Kurt Gödel's Ontological Argument - Part II

Anselm's ontological argument

It is a common conceit of our secular technological age that all medieval clerics were close-minded. Our picture of medieval monastic life reinforces that idea. We imagine some monk sitting in a darkened monastery endlessly reciting the aged dogmas of an ancient religion, and trapped in a social order that permitted no mobility. In truth, however, universities were started in Europe during the medieval period. Many of the principles of freedom of thought and speech were developed from the desire of early universities to maintain some independence from the authoritarian powers of the time.

Whatever validity there is to our picture of the Middle Ages, derived in part from movies, television and medieval romance literature, it is a vast oversimplification of the reality. The Middle Ages spanned a large stretch of time. To lump all these societies together would be comparable to confusing Jane Austen's England with Bill Gates' America. Moreover, many of the ideas we have about ecclesiastical institutions derive not from the Middle Ages, but from the dynamic changes of the Reformation and Counter-Reformation. Certainly there was not the clear line between the sacred and the secular as exists today. But it was not so much that ordinary life was more sacred. It was that ecclesiastical institutions were more secular. Violence was endemic to the society of the central Middle Ages, with much of the power in the hands of people who were actually local warlords. The justification for violence was not arbitrary, but was determined by the authority of the person who committed it. During Anselm's time the great cathedrals of Europe had yet to be built. But the need for fortified wooden structures was great, as banditry of one regional warlord against another (and his serfs) was commonplace.

In education, there was also no division of church and state as exists today. For a person of intelligence to receive a good education, it was necessary to be within the ecclesiastical structures and not an outsider. Thus the dividing lines between philosophers, scientists and theologians was not as great as in modern times. Opportunities were greater by far for men. However, anyone who has read the biography (Historia Calamitatum) of Peter Abelard (1101-1164), including the story of his tumultuous love affair with Heloise, knows that the love of learning by both genders is not solely a modern concept. (The reader of the Historia Calamitatum should not confuse the Anselm of that story with the Anselm who developed the ontological argument.) We know from her letters that Heloise was well trained in philosophy and logic: her letter show a keen awareness of the philosophical issues of the time. Abelard, on the other hand, was a strong believer in the principle that the true foundation for religion was not unthinking dogma, but vigorous scepticism. Only a faith founded on the possibility of doubt was worthy of respect in his eyes.

What can we say about Hildegard of Bingen (1098-1179)? In modern terms, she might be described as a polymath, i.e., an individual of encyclopaedic learning and talent, who was a major contributer to three spheres of medieval culture: art, music and religious writing.

Anselm of Aosta (1033-1109) should also be understood in this context. Like Abelard or Hildegard, Anselm was a brilliant man whose natural vocation was within the medieval Church. His biographers have referred to the transformation that he introduced as the "Anselmian revolution." While he is best known outside ecclesiastical circles for his ontological argument, his influence on religious writing is every bit as important. Anselm invented a new form of poetry -- the poetry of personal devotion written in rhymed Latin with intricate logical structure and antitheses.

Anselm was born in Aosta, in the kingdom of Burgundy. He entered the abbey of Notre Dame at Bec, and, at the age of thirty, replaced Lanfranc as prior. In 1078 he became abbot when Herluin died. In 1093 he became Archbishop of Canterbury, a position he held until his death. However, it is during his time at Bec that he wrote much of the material for which he is remembered, including the Prayers and Meditations and the Proslogion. It is the Proslogion which contains the basic argument for the existence of God which we know today as the ontological argument.

Calling Anselm's approach to the existence of God the ontological argument is probably better than calling it the ontological proof. In order for an argument to be a proof it must proceed by a sequence of statements each of which can be deduced from previous statements or can be regarded as axiomatic in nature. Mathematical proofs, for example, are usually highly convincing, and leave little room for doubt provided that they are not so complex as to overwhelm the intellect. Anselm's ontological argument, while very subtle, is not complex in the sense that, say, Wiles' proof of Fermat's Last Theorem is complex. Nevertheless, very few people have been converted from atheism or agnosticism to the theist position by the argument. The reason for this is that most people who start out as sceptics remain so because they have an objection to an assumption of the ontological argument or to one of its rules of deduction. Nor is it only sceptics who have objections. Many religious thinkers have found the basis for their spiritual lives in personal experience rather than dry logical argumentation.

