I have lately had the unenviable task of trying to explain Anselm’s ontological arguments to undergraduates. Over the years I have read many expositions of ontological arguments and many critiques. However, I had never sat down and gone through Anslem’s arguments line-by-line. Now I have done so in an effort to make a hand-out that will, I hope, things more tolerable for my students. I enjoyed doing it and I found it an instructive and challenging exercise, and I am appending it below.

There have been many attempts to reconstruct Anselm’s argument. Instead of adding another I am just going to quote Anselm’s own words, breaking up his argument into numbered segments (and omitting some unnecessary or redundant sentences). I will then explain and evaluate the argument with reference to those segments. I begin with the argument in Proslogion II:

1…we believe that thou [God] art a being than which nothing greater can be conceived.
2. Or is there no such [being] since the fool hath said in his heart there is no God? But, at any rate, this very fool, when he hears of this being of which I speak—a being than which nothing greater can be conceived—understands what he hears, and what he understands is in his understanding; although he does not understand it to exist.
3. Hence, even the fool is convinced that something exists in the understanding, at least, than which nothing greater can be conceived.
4…that than which nothing greater can be conceived cannot exist in the understanding alone. For, suppose it exists in the understanding alone: then it can be conceived to exist in reality; which is greater.
5. Therefore, if that, than which nothing greater can be conceived, exists in the understanding alone, the very being, than which none greater can be conceived, is one, than which a greater can be conceived. But obviously this is impossible.
6. Hence, there is no doubt that there exists a being than which none greater can be conceived, and it exists both in the understanding and in reality.

(1) Anselm begins with a definition of “God:” God is that than which nothing greater can be conceived. Let’s shorten this by saying that “God” means “the greatest conceivable being (GCB).” What does this mean? It means that we conceive of God as possessing every good quality to the maximum possible degree. For instance, God’s power is unlimited, that is, he is almighty or omnipotent. If anything is doable, God can do it. Similarly, God’s knowledge is unlimited. If anything is knowable, God knows it. A being that thus knows as much as any being can know is said to be “omniscient.” Further, God’s goodness is maximal; God possesses moral goodness to the highest possible degree. It follows from this definition that God cannot lack any perfection. If a property is a perfection, God must possess that property. It also follows that it is impossible to conceive a greater being than God, since a greater being would have to have a perfection that God lacks, and this, by definition, is impossible.

(2) Referring to the psalmist’s fool (Psalm, XIV) who says in his heart that there is no God, Anselm notes that at least the fool understands what he hears when he hears that God is the GCB, and what he hears exists in his understanding. What is in the fool’s understanding? It would not be right to say that the fool understands that God is the GCB, since this would seem to imply that the fool thinks that there is a God who is the GCB. However, Anselm notes that the fool does not think that God exists. What must be in the fool’s understanding then is the meaning of the concept “God.” “God” means “GCB.” By the way, in analyzing Anselm’s argument it is very important to keep clear at any given time whether you are talking about God, the putative being, or the concept of God, so when speaking of the concept, I’ll enclose the word G-o-d in parentheses, and when speaking of the putative being, I will omit parentheses. So, the fool understands the concept “God,” but does not understand (since he does not believe) that there actually is such a being. OK, so far so good (I think!).

(3.) This is a bit confusing. By referring to “something…than which none greater can be conceived,” if Anselm means God, the being, then obviously this premise is false since nobody, not even the fool, is going to hold that God Himself exists in the understanding. What exists in the understanding is concepts, ideas, notions, etc., not real, substantial, mind-independent entities (as God is held to be). A cat does not exist in your understanding, but in the external physical world. What exists in your understanding is the concept or idea of a cat. Similarly “God” may exist in your understanding, but not God. To speak of cats, or God, as existing in the understanding is to commit a gross category mistake. A category mistake occurs when we apply concepts appropriate for one type of thing to something to which those concepts cannot apply. F0r instance, you would commit a silly category mistake if you thought that the statistical abstraction “the average American” was the flesh-and-blood guy living next door. It is just as big a category mistake to speak of cats, or God, as existing in the understanding.

(4) This is where things get really confusing. When Anselm refers to “that than which nothing greater can be conceived” here is he referring to a concept or to an actual being? “God” or God? If he is referring to God, then, as we noted above, this sentence is just gibberish. You cannot even coherently suppose (second sentence) that God exists in the understanding alone, any more than you could suppose that you could live next door to the Average American Family with their 1.6 cars and 2.3 children.

