The Ontological Argument was poised in its paradigmatic form by St. Anselm of Canterbury in the 11th century [1]. It attempts to show that God exists necessarily due to what "god" means, that it is impossible for god not to exist due to the very structure of what is and isn't possible.

Types of ontological argument[]

Ontological arguments aim too high. Just like logical calculus cannot ascertain a specific basic proposition is correct, existential calculus should not be able to conclude that some specific being exists. Logical calculus can show that if a bachelor exists then a man exists (since a bachelor is, specifically, a man), and existential calculus might be able to show that if certain things exist then god exists. But ontological arguments try to prove that something (god) exists without committing to the existence of any specific thing. They can therefore be roughly divided into several types:- (a) ones that assume that god exists from the get-go, but disguise it in some way;
(b) ones that make an error in existential calculus, so are not sound;
(c) ones that are correct but trivial, e.g. showing that the sum of all things exists in some sense.

The hope of ontological calculus is that correct but not-trivial truths about the necessary structure of existence can be derived that will imply the existence of a specific being (god), but this should not be possible without committing to some structured set of existing things - which is not something ontological arguments are wont to do.

While there are many ontological arguments, this main article on the topic will deal Alvin Plantiga's modal argument. Other arguments are often rephrasing of the same ideas, so the treatment offered below should suffice for most ontological arguments. There are a few exceptions, e.g. mereological arguments that argue that the sum of all things exist.

Plantiga's Modal Argument[]

The most modern form of the argument is perhaps the modal argument due to Alvin Plantiga [2]. It goes something like this [3] Note: even if the argument were valid it could not prove the god of any specific religion. Restricting as maximally great being to what the Bible says or to what one specific Religion/Denomination says looks like an unacceptable restriction.

1 The concept of a maximally great being is self-consistent. A being is "maximally great" if it is omnipotent, omniscient, and omnipotent in all possible worlds.
2 If 1, then there is at least one logically possible world in which a maximally great being exists.
3 Therefore, there is at least one logically possible world in which a maximally great being exists.
4 If a maximally great being exists in one logically possible world, it exists in every logically possible world.
5 Therefore, a maximally great being (i.e., God) exists in every logically possible world.

The argument is discussed in terms of "possible worlds", which may be hard to follow for the uninitiated. It is perhaps best to present a simplified argument, following [2] [4]:

D1 (Definition 1) G is Necessary, which is to say that if it can exist then it does
A1 (Assumption 1) It is possible that G exists
T1 (Theorem 1) From D1 and A1, G exists
C1 (Conclusion) G exists

Plantiga's argument therefore falls under categories (a) and (b) above. It implicitly assumes that god exists in proposition (1) or (A1), and reflection will reveal that this it treats the possibility of G in conflicting ways (an error in existential calculus). D1 says that G is not contingent, that it can't be that god may happen to exist in our world or may not (in the sense that it may be that I'll be hit by a car tomorrow or it may not). A1 says that it is possible that god exists in that sense, i.e. that God's existence is a contingent fact. A proper formulation will say, perhaps,

D2 A Necessary-God is Necessarily True or Necessarily False
A2 To our knowledge, god may be a Necessarily-True Necessary-God
C2 We can't say whether or not god exists

Modal Logic[]

The ontological argument is always confusing. Plantiga's argument is couched in the language of possible worlds. I am not sure this aids the understanding, but I will proceed to discuss it in these terms. To accomplish this, I will spend this section on explaining what possible worlds are, highlighting certain finer points that the argument exploits to fool the reader.

Consider a set of elements (x,y,z...). Each element represents one putative element of reality (an atom, Rome defeating Carthage, whatever). Some of these contradict each other.

A possible world is some consistent partial set of these elements, i.e. some collection of elements that can fit together without any contradiction. An element in it is said to exist in that possible world. There is one possible world that is special - that is the Real World, our world. Elements in it are said to exist, plain and simple.

Each possible world is accompanied by a (transcendent) shadow-world of "propositions", or "properties" or "premises". The notation "Px" means that x has the property P in the particular possible world under discussion.

The argument revolves around another type of property - broad properties, properties that pertain to how elements are spread across all the possible worlds. Three key properties of this sort are possibility, contingency, and necessity. An element x is (logically) possible if it exists in at least one possible world, contingent if it also does not exist in at least one possible world, and necessary if it exists in all possible worlds.[4]

Normally when we speak of possibility, we speak of logical possibility. When we say x is possible we mean that there is at least one possible world where x exists; when we say Ox is possible we mean there is at least one shadow-world where Ox is true, and so on. But when it comes to these broad properties, things are different. When we say that it is possible that x is necessary we are not saying that "x is necessary" is correct in some possible world.[5] Rather, we are saying that we don't know whether the structure of possibilities is such that x is necessary. This kind of possibility is epistemic possibility, i.e. relating to our knowledge.

For concreteness, I'm going to use an example throughout the following. In Our Example, the only possible elements are whole positive integers. So the possible worlds look like {1,2}, {1,5,10}, and so on. A simple property O might be Odd, so that Ox is true if x=1 but not for x=2. A broad property might be Sx, "x is the smallest element in all worlds", which is almost true of x=1 (it is only true in worlds where the element "1" exists).

The Definitions[]

The argument is founded on two definitions.

An element of reality x is said to be Maximally Excellent (Ox) if it has that property, O. "O" is taken to mean things like "Omnipotent", "Omniscient", "Perfect", or so on - and may be defined in relation for other things in that possible world. For example, an integer x might be O if it is 8 (Octal), or if it is larger than any other element y in that possible world, or so on.
An element of reality x is said to be Maximally Great (Gx) if it is maximally excellent in every possible world.