Yet the argument has power. The well known religious writer C. S. Lewis, author of the Narnia children's books and other influential religious works, was reportedly converted to Christianity by the ontological argument. Some people have found the ontological argument to be absurd and have dismissed it. ("Existence," said Immanuel Kant, "is not a predicate.") Others have found the argument strangely compelling. One thing is certain, however. Those who would dismiss God as the product of irrational or superstitious cultures must contend with the intellectual framework that the ontological argument sets up. There have been many people, both past and present, who have been drawn to a faith in God, but have found their way blocked by certain intellectual barriers. For these people, faith has not been enough unless it was compatible with human reason. If the human heart leads us to God while the head howls its indignation, then a reconciliation between the two may end up being the starting point for a valid faith for these individuals. For those who have no intellectual difficulties with the existence of God, the ontological argument will probably remain a curiosity and nothing more.

As I argued on the previous page, a determination of the existence or nonexistence of any individual requires an examination of the essential properties of that individual. A claim that Santa, for example, does not exist must be backed up by arguing that wearing a red suit and a white beard does not make you into Santa Claus. The same would appear to be true of arguments for the existence or nonexistence of God. If two people have different beliefs about the existence of God, are their beliefs contradictory, or do they simply have different conceptions of God (i.e., God's essence)? Some people feel that they have outgrown the need for God. For these people, the Judeo-Christian God that they grew up with is far too human in characteristics to be more than a projection of human vanity onto the cosmic canvas. Other people, who believe in God, agree with this. They feel that God is a constant towards whom we move with better and better degrees of approximation. Still other people, who see the Bible as inerrant, argue that the Biblical passages give us a clear and accurate understanding of God inasmuch as God has seen fit to be revealed to us. All of these people vary, to a greater or lesser extent, in their understanding of the essential properties or essential qualities of God.

One of Anselm's basic conclusions was that existence is an essential property of God. This idea is obtained in the Proslogion by deduction from an understanding of God's essence. The Proslogion does not stop at the existence of God, but deduces many other attributes as essential to God: that God is the creator of everything, that God is omnipotent, compassionate yet beyond passion, limitless and beyond time, and other properties or attributes. Yet it is for the existence argument that we most remember Anselm.

The translation of the Proslogion that I shall use is that of Benedicta Ward, which can be found in the popular Penguin Classics edition The Prayers and Meditations of Saint Anselm with the Proslogion, published in 1973. The numbering of verses follows that edition.

Within his introductory passages of the Proslogion, Anselm asks

By what signs, by what forms, shall I seek you [i.e., God]? (27)

I take this question to be an inquiry into the essential qualities [i.e., properties] of God, and perhaps an inquiry in the essence of God. Anselm realises that without some understanding of the essential qualities of God, God will be completely inaccessible.

We believe you [i.e., God] are that thing than which nothing greater can be thought. (160)

This is Anselm's famous statement about God. It is either to be understood as a statement of God's essence, or as a statement about an essential property of God from which many other essential properties are to be deduced. The expression "can be thought" is not meant as in the psychological sense, but as a limitation on possibility: God is that for which it is impossible that there be a greater being in any respect.

But when the fool hears me use the phrase
"something than which nothing greater can be thought"
he understands what he hears ... .
So the fool has to agree that
the concept of something than which nothing greater can be thought
exists in his understanding
(163, 173)

The word "fool" is not to be understood in its modern sense. Anselm means an atheist by this term. This usage derives from the Bible, in which the Hebrew word translated as "fool" denotes a morally deficient person, rather than someone who is stupid in the intellectual sense. Once again, by the "understanding", Anselm is referring to the modal realm of possibility. That which exists in the understanding is that which is possible. So Anselm is asserting that even the atheist would agree that it is possible that God exists. Even though the atheist asserts that God's existence is false, the atheist surely would agree that this is a contingent falsehood. Like a painter who imagines a picture before painting it, the atheist can conceive of a world in which God exists even if that world is not the true world. This leads us to the first of Anselm's axioms, namely

