What it means to think of something in the context of Anselm’s famous ontological argument is not clear, even though the argument itself depends on the fact that we are able to think of certain objects. From reading the argument itself it may seem that by “thought” Anselm means simply that which can be described or picked out by properties. However, later he describes god as greater than can be thought because god cannot be comprehended, which may seem to indicate that by “thought” he means able to be understood or comprehended completely. Obviously Anselm is aware of this distinction, as he says “In one sense of word, to think a thing is to think of the word that signifies that thing. But in another it is to understand what exactly the things is.” (Proslogion) Obviously there are many things that do not have a single word that signifies them, but we can charitably understand this definition of thinking as including descriptions which pick it out objects; the first possibility. Anselm goes on to indicate that it is the second meaning that he is working with in his ontological argument, but, as I will demonstrate, neither of these understandings of what can be thought, nor any other that can be derived from the text, works, in the sense that the ontological argument can’t go forward under any one of them.
The problem with defining what can be thought as what can be described or picked out is that it is inherently contradictory. For example, you could think of something that cannot be thought of in this way, which is a contradiction because if you are actually thinking of it, in this way, then it is not the case that you cannot be thinking about it, in which case you aren’t thinking about it after all. Another problem is that given any object you can always think about another, non-identical, object that is greater than it (using that very description). Which means that even given “that than which no greater can be thought” you can still describe some other object that is greater than it, and obviously taken together these are a contradiction. And given that there are contradictions that arise from this definition of what can be thought the ontological argument, if it is to be valid, cannot rest on it, since from a contradiction everything can be proven.
And to define what can be thought as what can be completely comprehended or understood isn’t a solution either. To begin with Anselm points out, in chapter 14, that god is not in fact thinkable in this way “… you are not merely that which a greater cannot be thought; you are something greater than can be thought. For since it is possible to think that such a being exists, then if you are not that being, it is possible to think something greater than you. But that is impossible” (Proslogion). And this means that god is not in fact that than which no greater can be thought, because there would be something less that filled that role, since god himself cannot be thought, and thus cannot serve as a counterexample to that thing. But then we would not have proved the existence of god, only the existence of that than which no greater can be thought, which may even vary from person to person on this interpretation of what can be thought, since the greatest thing that can be comprehended will vary depending on individual mental facilities. Another way to see this is to consider a hypothetical individual who can only hold twelve properties in mind at the same time. Thus at best they will be able to comprehend things that are defined in twelve properties or less. And this means that for them there will be object, defined as a collection of the twelve best properties, that is greater than every other object they can comprehend, since all other objects have at least one of these twelve best properties replaced by a lesser property, and thus that than which no greater can be thought, for them. But it isn’t god, since god has every perfection, not just twelve of them, and thus god cannot be comprehended by that person. And so the ontological argument would then fail to prove the existence of god.
Now some might object to this, and argue that god is that than which no greater can be thought for every possible thinker. And let us further suppose that everything else but god can be thought by some thinker. But this might seem questionable, because then there would not be the only thing than which no greater can be thought. At the very least it would describe both god and the greatest of possibly thinkable things, and thus imply that they both exist. But perhaps we can accept even this conclusion, since the ontological argument is attempting to show that at least god has necessary existence it might not be impossible for other objects have this property as well. But instead of answering this objection let us consider yet another, since both this objection and the next can be answered in the same way. The second possible objection is to argue that god is not unthinkable by every possible thinker, but that god is simply unthinkable by us, but perhaps thinkable by some sufficiently advanced thinkers, and perhaps those thinkers cannot think about “something greater than can be thought” (thinking about which motivated Anselm to say that god was greater than could be thought), for the same reasons that Anselm claims that we can’t think of god not existing, because to truly understand god is to realize that he can’t not exist. So perhaps to truly understand god is also to realize that there is nothing that is greater than can be thought, given that you can think of god. But both of these objections would put god outside of our understanding, and thus outside of what we can think, and because of this we can’t make the argument go forward, because we can’t arrive at a contradiction that forces us to conclude that god exists. Since we can’t think of that than which no greater can be thought, we certainly can’t think, in terms of understanding, of something greater than it. And so even if we think of god as not existing it does not violate his property of being that than which no greater can be thought, because the new entity we postulate to attempt to arrive at the contradiction, god plus existence, cannot be thought either (since god cannot be thought, and adding properties doesn’t make him easier to understand). But if it can’t be thought then there is no contradiction, since god is that than which no greater can be thought; and this supposedly greater thing can’t be thought.
