7

Can We Prove God Exists by Pure Logic?

Some people used to think we can prove the existence of God by pure logic. The argument they thought could do this trick is called the Ontological Argument.

The Ontological Argument is terrific fun. In its simplest form, it goes like this:

1. Nothing greater than God can possibly be thought of.

2. A God who exists is greater than a God who does not exist.

3. Therefore: God exists.

At a quick glance, this argument seems to work! Conclusion 3 does seem to follow from Premisses 1 and 2. If it’s part of the definition of God that we cannot possibly think of anything greater, then apparently a God who does not exist must be ruled out, as a God who exists appears to be obviously greater than a God who does not exist.

I’m going to criticize this argument on three grounds:

a. The Argument is unsound because it relies on switching the meaning of the word ‘greater’.

b. The greatest thing we can possibly think of cannot be God without the universe, but must be the whole universe. However, even the claim that the whole universe, including God, could be the greatest thing we can possibly think of is a bit dubious.

c. The greatest thing we can possibly think of, if it could exist, would have to contain an infinite collection of items. There are reasons for not accepting the existence of such an entity, and most theists wouldn’t want to accept it.

The Argument is sometimes worded in terms of ‘most perfect’ instead of ‘greatest’. This makes no difference. The Ontological Argument, as I have phrased it, does not explicitly state that the greatest thing we can possibly think of is a person, but this is taken for granted by everyone who wants to defend the Argument. The Argument in effect assumes that we can think of nothing greater than an all-powerful, all-knowing person.

A Flaw in the Ontological Argument

Suppose someone were to say, ‘Sherlock Holmes is certainly a great detective, but he would be so much greater if he had actually existed’. Either this is a joke or it’s a muddle. Someone who wanted to persist with this might say, ‘But look, if Sherlock Holmes really existed, he would be able, in the real world, to solve crimes. The fictitious Sherlock Holmes has never really solved any crimes at all, because, poor thing, he is fictitious. It’s indubitably the mark of a great detective to be able to solve real crimes in the real world. So you have to admit that a real Sherlock Holmes would be a greater detective than the fictitious Sherlock Holmes.’

However, if we adopt that way of talking, then the fictitious Sherlock Holmes has no claims to greatness as a detective, or even to being a detective at all. If we want to say both that the fictitious Holmes is greater than the fictitious Lestrade, and that a historical Holmes would be greater than the fictitious Holmes, then we’re switching the meaning of the word ‘greater’. For, in precisely the sense in which a historical Holmes would be greater than the fictitious Holmes, the fictitious Holmes is not one eentsy bit greater than the fictitious Lestrade. And, in precisely the sense in which the fictitious Holmes is greater than the fictitious Lestrade, the real Holmes is not one smidgen greater than the fictitious Holmes. As a real-life detective who can solve real-life crimes, the fictitious Holmes is a hopeless failure, a nullity, a cipher. There’s not an ounce of greatness in him.

What we have here is a confusion of two entirely different usages of ‘greater’. When we first look at the Ontological Argument, we easily overlook the fact that the expression ‘greater then’ means something quite different in Premiss 1 than it means in Premiss 2. We tend to suppose that a nonexistent God would be quite great but an existent God would be even greater. But this is misleading, for, in the precise sense in which an existent God is greater than a nonexistent God, a nonexistent God is bereft of any particle of greatness.

In the history of discussions of the Ontological Argument, this weakness in the Argument has mainly been addressed in discussions of whether existence is a property (or whether existence is a predicate). Immanuel Kant was the first philosopher to claim (in opposition to the Ontological Argument as advanced by René Descartes) that ‘existence is not a predicate’. Discussion of whether existence is a property has been of great use in clarifying some of the basic principles of logic. However, it’s still not quite resolved by philosophers of logic whether existence can be a property. What we have seen above is that from the standpoint of criticizing the Ontological Argument, whether existence is a property is not the crux of the matter. Even if existence is a property, the Ontological Argument still fails.

The flaw in the Ontological Argument is more fundamental and more simple: the Argument contains a fallacy of equivocation. A fallacy of equivocation is a feature of an argument which is unsound because one of the terms switches its meaning in the course of the argument (and the conclusion depends on this switch of meaning). The fallacy of equivocation in the Ontological Argument occurs with the term ‘greater than’. ‘Greater than’, in the sense in which God is held to be greater than (for instance) the greatest human being, means something entirely different from ‘greater than’, in the sense in which an existing God is held to be greater than a purely imaginary God.