Axiom 1: g

where g is the proposition that God exists. When we consider Gödel's ontological argument, we shall see that this statement will not be taken as axiomatic. Indeed, Gödel's contribution to the ontological argument is partly a deeper analysis of this statement, which is axiomatic here. We have to say that Anselm does provide an argument for Axiom 1. However, his argument is not convincing, because he does not distinguish between propositions which are conceivable, and propositions which are possible. (We take him to mean the latter.) The distinction is very important. For example, I can conceive of an even integer greater than two which cannot be written as the sum of two prime numbers. However, the fact that I can conceive of such an integer does not mean that one possibly exists. Mathematical objects such as integers are usually understood as having only essential properties and not accidental ones. (At least, this is the usual understanding of the mathematical Platonist.)

Something than which nothing greater can be thought so truly exists
that it is not possible to think of it as not existing

With this statement, Anselm rules out the contingent (i.e., accidental) existence of a deity. We can formalize this in modal logic as

Axiom 2: g g

Anselm's argument is that if God existed only in a contingent sense, then we could imagine a greater being, namely one which whose existence is necessary. By our understanding of the essential properties of God as developed above, it is impossible to conceive of a being greater than God. Perhaps a more modern language would help in thinking about Axiom 2. Anselm is saying, in effect, that if God existed contingently, but not necessarily, then we could ask questions like "If God made the universe, then who made God?" A being invoked to explain the existence of the universe, but whose existence is in need of explanation, is not great enough for Anselm to call God.

Anselm's basic idea was that God has a property that other individuals lack, namely maximal greatness or perfection. Imagine that you, like the great Italian poet Dante, were lifted up in a dream for a personal interview with God. Having explained to you how she created the universe, and explained the answers to mysterious questions in quantum mechanics and particle physics, you finally get around to asking the following question: "OK God, you've explained to me how you made the universe. So I know where the universe comes from and why it is the way it is. But where do you come from?" If God answered "Gee, I don't know" you would feel disappointed I suspect. You would feel that God is not as knowledgeable as you would like. Anselm's insight was that a God who could answer this question (even if we did not understand the answer) would be a greater God than one who could not. Since God, by definition, is maximally great, she should have an answer. To say there can be no answer is to say that Anselm's God necessarily does not exist. To say that there is an answer is to say that necessarily Anselm's God exists. (Note that I am careful to speak only of Anselm's God here. Others have seen God in different terms.) Since God is maximally great, she must be self-creating, i.e., have the source of all existence (including her own existence) within herself.

[As a technical aside, we should note the affinity between this step of Anselm's argument, and certain reasoning in the cosmological argument for God's existence. The principle of sufficient reason implies that in order for the universe to have a first cause, namely God, the first cause must be a "self-causing cause." If not, then God would have a prior cause. But a self-causing cause -- one which contains the explanation for its own existence -- is rather like a necessarily existing being. The similarity between the arguments should not be overstated: belief in one argument does not entail belief in the other. For example, Thomas Aquinas accepted the cosmological argument but not the ontological argument for God.]

Now let us consider the ramifications of Axioms 1 and 2 within the modal logic S5.

1. g g (Axiom 2)

2. ~g ~g (Becker postulate 2)

3. g ~g (law of excluded middle)

4. g ~g (2, 3 using substitution rule)

5. ~g ~g (contrapositive of Axiom 2)

6. ( ~g ~g) (necessitation postulate on 5)

7. ~g ~g (modus ponens on 6)

8. g ~g (substitution rule on 4 and 7)

9. ~~g (Axiom 1)

10. g (8 and 9)

This is the modern form of Anselm's ontological argument, due to Charles Hartshorne. Anselm himself did not have the full power of S5 available to him. So his argument is not exactly the same as this. However, the basic idea of both arguments is that necessary existence is an essential property of God entailed by God's greatness.