Thus, if the ontological argument is to go forward we must define what can be thought in some other way. I think Anselm provides us with such a third possibility in chapter 4, specifically in response to the fool who says in his heart that god doesn’t exist, and thus supposedly thinks that god doesn’t exist even though we have supposedly just shown that it is impossible to think of god as not existing. Anselm phrases his response in terms of understanding, which we have already rejected as a possible definition of what it means to think something. But he also implies in that passage that contradictions cannot really be thought, and that we just think that we can think them because we don’t possess sufficient understanding. If we simply adopt this of our definition of what can be thought, as that which can be described without contradiction, then we have a definition that avoids at least some of the problems posed so far, since it neither requires full comprehension, which we can’t have, and eliminates contradictions, by definition.
But to adopt such a definition of what can be thought introduces a new problem, which is to show that we can think of “that than which no greater can be thought” in this way. It might not seem problematic, after all there is no contradiction within the statement itself. But it also seems apparent that we can think about the object that is greater than another arbitrary object. And as mentioned previously when both of these are put together a contradiction results. Thus one or the other must be discarded, but deciding which to discard is problematic, since neither is a contradiction by itself, and it seems like both principles reflect what we can think. We can understand both principles as corresponding to different kinds of structures. Accepting the principle that for any object we can think of an object greater than it is analogous to treating the objects we can think of as a range of real numbers, say from 1 to 2, that excludes the limiting points. Thus for any number in this range, say 1.9999, there is always some number that is greater (closer to 2), and because of the nature of the structure there is no greatest number. And accepting the principle that there is some thing which than no greater can be thought is analogous to treating the objects we can think of as a range of real numbers, say 1 to 2 again, that includes the limiting points. In such a range there is indeed a number than which no other is greater, 2, although the principle that for every number there is one that is greater no longer holds. The fact that both of these analogous situations are consistent reveals that either treatment of what we can think about is consistent, which means that we cannot choose one or the other of them on apriori grounds.
But, for the sake of the ontological argument, let us simply assume that it is the case that there is something than which nothing greater can be thought (assuming that somehow we have discovered in some way that this reflects the structure of what we can think). Even so, there is a further problem, the one raised by Kant*. Kant argued that existence was not an attribute that we can apply to things in thought, precisely because if that were the case then to think of something, anything, as non-existent would be a contradiction. And since we have defined what can be thought as that which can be described without contradiction we would be not be able to think of anything that didn’t exist, and thus the ontological argument would not be able to go forward (or, more precisely, it would show that everything must exist, which is nonsense). We can justify the claim that thinking of something as non-existent is a contradiction within Anselm’s system as follows: if something doesn’t exist then it is nothing, and if it is nothing it has no properties (except perhaps non-existence, since we are still working on the assumption that things can be said to not exist). But then we are thinking something contradictory, because we would then be thinking that something both did and didn’t have certain properties. For example, if I thought that unicorns were non-existent I would be simultaneously thinking of them as having horns, because that is part of the essential nature of unicorn-hood, and as not having horns, because they don’t exist. And therefore we can conclude that to say that something has the property of not existing is a contradiction.
