiums, companions, encylopedias, and the like. So, for example, there are review discussions of ontological arguments in: Leftow 2005, Matthews 2005, Lowe 2007, Oppy 2007, and Maydole 2009. While the ambitions of these review discussions vary, many of them are designed to introduce neophytes to the arguments and their history. Given the current explosion of enthusiasm for compendiums, companions, encylopedias, and the like, in philosophy of religion, it is likely that many more such discussions will appear in the immediate future.

Some recent discussions of ontological arguments have been placed in more synoptic treatments of arguments about the existence of God. So, for example, there are extended discussions of ontological arguments in Everitt 2004, Sobel 2004, and Oppy 2006. Sobel’s examination of ontological arguments is exemplary. He provides one chapter on ‘classical ontological arguments’: Anselm, Descartes, Spinoza, and Kant’s critique of ontological arguments; one chapter on ‘modern modal ontological arguments’: Hartshorne, Malcolm and Plantinga; and one chapter on Gödel’s ontological argument. His analyses are very careful, and make heavy use of the tools of modern philosophical logic.

There has been one recent monograph devoted exclusively to the analysis of ontological arguments: Dombrowski 2006. Dombrowski is a fan of Hartshorne: the aim of his book is to defend the claim that Hartshorne’s ontological argument is a success. While Dombrowski’s book is a useful addition to the literature because of the scope of its discussion of ontological arguments—for example, it contains a chapter on Rorty on ontological arguments, and another chapter on John Taylor on ontological arguments—even reviewers sympathetic to process theism have not been persuaded that it makes a strong case for its central thesis.

Swatkowski (2012) is the most recent collection of papers on ontological arguments. A significant proportion of papers in this collection take up technical questions about logics that support ontological derivations. (Those interested in technical questions may also be interested in the topic taken up in Oppenheimer and Zalta (2011) and Gorbacz (2012).)

Finally, there has been some activity in journals. The most significant of these pieces is Millican 2004, the first article on ontological arguments in recent memory to appear in Mind . Millican argues for a novel interpretation of Anselm’s argument, and for a new critique of ontological arguments deriving from this interpretation. Needless to say, both the interpretation and the critique are controversial, but they are also worthy of attention. Among other journal articles, perhaps the most interesting are Pruss 2010, which provides a novel defence of the key possibility premise in modal ontological arguments, and Pruss 2009, which kick-started recent discussion of higher-order ontological arguments. There is also a chain of papers in Analysis initiated by Matthews and Baker (2010)

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How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up this entry topic at the Indiana Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Kurt Gödel’s Ontological Argument (Christopher Small, University of Waterloo)
• Medieval Sourcebook: Philosophers’ Criticisms of Anslem’s Ontological Argument for the Being of God (Paul Halsell, Fordham University)
• Handout for a Talk on the Ontological Argument (J. R. Lucas, Oxford University)
• Ontological Argument Revisited by Two Ottoman Muslim Scholars (Umit Dericioglu)
• The Ontological Argument (Kenneth Himma, University of Washington)
• Anselm’s Ontological Argument (Gideon Rosen, Princeton University)
• Hegel and Kant on the Ontological Argument (Maria de Lourdes Borges, Federal University of Santa Catarina)
• Ontological Argument (links to papers on ontological arguments)
• “ Formalization, Mechanization and Automation of Gödel’s Proof of God’s Existence , unpublished manuscript.
• “ Automating Gödel’s Ontological Proof of God’s Existence with Higher-order Automated Theorem Provers , published in ECAI 2014, T. Schaub et al . (eds.), IOS Press.

Anselm, Saint [Anselm of Bec, Anselm of Canterbury] | a priori justification and knowledge | Descartes, René | existence | God: concepts of | Gödel, Kurt | Hegel, Georg Wilhelm Friedrich | Kant, Immanuel | logic: informal | logic: modal | Meinong, Alexius

Copyright © 2016 by Graham Oppy < Graham . Oppy @ monash . edu >

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