We must either accept the conclusion of the argument, namely that God exists necessarily, or find fault with an axiom or postulate. Critical scrutiny has been brought to bear on the following parts of the argument:

While there was initially some suspicion of Becker's postulate and the modal logic S5, the dust seems to have settled on this. We now know that there is nothing particularly remarkable about the assumptions of S5. While Becker's postulate is not required in other types of modal logic, Saul Kripke's possible world semantics have interpreted Becker's postulate for us. If we imagine that two individuals from different possible worlds were able to compare their understanding of the meaning of , then Becker's postulate is a statement of equivalence in their interpretations of . We might imagine, for example, that a Napoleon who won the battle of Waterloo was able to sit down at the table with a Napoleon who lost the battle of Waterloo to discuss how the battle went. If Becker's postulate is satisfied, then they should be able to agree about which contingencies are truly possible. This seems to be most closely in accord with our naive views of modal possibility. That is not to say that it is forced upon us.

In the absence of anything outrageously wrong with S5, we can turn to the two axioms. Axiom 2 has come under a lot of scrutiny. However, many of the arguments have lost their force. In the original version of the argument, Anselm claimed that contingent existence is greater than possible existence, and that necessary existence is greater than contingent existence.

This seems to treat existence as just another property of individuals, such as whether they are wearing red suits or have white beards. Immanuel Kant (1724-1804) criticized the ontological argument by stating that existence is not a predicate. That is, existence is not a property of individuals in the same way that being short or red is. It is certainly true that we have to be careful here. If we can arbitrarily add existence as a defining property for an individual, there seems to be no limit to what we can prove to exist. For example, we might define a unicorn as follows:

Definition: A unicorn is a four-footed beast resembling a horse having a horn on its head and existing.

Thus unicorns exist. By definition.

However, this is a parody of Anselm's argument, and doesn't stand up under close examination. Any good mathematician will allow you (within reason) to define your terms any way that you like. So there is nothing wrong with the definition. Can we really show that unicorns exist using this argument? The answer is no. Our definition of a unicorn would only seem to imply that all unicorns exist, or equivalently, that for all x, if x is a unicorn then x exists. However, this statement is trivially true, because it is vacuously satisfied.

Anyway, the form of the ontological argument that we have used does not explicitly assume that existence is a predicate. It assumes that the modal status of an individual (the Eiffel tower, say, or the number 17) can be regarded as a property. A number between 16 and 18 exists necessarily, whereas the Eiffel tower exists contingently, and the distinction between the two can be regarded as a property of each. The statements

(x) (16 < x < 18) (x) (16 < x < 18)


(x) (18 < x < 16) (x) (18 < x < 16)

are both true in the domain of natural numbers because natural numbers are Platonic objects. (The latter is also true vacuously.) Both these statements are reasonable mathematically, and parallel to Axiom 2. Therefore Axiom 2 cannot be easily dismissed.

For my money, the real problem with Anselm's ontological argument is Axiom 1. I have argued that the reasons given for believing Axiom 1 are not correct because they confuse statements which are possibly true with statements which are conceivably true. If we remove Axiom 1 from the argument, then we are still able to prove the statement

g g

namely, that the possibility of God's existence implies the necessity of God's existence. But we can prove no more than this. Many philosophers who have studied the ontological argument are willing to accept that Anselm's reasoning can take us this far and no further. While the jump from possible existence to necessary existence would seem to be useful, we should remember that if the statement g is interpreted as any mathematical statement, true or false, about the natural numbers, then the statement is true as well. It may be a bit of a stretch to declare that Anselm has demonstrated that God, like the smallest even integer greater than two which cannot be represented as the sum of two primes, is a Platonic entity. However, in the broad sense, that is what he accomplished.

It was Gödel who tackled the problem of possible existence head on. Before we consider his proof, it should be pointed out that much of Gödel's argument is inspired by the writings of Leibniz on the subject. Leibniz argued that the weakness of Anselm's argument is in the statement of God's possible existence. Gödel's argument follows Leibniz in bolstering this step with additional argumentation. Reader's who are familiar with Leibniz's ontological argument may find echoes of Leibniz's method of affirmative simples in Gödel's work.

[BACK] Back to Christopher Small's home page .