Nevertheless, Anselm seems to suppose that God can have two modes of being, one in the understanding and one in reality, and that the mode of existing in reality is superior to the mode of existing in the understanding. Again, though, it is very hard to see what it could mean to say that one mode of God’s existence is as a concept, unless this is just a very roundabout and confusing way of asserting that there is a concept “God,” where “God” is defined as “the GCB,” and that people, including the fool, understand that concept. In any case, in denying that God exists, the fool is not attributing some inferior mode of existence to God. He is not saying that God exists but only in an inferior way as a mere concept. If that is what the fool was saying, then, he would indeed be saying something foolish. He would be saying that God exists, but only in an inferior mode. However, God, if he exists at all, cannot exist only in an inferior mode. If the GCB exists in some mode, he must exist in the best mode, or he is not the GCB! What, then, is the fool saying when he says that God does not exist?

When we say that a being is the GCB, the phrase “the GCB” is what philosophers call a definite description. A definite description is one in which the description picks out one, unique entity as the bearer of a property. For instance, if we describe something, x, as “the smallest prime number,” or “the meanest man in Texas,” or “the GCB,” each of these descriptions picks out something, x, as being the unique bearer of the stated property.

Eminent English philosopher Bertrand Russell addressed puzzles about definite descriptions in a famous paper “On Denoting” published in 1905. For instance, how do we understand a sentence like “The present King of France is bald”? There is no present king of France (the French Revolution took care of that). So do we say that the sentence is false? But to say that it is false seems to mean that there is a present King of France who is not bald. If, on the other hand, we say that the sentence is true, we seem to be saying that th

ere is a present King of France who is bald. Either way then, whether we regard the sentence as true or false, we seem to be committed to saying that something, the present King of France, exists when it does not exist. How can we sensibly talk about something that, we all admit, does not exist without apparently positing its existence? When we make such statements are we, as some philosophers thought, referring to mysterious non-existent objects (e.g. mermaids, golden mountains, and honest politicians) that are in some sense real (since we can refer to them) even though they do not exist?

Russell’s brilliant analysis avoids appeal to such mysterious quasi-entities. As Russell interprets it, to say that some unique F is G, is to say, (1), that there exists something which is F, (2) that there is nothing else which is F, and (3) that if anything is F it is also G. So, to say “The present King of France is bald” is to say (1) There exists something which is the present King of France, (2) Nothing else is the present King of France, and (3) If something is the present King of France, it is bald. In the language of predicate logic, what we are saying is this:

(∃x) {Fx & [(∀y) (Fy → y = x) & Gx]}

Further, on Russell’s analysis, to say that the present King of France exists is just to assert (1) and (2) above. To deny that the present King of France Exists is to say either that nothing is the present King of France, or that more than one thing is the present King of France. Hence, to deny that the present King of France exists is not to say that there is a present King of France who has the property of not existing. It is to say that there is nothing which is the present King of France or that more than one thing is.

On this analysis, to say that the GCB exists is to say, in language of predicate logic, and where “G” stands for “is the GCB,” is this:

(∃x) [Gx & (∀y) (Gy → y = x)]
That is, in English: “There is an x that is the GCB, and, if there is a y that is the GCB, then y is identical to x.”
To say that the GCB does not exist is to say:
~(∃x)[Gx & (∀y) (Fy → y = x)]

Which is equivalent to:

(∀x) [~Gx v (∃y) (Fy & y ≠ x)]

Or, in English: “Either there is nothing that is the GCB or there is more than one GCB.”

However, there cannot be more than one GCB, so to deny that the GCB exists is really to say “Nothing is the GCB.”

On Russell’s analysis, then, when the fool denies the existence of God he is merely saying that nothing is the GCB. He is not saying, as Anselm seems to think, positing a GCB and then attributing to it an inferior mode of existence. He is saying that there is no such thing as the GCB. The fool has the concept “GCB,” he just holds that there is nothing, no real thing, that instantiates or exemplifies that concept.

(5) What, then, is the absurdity to which Anselm thinks the fool is committed? The fool would indeed be committing an absurdity if he were speaking of God as having some inferior mode of existence. However, as we just saw, this is not what the fool is saying when he denies the existence of God. The fool is not absurdly saying, or at least need not be taken as saying, that God has the property of existing only as a concept. Rather, he is saying that there does not exist an x such that x is the GCB, i.e. there does not exist an x such that x is God. Where is the absurdity?

Maybe Anselm suspects that there is something fishy about this argument because he immediately moves to a second argument in Proslogion III, which is so succinct that I will present as a whole:

So truly does such a thing [the GCB] exist that it cannot be thought of as not existing. For we can think of something as existing which cannot be thought of as not existing, and such a thing is greater than what can be thought not to be. Wherefore if the thing than which none greater can be thought could be conceived of as not existing, then this very thing than which none greater can be thought is not a thing than which none greater can be thought. But this is not possible. Hence something greater than which nothing can be conceived so truly exists that it cannot be conceived not to be.