The equivocation is difficult to spot because of the vaguely inclusive nature of the word ‘great’ (or in some versions of the Argument, ‘perfect’), to refer to a rag-bag of different qualities. If only one quality were being referred to, the fallacy would be more obvious:

1. Think of the biggest elephant you can possibly think of.

2. An elephant that exists is bigger than an imaginary elephant.

3. Therefore the biggest elephant you can possibly think of must exist.

Could God Be the Greatest Conceivable Thing?

I’ve been using the phrase “greatest thing we can possibly think of.” I’m now going to shorten that to ‘greatest conceivable thing’.³³ However, God cannot be the greatest conceivable thing, because God plus his creation (all the things he has created) are together greater than God alone. (In other words, the metaverse must be greater than God, who is only part of the metaverse.) So God is not the greatest thing that exists.

Even if ‘greatness’ were to be confined to persons, and denied to unthinking rocks and stars, then this criticism still holds. God plus Robert Schumann is a lot greater than God without Robert Schumann. It just won’t do, of course, to respond that anything done by Schumann is ‘really’ done by God. For several reasons, notably the Problem of Evil (which we’ll look at in Chapters 14 and 15), classical theists are most anxious to insist that what individual persons (angels, jinns, or humans) other than God do is not done by God. They may even contend that God bears no responsibility for what these other persons do.

What could the proponent of the Ontological Argument say against the criticism that the entire metaverse must be greater than God? There are basically two answers he might have to it. One is to say that everything God has created is part of God. The other is to deny that God plus his creation is a thing. Let’s look at each of these in turn.

In classical theism (in the orthodox theologies of Christianity, Judaism, and Islam), God’s creation is not God, nor is it part of God, nor is any part of it God or part of God. Thus, for instance, you and I are not God, nor are we parts of God. The Devil is not God, nor is he part of God. The opposite view, that you and I are God, or parts of God, and that the Devil (if he exists) is God, or part of God, is called, in one form of the idea, pantheism, in another form, panentheism. So in taking this line, the theist gives up classical theism and embraces either pantheism or panentheism.

The other way of answering my criticism would be to say that the greatest conceivable thing has to be a single entity, and God plus his creation is not a single entity. This answer can only be sustained if the Ontological Argument automatically excludes collections of things and applies only to individual things. But if we reword the Ontological Argument so that it refers to ‘the greatest conceivable thing or collection of things’, it doesn’t lose any of the sense or force it had when it referred to ‘the greatest conceivable thing’. If ‘greatness’ is applicable to God, then ‘greatness’ is applicable to the metaverse, God plus his creation. For example, imagine God plus his creation and call this M. Now suppose that God had made his creation twice as great as in the case of M. Call this second possibility M2. Then M2 is greater than M.

Is the Greatest Conceivable Thing Conceivable?

My third ground for criticizing the Ontological Argument is to point to a difficulty with the very idea of the greatest conceivable thing. Since the greatest conceivable thing is a knowledgeable person (we’re accepting this for the sake of argument), being the greatest conceivable thing implies knowing as much as we can conceive anyone knowing. However, some kinds of knowledge are such that they are inherently unlimited. What this means is that however much knowledge we can conceive of someone having, it’s always possible to imagine someone having more.

Consider the series of positive integers (an integer is a whole number). It begins 1, 2, 3 . . . Now, what’s the highest integer? The correct answer is: There can be no highest integer. Anyone who thinks there could be a highest integer has merely not understood the concept of ‘integer’. You can always add 1 to any integer, however big. The series of integers goes on for ever.

You might ask God to name the highest integer, and an omniscient God just could not do this, because there is no highest integer to be named. To ask for the highest integer is as pointless as asking for a square circle. It is part of the definition of an integer that we can add 1 to any integer, so no integer can ever be the highest.

But what this means is that, if God knows any integers, the highest one he has ever specifically thought of must be a finite integer, and we can conceive of an imaginary God who has specifically thought of a higher integer than the real God has thought of. Let’s consider a specific application.

