The point of departure of the argument is a Biblical passage which describes how a fool says in his heart that there is no God (Psalm 14:1; 53:1). Anselm claims that the fool is capable of understanding the notion of something than which no greater can be thought. (Hereinafter this being will be be called "the NGT" when greatness is at issue. The being will be referred to as "God" when more traditional theological considerations are under discussion, as in this case, where it is denied that God exists.) As Aquinas noted, it is not clear that everyone would understand this complicated concept, but a more sophisticated atheist might, if indeed it can be undestood at all.
The next step is to make a claim about the existence of what is understood. Anselm maintains that the NGT has an existence in the understanding of someone who understands it. Given that the atheist understands what the NGT is, the NGT exists in the understanding of the atheist who understands it. What remains to be shown is that the NGT exists outside the atheist's understanding. For example, a time-machine might exist in my understanding, in that I think of a device that would allow me to travel back and forth through time. Such a thing has been depicted in various works of fiction. But it does not exist outside my understanding and that of others who have conceived of one. So what makes the NGT any different from a time machine with respect to existing outside the understanding?
The Ontological Argument
The argument proper consists of a reductio ad absurdum. That is, it begins by assuming that the NGT exists only in the understanding and proceeds to derive a contradiction, concluding that the assumption is false. If the NGT does not exist only in the understanding, then it exists outside of it. Here is a regimented version of the argument.
The argument depends crucially on what the word "greater" means in premises 2 and 3. There may be some intuitive senses in which we would say that a being that exists outside the mere understanding is "greater than" a being that exists only in the understanding. For example, a million dollars in my own bank account seems in some sense to be "greater" than a million dollars that I only dream of having. This is because the externally-existing million can be used to buy the things I want and need, while the merely-thought millions cannot.
There is another sense, though, in which a million dollars in the pocket is no greater than the million in thought. Each is worth exactly the same amount, represented, say, by a ten-thousand one-hundred dollar bills. There is no greater quantity involved in either case.
What about the alleged "something than with nothing greater can be thought?" Why would the existence of this being merely in thought be less great than the existence of this being outside of thought as well? Perhaps Anselm would answer by saying that any object existing only in thought cannot do anything (the way an imaginary million dollars cannot buy a yacht). It would be deficient in power, and hence not the greatest conceivable being. It could be argued, however, that the pure object of thought can do exactly the same things as can the object that exists outside of thought. If you leave out of conceived object something that it could do (such as buy a yacht), then you are not talking about the same thing as the externally-existing object. You must think of the million dollars as money that has purchasing power, or you would not really be thinking about a million dollars. In the case of God, if you did not endow the content of your mere thought of the NGT with all the power possible, you would not be thinking of the NGT.
The line of argument just advanced is adapted from the eighteenth century philosopher Immanuel Kant. The upshot of the argument is that there is no way to distinguish a thought-being from a thought-independent-being by virtue of anything specific you might say about it, such as what it can do. The only way of distinguishing them is through the bare fact that one exists in thought and one exists outside of thought. Then Anselm could be asked why this makes the latter greater than the former.
This second version of the argument highlights an objection made by Thomas Aquinas.
Perhaps not everyone who hears this name God understand it to signify something than which nothing greater can be thought, seeing that some have believed God to have a body. Yet, granted that everyone understands that by this name God is signified something than which nothing greater can be thought, nevertheless, it does not therefore follow that he understands that what this name signfies exists actually, but only that it exists mentally. (Summa Theologica, Part One, Question II, First Article)Here is a way of interpreting Aquinas's point. Suppose the NGT cannot be thought not to exist, since that is contrary to the thought of the NGT. Still, the fact that we cannot think of its non-existence does not imply that it actually exists. That is, step 5 is false, or at least not justified by the argument. Our failure to be able to think of the NTG as not existing may be regarded as simply that--a failure on our part.
But Anselm may not be on very solid ground himself here, due to a problem was raised by the seventeenth century German philospher, Gottfried Willhelm Leibniz. He pointed out that the NGT must be thinkable in the first place, that it does not harbor any hidden contradictions. The "thought" of a speed than which none greater can be thought is self-contradictory: one can always think of a speed faster than any given speed. So how do whe know that there is (in the understanding) a being than which none greater can be thought? This may be just a string of empty words, especially since there is no content to the "thought" beyond that of how great it is. So if the fool's "thought" is incoherent, it may be that the believer's "thought" is empty--the opposite error, but one that is of no use by itself in proving that God exists.
Gaunilo's Reply
The monk Gaunilo came up with a number of objections to Anselm's argument. One objection is widely known. Gaunilo tried to show that the argument must be incorrect because its format allows the proof of the existence of the "greatest" being of any category. For example, the existence of a "Lost Island," more abundant than any known existing island, could be demonstrated as follows.
If a thing exists through another thing, perhaps as being created by it, then such a thing is not the greatest thinkable, since it depends on something else for its existence. So the NGT exists through itself alone. Moreover, Anselm claims that it has created all other things from nothing. Presumably the reason for this is that if something else had created other things, the NGT would not be the greatest conceivable being. And creation from nothing would be greater than creation from something pre-existing.
The greatest thinkable being would also have to be the highest good. If something were better than it, the NGT would not be the greatest conceivable. One would have to add that good is better than its opposite evil, which must have been self-evident to Anselm. As special cases of the NGT's goodness, we have truth, justice and happiness. In each case, it would be better for a being to have this property than its opposite.
A more specificially theological conflict lies in the fact that the NGT should help people as much as possible, yet some people are undeserving of help. It would seem that all such people ought to be punished for their bad deeds, as this would serve justice. So if the NGT must be just (as seems to be the case), then the NGT should punish the wicked. But God may choose to spare them instead. This would be explained by the maximization of goodness, in which case God's action would be just. But it also promotes goodness for God to punish the unjust because they deserve it. Anselm wants to have it both ways. Whatever God does toward the wicked serves justice.
One might argue against this view as follows. There are two possibilities: one of the two possible actions, sparing the wicked or punishing them, is the greater actions, or else neither is greater than the other. If one is greater, then God ought only to do the one which is greater, whichever it is. If neither is greater than the other, then God acts out of indifference. But the NGT would never act out of indifference, but only for the best. (This position is similar to the one taken by Socrates against Euthyphro.) The reason this problem is more theological than philosophical is that the description of God's behaving in both ways toward the wicked in described in the scriptures.
There is still a residual puzzle, though. The series of time and the extension of space themselves consist of parts. One time is before another time and after yet another. If God exists at all times, then God's existence could be chopped up into what we would now call "time slices." Similarly, if God exists in all places, then we would seem to be able to isolate parts of God. To solve this problem, Anselm denies that God is in time or space. Time and space, as well as everything in them, exist instead in God.
A final philosophical issue is the relation of God to other things that do not have an end. Presumably, what Anselm has in mind here are immortal souls created by God. But God is greater than these eternal things for several reasons. They depend on God for their existence, and would be nothing without God. Moreover, though God created them to exist forever, we can think their non-existence. And these creatures are not yet in their futures, while God is in all of eternity as a unitary being, as argued above.