But obviously we must be able to recover the thought that unicorns don’t exist in some way, as it doesn’t seem like a contradiction, since not only do I seem to be able to think it, but I can draw conclusions from it as well, and communicate the idea to others. One way to do this is to think of existence as instead characterizing our conception of a set of “real” things. Thus when I think to myself that unicorns don’t exist I am really thinking about two things, unicorns, and the set of real things not containing any of them. And likewise to think of something as existing would be to think of it and the set of real things as containing it. From this analysis of existence it might be possible to recover the ontological argument, at least in an altered form. What needs to change is the idea that the objects of thought can be compared to each other based solely on their intrinsic properties. Instead they must be compared to each other in terms of their intrinsic properties and in terms of whether our conception of the set of real things has them as in it. And thus the great making principle would be recast as “it is greater for something to be in our conception of the set of real things than not”.
Now before I examine whether this version of the argument can go forwards I need to address the possible objection that the great making principle should be instead recast as “it is greater for something to be contained in the actual set of real things than not”. The problem with this is that it makes the ontological argument an aposteriori one rather than an apriori one. More precisely, it would render us unable to compare the greatness of the objects of thought without examining the actual set of existing things first to see if it contained them. And this means that it would make the idea of “something than which no greater can be thought” nonsensical. Its very status as being greater than every object of thought would be then dependent on whether it really existed, in addition to the properties it is conceived of as having, meaning that whether it actually is that than which no greater can be thought can’t be known without also knowing whether it actually exists, and thus to reason from that supposed fact to the fact that it indeed exists would be to beg the question. Another way to illustrate this problem is to consider some imperfect thing that actually existed, say an actual imperfect cone of ice-cream. Now we might think to ourselves: the same cone, with the ice-cream in a perfectly spherical glob would be greater. But just because we think in our minds of it as being greater doesn’t mean that it actually is, given this version of the great making principle, because no perfectly spherical glob of ice-cream exists, and hence it is at least in one way actually inferior to the actual imperfect cone of ice-cream, and so we couldn’t proceed to conclude that the greater ice-cream cone that we conceived of was necessarily actually greater, despite the fact that it being greater was part of our conception of it. So not only does this understanding of the great making principle prevent the ontological argument from going forward, it also undermines the very basis of the argument, that if something is conceived of as greater then it is actually greater.
So let me return then to the question of whether the ontological argument with a great making principle “it is greater for something to be in our conception of the set of real things than not” is valid. And furthermore let us assume that we have in mind “that than which no greater can be thought”. Is it a contradiction if it doesn’t exist in the set of real things as we conceive of it? In this case it isn’t. Certainly it doesn’t necessarily make it less than everything else that can be thought, because we have assumed that objects of thought are compared on the basis of their internal properties, as well as whether they are members of the set of real things as we conceive of it, and so it could be sufficiently great that it could beat out every thing we do think of as in the set of real things simply by being sufficiently greater than them in other ways. Nor does it imply that we can think of something greater by thinking of something that exists. This is because to think of the same thing as existing would not be simply to take our existing object and add existence, resulting in a new object of thought, instead we would have to change our conception of the set of real things to include it. But even if we did this it wouldn’t be a new object greater than the one we were previously thinking of, we have simply changed our conception of the set of real things in order to make it greater, and hence there is no contradiction. Even though it could be made greater the changes that make it greater are not in the object of thought itself, and so it is still that than which no greater can be thought. And thus we can’t conclude that it necessarily exists, because there is no contradiction in supposing that it doesn’t.
This exhausts the three possible interpretations of what it means to be able to think of something in Anselm’s Proslogion. Obviously there are other possibilities, such as an object having the right kind of causal connection to the concept (the causal theory extended to cover everything instead of just proper names), but to address every possibility would take forever. Furthermore the ontological argument working with such understandings of what can be thought can usually be objected to for reasons already raised (for example, under the extended causal theory existence can’t be a property of the objects as thought, since one cannot be causally related in any way to a non-existent object). Finally, I feel that it is sufficient to rest my case that the ontological argument, as presented in the Proslogion, does not successfully prove that god necessarily exists.
* Admittedly Kant was responding to Descartes’ version of the ontological argument, which was much simpler in form. So the positions I attribute to Kant here are really adaptations of his positions.