The decimal places of pi constitute an infinite series. Suppose God exists. If we ask God to name the last decimal place value of pi he has thought of, the answer can only be a finite number. However, for any finite number, we can always name a greater number. Therefore we can always conceive of a hypothetical nonexistent God who can name a later decimal place value of pi. ‘Great’ includes ‘knowing a lot’ and ‘knowing more than’ implies (other things being equal) ‘being greater than’. Therefore, if God exists, we can always conceive of a nonexistent God who is greater than the real God. Therefore, God cannot be the greatest conceivable thing.

Pi is a number used in calculating the measurements of circles and spheres. It’s approximately 3.14159265. For most practical purposes, we only need to know pi to a few decimal places. Mathematicians using computers have now calculated pi to many thousands of decimal places. Unlike, say, the decimal expression of one-thirteenth, which is 0.076923 . . ., with the ‘076923’ repeating indefinitely, the decimal places of pi display no regular pattern. There is no short cut, having been given pi to a hundred decimal places, to quickly find what numeral occupies the two hundredth decimal place. You just have to keep on working out all the decimal places until you get as far as you want to go.

The decimal places of pi constitute an infinite series. There are many other such numbers, for instance the ‘exponential number’ known as e, which is roughly 2.71828183. Such numbers are called ‘irrational numbers’, and it has been proved that for all irrational numbers the decimal places will never repeat or terminate.

Infinity is not ‘a big number’. Infinity means you can always add more and still not get there. No matter how big a number you have, you can still add 1 (or, for that matter, multiply by fifty trillion), and you are no nearer the end, because there is no end. It is, therefore, impossible for anyone to ‘know’ the whole of an infinite series like the decimal places of pi.

A theist might respond like this.

God’s thoughts are so much higher than our thoughts that he can dispense with our tools of thinking. Since God can directly perceive the precise ratios involved in circles and spheres, he doesn’t need to calculate any decimal places of pi. He doesn’t have recourse to pi at all.

By analogy, we might consider a domestic dog’s ‘beliefs’ about where its food comes from, as contrasted with its owner’s beliefs. The owner’s knowledge of where dog food comes from and how it gets into the dog’s bowl is so much superior to the dog’s that whatever ‘concepts’ the dog may have are just ridiculously inadequate, especially if the owner happens to be production manager at a dog food factory. Similarly, our mathematics, including our need for pi, and perhaps for any numbers, would appear just as ridiculously inadequate by comparison with God’s way of thinking. God’s way of thinking has no need for such mathematical devices as recurring decimal places and infinite series.

Accepting all that, it’s still a part of God’s knowledge how much he knows of the content of the theories devised by humans, just as it is part of our knowledge how much we know of the content of a dog’s consciousness. To know everything that it’s logically possible for anyone to know, God has to know what we know, and he also has to know those logical implications of our intellectual tools that are beyond us. He must know—what no human poker player can possibly know—all those precise circumstances in which it’s correct to raise pre-flop with a pair of nines.

Consequently, since it remains necessarily true that any God must know a finite number of decimal places of pi (counting zero as a finite number), and since we can always conceive of a non-existent God who knows more decimal places of pi than any given God, there cannot be a God who knows the most conceivable decimal places of pi, and therefore there cannot be a God who is the most knowledgeable conceivable person, and therefore there cannot be a God who is the greatest conceivable thing.

Traditionally God has been depicted as possessing some proficiency in arithmetic. According to Matthew’s Jesus (10:30), “The very hairs of your head are numbered.” Unfortunately for God’s math skills, 1 Kings 7:23 (repeated at 2 Chronicles 4:2) strictly implies that pi is equal to 3. Oops.

God Containing an Infinite Collection of Items

In response to the above, a theist might simply claim that God is instantly aware of all the decimal places of pi. The theist could say that God’s mind is infinite, and that he knows all the elements in an infinite series. Thus God knows every decimal place of pi, and he knows every integer. He doesn’t have to look them up; he’s instantly aware of each number in every series. It’s true that God couldn’t name the last decimal place of pi, because there is no such numeral, just as he couldn’t name the highest integer, but he still knows all decimal places of pi (and all integers) because he is infinite. Augustine took essentially this position when he claimed that God knows the identity of every number, even though the number of numbers is infinite. I suspect the position is logically incoherent but I don’t know how to prove that.

This may look like a satisfactory position for the theist to take. Haven’t theists often said that God is infinite? However, it’s a position few theists will want to take. Theists who have said that God is infinite have generally also said that God is simple. It is one thing to say that God is infinite and another thing to say that God contains an infinite collection of items.

