Among philosophers who have claimed to prove that God exists, a small minority has claimed to prove that God exists without appealing to anything more than an understanding of what God would be were God to exist.
The first such minority philosopher is St. Anselm. In 1078, in his Proslogion, he argued as follows:
“We believe that You are that than which no greater can be conceived. Or can it be that a thing of such a nature does not exist, since ‘the Fool has said in his heart, there is no God’? But surely, when this same Fool hears what I am speaking about, namely ‘that than which no greater can be conceived’, he understands what he hears, and what he understands is in his understanding. … Even the Fool, then, is forced to agree that that than which no greater can be conceived exists in the understanding. … And surely that than which no greater can be conceived cannot exist in the understanding alone. For, if it exists solely in the understanding, it can be conceived to exist in reality also, which is greater. If, then, that than which no greater can be conceived exists in the understanding alone, this same that than which no greater can be conceived is that than which a greater can be conceived. But this is obviously impossible. Therefore, there is absolutely no doubt that that than which no greater can be conceived exists both in the understanding and in reality.”
In 1641, in his Meditations on First Philosophy, Descartes argued in a similar fashion, as follows:
“I clearly see that existence can no more be separated from the essence of God than can its having three angles equal to two right angles be separated from the idea of a rectilinear triangle, or the idea of a mountain from the idea of a valley; and so there is no less repugnance to our conceiving a God (that is, a Being supremely perfect) to whom existence is lacking (that is to say, to whom a certain perfection is lacking) than to conceive of a mountain which has no valley. … From the fact that I cannot conceive God without existence it follows that existence is inseparable from Him, and hence that He really exists.”
Why do most philosophers find these two arguments uncompelling? I think it is because most philosophers think that, were these arguments compelling, we would have equally compelling arguments for the existence of things that we know do not exist. In 1079, Gaunilo, a monk at the same monastery as St. Anselm, claimed that, following St. Anselm’s lead, you can construct an equally compelling argument for the existence of that island than which no greater island can be conceived. And, in 1641, Johan de Kater claimed that, following Descartes’ lead, you can construct an equally compelling argument, from the mere idea of an existent lion, to the existence of at least one lion. Since the details of the objections from Gaunilo and de Kater are controversial, I shall rework their objections using a different example.
Suppose that I have just completed a test on which the maximum conceivable score — and the maximum possible score — is 100%.
Consider, first, that score by me on the test than which no greater score by me on the test is conceivable. When you hear the words “that score by me on the test than which no greater score by me on the test is conceivable”, you understand them. So, according to Anselm, that score by me on the test than which no greater score by me on the test is conceivable exists in the understanding. But, by Anselm’s lights, that score by me on the test than which no greater score by me on the test is conceivable cannot exist in the understanding alone. For, if it exists solely in the understanding, it can be conceived to exist in reality also, which is greater. If, then, that score by me on the test than which no greater score by me on the test is conceivable exists in the understanding alone, this same that score by me on the test than which no greater score by me on the test is conceivable is that score by me on the test than which a greater score by me on the test is conceivable. But this is obviously impossible. Therefore, by Anselm’s lights, there is absolutely no doubt that that score by me on the test than which no greater score by me on the test is conceivable exists both in the understanding and in reality. Which is to say that, by Anselm’s lights, there is no doubt that I got 100% on the test (after all, that’s just what it is for that score by me on the test than which no greater score by me on the test is conceivable to exist in reality). But that is absurd; it is perfectly consistent with everything I told you that I crashed and burned on the test.
Consider, second, a supremely perfect score for me on the test. Such a score has two dimensions: first, it is a maximal score (100%); and, second, it really exists (for example, it is not just a figment of my imagination that I got 100% on the test). By Descartes’ lights, it seems, we can see that existence can no more be separated from the essence of a supremely perfect score for me on the test than the idea of a mountain can be separated from the idea of a valley: there is no less repugnance to our conceiving a supremely perfect score for me on the test to which existence is lacking than to conceive a mountain which has no valley. By Descartes’ lights, it seems, from the fact that I cannot conceive a supremely perfect score for me on the test which lacks existence, it follows that existence is inseparable from it, and hence it really exists. That is, by Descartes’ lights, I got 100% on the test. But, as we noted above, that is absurd: it is perfectly possible that I bombed on the test.
The objection that I have just set out is subtle and often misunderstood. In particular, it is important to understand what is not being claimed in the setting out of this objection.