Some recent philosophers such as Norman Malcolm and Charles Hartshorne think that this is a very different argument, and one which is much superior to the argument from Proslogion II described and critiqued above. What, then, is Anselm saying here?

Anselm once again seems to be making a distinction between two modes of existence, but it is not the same contrast as the one made in Proslogion II between existence-in-the-mind and existence-in-reality. Here the contrast is between a mode of existence such that anything that exists in that way could not conceivably not exist and a mode of existence such that anything that exists in that way could conceivably not exist. What does it mean to say that something, A, cannot be conceived not to exist? Anselm seems to mean that if we attempt to entertain the proposition “It is conceivable that A does not exist” we immediately see that this proposition entails a contradiction, an assertion of the form “p & ~p.” Therefore, it is not the case that it is conceivable that A does not exist, i.e. the non-existence of A is inconceivable. But if the non-existence of A is inconceivable, then we must conceive of A as existing, and, hence, A exists.

Anselm’s argument here therefore can be set out as below. Note that the argument has the form of what philosophers call a “reductio ad absurdum,” or “reduction to the absurd.” Such an argument works by making an assumption and then showing that the assumption leads to an explicit contradiction (something of the form p & ~p). We then reject the assumption and accept its contradictory.

1) “God” is defined as “The GCB.” (premise).
2) Something that cannot be conceived not to exist is greater than something that can be conceived not to exist (premise).
3) The GCB can be conceived not to exist (assumption for reductio).
4) If the GCB can be conceived not to exist, then it is not the GCB (because, in that case, an even greater being could be conceived, namely, one that cannot be conceived not to exist).
5) It (the GCB) is not the GCB (from 3 and 4 by modus ponens).
6) The GCB cannot be conceived not to exist (rejection of assumption; 5 is an explicit contradiction).
7) God cannot be conceived not to exist (from 1).
8) God must be conceived of as existing (Anselm appears to hold that if “not-p” is inconceivable, we must conceive that p).
9) God exists (Anselm does not explicitly draw this conclusion, but he apparently assumes that if we must conceive that God exists, then God exists).

This argument avoids making the fishy distinction, made by the argument in Proslogion II, between existence-in-the-mind and existence-in-reality. Of course, the distinction between existence that cannot be conceived not to exist and existence that can be conceived not to exist might be equally fishy (I think it is). However, for the sake of argument, let’s accept it.

To make things less cumbersome, let’s say that a mode of existence such that anything that exists in that way could not conceivably not exist is “necessary existence.” Anselm seems to be saying that necessary existence is an essential part of what it means to be the GCB just as having eight sides is an essential part of what it means to be an octagon. The fool who thinks that the GCB does not necessarily exist is like someone who asserts that an octagon does not have eight sides. Such a person either does not grasp the concept or willfully endorses a contradiction in terms. Hence, the fool who understands what “the GCB” means comm

its himself to a contradiction if he then denies that necessary existence is included within that concept. The fool in that case is just as ridiculous as one who says that the idea of an octagon does not include the idea of eight-sidedness.

However, the fool can say, just as much as Anselm can, that necessary existence is a part of the concept of the GCB, just as eight-sidedness is a part of the concept of the octagon. That is, if the fool were asked to explicate the meaning of “the GCB,” then, just like Anselm, he could say “The concept of the GCB is the concept of a being that is omnipotent, omniscient, perfectly good, necessarily existent…etc.” However, the fool commits no self-contradiction nor any other absurdity if the then goes on to say “However, there is no actual entity that exemplifies the properties of omnipotence, omniscience, perfect goodness, necessary existence…etc.” This is no more a self-contradiction than to correctly define “octagon,” but then to deny that there is any real thing that exemplifies the property of eight-sidedness. You do contradict yourself if you affirm a concept but then deny that the concept contains one of its essential, defining predicates. However, this is not what you are doing if you declare that the concept, as a whole, is not instantiated.

In other words, the fool, just as much as Anselm, admits that “necessary existence” is part of the concept “the GCB,” here merely is making the further judgment that there is no actual entity that exemplifies the properties that constitute that concept. There is no contradiction at all in affirming that necessary existence is part of the concept “the GCB” and then denying that “the GCB” is instantiated as an actual entity. Put yet another way, there is no contradiction at all in saying that God, the GCB, exists necessarily…if he exists.

Throughout this discussion we have retained Anselm’s invidious characterization of his opponent as “the fool.” At this point, though, the fool is looking pretty smart.