Some theists have actually denied that an infinite collection of items can exist, even in God’s mind. This is the basis for the Kalam Argument, favored by William Lane Craig. We can now see that the Ontological Argument is incompatible with the Kalam Argument. Anyone who does maintain that God, being infinite, can know all the numbers in an infinite series, cannot also accept the Kalam Argument (which we looked at in Chapter 6). It’s essential to the Kalam Argument that the whole of an infinite series cannot have any actual existence outside mathematical theory. When someone objects to the Kalam Argument that if there is no actual infinite, God can’t exist, since God is supposed to be infinite, the advocate of the Kalam Argument replies that although God is infinite, he is simple. He does not contain an infinite series of items.

Yet according to classical theism, God must contain at least one set of items: ‘things God knows’. Or even, ‘real numbers God has specifically thought of’. Is this set finite or infinite? If, for instance, God knows all the decimal places of pi, then there’s an actual infinite collection of pieces of knowledge in God’s mind, and this contradicts the Kalam Argument. A proponent of the Kalam Argument must accept that the number of pieces of knowledge in God’s mind is finite. If God does not know all the decimal places of pi, he knows a finite number of decimal places of pi, and we can ‘possibly think of’ someone greater than God, a hypothetical imaginary God who knows one more decimal place of pi than the hypothetical actual God. Thus, the entity whose existence is claimed to be proved by the Kalam Argument cannot be the greatest (or most perfect) conceivable entity. Therefore, no one can accept both the Kalam Argument and the Ontological Argument.

The argument I have given here is simply one way to pin down a broader insight. For any quality that has no inherent limits, there cannot be any value for that quality which cannot imaginably be exceeded. Whatever value that quality has, one can always imagine that quality with a greater value.

Suppose that the universe had existed in every way just like it has, with one difference: Beethoven wrote twice as many symphonies (and wrote all his other actual pieces too). The almighty God of classical theism could have accomplished this entirely costlessly, by an effortless miraculous intervention, without withdrawing resources from any other project. And it won’t work to say that God has already done this in a parallel universe—we’ll just make it part of the supposition that in that parallel universe, in our hypothetical example, Beethoven wrote four times as many symphonies.

A God who created a universe in which Beethoven wrote twice as many symphonies would have to be greater than the God who actually exists (if he exists). Greatness is as greatness does. This example brings out, once again, that there can be no upper limit to the greatness of something ‘possibly thought of’. The greatest being we can ‘possibly think of’ is just incoherent and threrefore absurd, like the highest integer or a square circle.

A theist might reply that God’s greatness is only potential, not actual. He can do anything he likes, but it is no diminution of his greatness if he doesn’t do everything he can. Yet if I were to say that I am a greater poet than Arthur Rimbaud, because, had I put my mind to it, I could have written poetry even better than his, some literary pedant might suggest that my usage of the word ‘greater’ was eccentric.

If it’s true, as I believe, that the greatest conceivable thing cannot exist, then this disposes of the Ontological Argument. It doesn’t show that God doesn’t exist, for God might exist and not be the greatest conceivable thing. The list of God’s ten qualities I gave in Chapter 1 does not include being the greatest conceivable thing. My conclusions, then, are: 1. that the Ontological Argument fails to show that God exists, and 2. that if there were a God, we would be able to conceive of a greater God than that actual God—and this fact should not trouble the believer in God.

Every so often, some philosopher dusts out the Ontological Argument and gives it a new twist, hoping to find some version of it which is sound. This was done by neo-Hegelians around the end of the nineteenth century. It was done again by several philosophers, notably Norman Malcolm and Charles Hartshorne, in the mid-twentieth century. The neo-Hegelians were poor logicians whereas Malcolm and Hartshorne were highly expert logicians.

The arguments of Malcolm and Hartshorne both proceed by showing that if the greatest conceivable thing exists, then it exists necessarily, and if it does not exist then it necessarily does not exist. In other words, if it exists, it must exist, while if it doesn’t exist, it can’t possibly exist. They then claim that there is nothing to show that God logically can’t exist, and so we’re left with the alternative, that God must exist.

But there are several good arguments showing that God (let alone the greatest conceivable thing, which is a more ambitious concept than God) cannot possibly exist. I look at some of those arguments in Chapters 13-18 of this book.