It is certainly not being claimed that we should accept the conclusion of the arguments for the existence of that score by me on the test than which no greater score by me on the test is conceivable and the existence of a supremely perfect score for me on the test. We can stipulate, if we like, that I got 32% on the test. Given that stipulation, the conclusion of the parallel arguments is false. In particular, then, these arguments cannot be proofs of their conclusions. And, by parity of argumentation, the original arguments due to St. Anselm and Descartes cannot be proofs of their conclusions.
It is also certainly not being claimed that there is reason here to suppose that God does not exist. Had I scored 100% on the test, then there would have existed, in reality, that score by me on the test than which no greater score by me on the test is conceivable, and there would have existed, in reality, a supremely perfect score by me on the test. But my getting 100% on the test would not have made the parallel arguments into good arguments. The important point here is that the arguments are not compelling; nothing at all is being said about whether the conclusions of those arguments are true. Rejection of the claim that these ontological arguments are proofs of the existence of God is completely independent of the stance that one takes on the question whether God exists.
There is a common reply to the objection that I have set out that is worth discussing. Perhaps Anselm could say that whether the expression “that score by me on the test than which no greater score by me on the test is conceivable” is understood depends upon whether that score by me on the test than which no greater score by me on the test is conceivable exists in reality. Perhaps Descartes can say that whether we have the idea of a supremely perfect score by me on the test depends upon whether there is, in reality, a perfect score by me on the test. If they could say these things, then Anselm could deny that there is, in the understanding, a score by me on the test than which no greater score by me on the test can be conceived, and Descartes could deny that we have the idea of a supremely perfect score by me on the test.
Alas, this won’t do. After all, if Anselm and Descartes were entitled to go this way, then those who deny that there is in reality that than which no greater can be conceived would be entitled to deny that that than which no greater can be conceived exists in the understanding, and those who deny that there is in reality a supremely perfect being would be entitled to deny that they have the idea of a supremely perfect being. It is not as if the words “that than which no greater can be conceived” and “supremely perfect being” have magic properties that the words “that score by me on the test than which no greater score by me on the test can be conceived” and “supremely perfect score by me on the test” do not have.
Contemporary proponents of ontological arguments typically do not defend either Anselm’s Proslogion II argument or Descartes’ Meditation V argument. Instead, they typically defend a modal ontological argument or a higher order ontological argument.
Modal ontological arguments depend upon widely — but not universally — accepted claims about possibility, contingency, necessity, and actuality.
Among actually true claims, we can distinguish between contingently true claims — that happen to be true but could have been false, for example, that I had Weetabix for breakfast this morning — and necessary true claims — that could not have been false, for example, that 2+2=4. And, among actually false claims, we can distinguish between contingently false claims — that happen to be false but could have been true, for example, that I had porridge for breakfast this morning — and necessarily false claims — that could not possibly be true, for example, that 2+2=5. Every claim has one of the just identified modal statuses: some claims are necessarily true; some claims are necessarily false; and some claims are both possibly true and possibly false. Claims that are necessarily true are also both possibly true and actually true; and claims that are necessarily false are both possibly false and actually false. For example, since it is necessarily true that 2+2=4, it is also possibly true that 2+2=4 and actually true that 2+2=4. It is worth noting that necessity and possibility are interdefinable: a claim is necessary just in case it is not possible that it is false; and a claim is possible just in case it is not necessary that it is false.
It is widely — but not universally — accepted by philosophers that modal claims, that is, claims of the form it is necessary that such and such and it is possible that so and so, are themselves either necessarily true or necessarily false. So, for example, it is necessary that it is necessary that 2+2=4, and necessary that it is possible that I had porridge for breakfast. Further, the claims that we have tabled to this point entail that, if it is possible that it is necessary that such and such, then it is necessary that such and such. So this, too, is widely — but not universally — accepted by philosophers.
Given the above claims about necessity, contingency, possibility and actuality, we can construct an argument for the existence of God. The core idea behind the argument is that God’s existence cannot be a contingent matter: necessarily, if God exists, then it is necessary that God exists. But, given that God’s existence cannot be a contingent matter, if it is possible that God exists, then it follows that it is necessary that God exists, and then, of course, it follows that God actually exists.
Can we reasonably suppose that this is a proof of the existence of God? No, we cannot. Given that God’s existence cannot be a contingent matter, if it is possible that God does not exist, then it follows that it is necessary that God does not exist, and then, of course, it follows that God does not actually exist. Given that God’s existence cannot be a contingent matter, there are two available positions: on the one side, there is the view that says that God does not and cannot exist; and, on the other side, there is the view that says that God does and must exist. It would be an obvious error to suppose that one of the two arguments constitutes a proof of one of these two positions: this is a symmetry that cannot be broken by mere fiat. Although there is not universal agreement here, most philosophers accept that you cannot use modal ontological arguments to establish anything interesting about the existence of God.
Higher order ontological arguments require more complex logical machinery than other ontological arguments. While the leading ideas behind higher order ontological arguments go back to Leibniz, the first careful formulation of a higher order ontological argument is due to Kurt Gödel, arguably the greatest logician who ever lived. Gödel formulated his argument in about 1941; “unofficial” versions began to circulate in the 1970s. The past thirty years has seen intensive study of higher order ontological arguments. I shall do my best to give an intelligible presentation of a simple higher order ontological argument.
The most fundamental notion in higher order ontological arguments is the notion of a property. Things have properties. There is a cup on my table. It has the property of being blue, or, equivalently, the property of blueness. When I say that my cup is blue, I can be taken to be attributing the property of being blue, or, equivalently, the property of blueness, to my cup. Philosophers differ in their attitudes towards talk of properties: some deny that we should accept that there are properties; others do not. Presentations of higher order ontological arguments take it for granted that there are properties.
Given that there are properties, there are different kinds of properties and different ways in which things can possess properties. In particular, for any given thing, we can distinguish between its essential properties — the properties it cannot fail to have if it exists — and its accidental properties — the properties that it could have failed to have while yet existing. While, if my life had gone differently, I might not have had the property of being a philosopher — so that being a philosopher is one of my accidental properties — no matter how my life went, I would have been a human being, an occupant of spacetime, and larger than an electron — whence these are all among my essential properties.
Our simple higher order ontological argument begins with the idea of a positive property. Intuitively, the positive properties are God’s essential properties — the properties that God must have if God exists. Candidate examples of positive properties include: omnipotence, omniscience, perfect goodness, and necessary existence. We make two assumptions about positive properties:
1. If a property A is positive, then the negation of that property is not positive.
2. If a property A is positive, and property A entails property B, then property B is positive.
The negation of a property A is the property ~A had by exactly everything that does not have the property A. For example, the negation of the property of being blue is the property of not being blue. A property A entails a property B just in case, necessarily, everything that has property A has property B. So, for example, the property of being blue entails the property of being coloured.
We make one further definition: a thing is God like just in case the essential properties of that thing are exactly the positive properties. And then we introduce two new assumptions:
1. The property of being God like is positive
2. The property of existing necessarily is positive.
The key to our simple higher order ontological argument is the following claim: given 1 and 2, any pair of positive properties is such that it is possible for something to have them both. So, for example, given that being God like and existing necessarily are positive, it is possible for there to be something that is both God like and necessarily existent. We argue for this key claim as follows.
Suppose that A and B are positive. Suppose, further, that it is not possible for there to be something that is both A and B. Then, necessarily, everything that has A does not have B. So A entails ~B. But, by 2, this means that ~B is positive. And that contradicts 1, given that B is positive. So it is not the case that it is not possible for there to be something that is both A and B; that is, it is possible for there to be something that is both A and B.
The rest of the argument is straightforward, and follows the path of the modal ontological argument that we discussed earlier. Suppose that it is possible for there to be something that is God like and necessarily existent. Since it is possible that it is necessarily existent, it follows that it actually exists. And, since it is God like, it has all of the positive properties essentially, which means that it actually has these properties. So there is something that is necessarily existent, essentially omnipotent, essentially omniscient, essentially perfectly good, etc. In other words: God exists.
Is this higher order ontological argument a successful proof of the existence of God? No. To see why not, we need to note the following fact: properties that are not possibly instantiated entail all properties. If nothing can have property A, then, for every property B, A entails B.
Consider a competing view, according to which it is necessary that no God like thing exists: necessarily, nothing is omnipotent; necessarily, nothing is omniscient; necessarily, nothing is perfectly good; etc. Focus on omnipotence. If it is necessary that nothing is omnipotent, then, for any property at all, it is necessary that anything that is omnipotent has that property. So, in particular, it is necessary that anything that is omnipotent is not omnipotent. So being omnipotent entails not being omnipotent. So it cannot be that the property of being omnipotent is positive. Whether the property of being omnipotent is positive stands or falls with whether it is possibly instantiated. So we cannot use the assumption that it is positive to prove that it is possibly instantiated. And likewise for the assumptions that God likeness and necessary existence are positive.