For concreteness, let us consider Our Example. In Our Example, the elements of reality are integers. If we take O to mean "larger than any other element in that world", then x=10 is Ox in the possible world {1,5,10}. It is clear that there is no element x that is Gx in Our Example, for two reasons: (a) for any x, there are possible worlds that won't contain x, so specifically Ox will not hold there; and (b) for any x, there will always be a world with x+1 and x, and Ox won't be true in that world. Our Example alone suffices to show that the argument is not convincing - we can imagine scenarios where there is no x so that Gx, so even if the argument was valid we would be wise to reject its premises if this seems more reasonable than accepting that god exists (which, for an atheist, would be just about any premise).

The Argument[]

With all this in mind, let us return to Plantiga's argument, in its full formulation. I shall begin with point (4),

4 If a maximally great being exists in one logically possible world, it exists in every logically possible world.

While true, it is circular. A maximally great being is an element x that is Ox in all possible worlds, which requires that it exist in all possible worlds. (4) is therefore not much of a point, it is just elaborating the fact that Gx defines that x is necessary. Therefore (4) is equivalent to (D1), it is stating that we are talking about a Necessary-God.

1 The concept of a maximally great being is self-consistent.

What does "self-consistent" means here? A thing is self-consistent if you cannot, by assuming it, derive a contradiction. Let us assume that Gx in Our Example. Since this is mistaken, one can derive a contradiction from it; it is therefore not self-consistent. For Necessary Truths, self-consistency is the same as saying that they are true. What is going on here?

Plantiga seems to take "self-consistent" to mean that it is logically possible that Gx. But as shown above this is just not a correct way to treat the possibility of a necessary truth (or any broad property). The proper sense of "possible" here is epistemic possibility.

In Our Example, it is not correct that there is an x so that Gx, so Gx is False and we may be indeed tempted to say it is "not self-consistent". But this is equating "self-consistent" with "true" for necessary propositions. If considered epistemically, however, saying that Gx is self-consistent is saying that we can imagine that there is a set of possible worlds where it is correct. Perhaps a greater understanding of what is possible will make the apparently impossible possible (like "a triangle necessarily has 200 degrees", that becomes possible in a non-Euclidean geometry), or perhaps we aren't sure and want to cover all options. At any rate, treating "self-consistent" as "true" for a Necessary thing is not appropriate.

Far from being a modest assumption of god being at least possible, Plantiga's interpretation makes (1) the assumption that theism is not only true, but that it is necessarily true; it is the assumption that there is no way atheism is correct. All of this should simply be avoided, and the epistemic interpretation followed.

Assumption (1) therefore is (A1) in the shorthand analysis: it is possible that god exists. And while the original talks about logical possibility, this should be replaced with epistemic possibility. The proper assumption is

1B It is epistemically possible that there is an x so that Gx

Is this a reasonable assumption? I'd say it isn't, not given that Plantiga puts Omniscient, Omnipotent, and Perfectly Moral as his Ox. Unpacking these terms is an entire side-trek which I won't go into, but I will say that nothing can be Omniscient because information is physical which makes self-reference an insurmountable difficulty, that nothing can be Omnipotent since that would mean it has no nature, and that nothing can be Perfectly Moral since morality is composed of conflicting elements. Even if all these objections were set aside, I think a possible world (indeed many) can be built without even a being as wise, powerful, and moral as a human - let alone god. So I don't think 1B is a reasonable assumption.

2 If 1, then there is at least one logically possible world in which a maximally great being exists.
3 Therefore, there is at least one logically possible world in which a maximally great being exists.

The falsity of (2) is now apparent. That it is epistemically possible that Gx (i.e. 1B) does not imply that x exists in one possible world. The situation is that either x exists in all worlds and is Ox in them (Gx True), or that it doesn't exist in all worlds and/or isn't Ox in some (Gx False); both are possible in the sense that we don't know how the space of possibilities is built, not in the sense that there is an x in a possible world so that "x exists in all worlds and is Ox in them" (logical possibility) - that's a category error, x existing in a possible world cannot represent the possibility that x exists in all possible worlds.

In Conclusion[]

It has been revealed that the argument is problematic on four grounds:

A) It defines God as a Necesary being which is necessarily O, and then "derives" that god is a necessary being. This isn't so much a flaw as it is misleading.

B) It assumes that which it seeks to prove, namely it assumes that the Necessary-God is logically possible, which denies the very possibility that god doesn't exist. Instead of wrestling with the claim that god doesn't exist and showing it is false, the argument a-priori assumes that it is false.

C) It relies on a confusion between epistemic and logical possibility. As a broad proposition the Necessary-God needs to be treated as an epistemic possibility, leading to a weakened 1B which cannot support the rest of the argument.

D) Even the weakened 1B should not be accepted.

Like all Necessary postulates, that a Necessary-God exists is either true in all possible worlds or not true in all possible worlds. The argument doesn't advance the position that it is true in all possible worlds.

Plantiga doesn't claim the argument shows god exists. Rather, he claims that it shows that it is reasonable to believe it does since assumption (1) is reasonable. We have shown that assumption (1) is anything but reasonable - as Plantiga understands it (relating to logical possibility), it is the assertion that atheism is a priori not possible, not the much milder assumption that it is merely possible that god exists. If (1) is taken in its proper (epistemic) sense (1B), it is nothing but admission of ignorance and doesn't advance the argument one iota. And even this weakened assumption ought to be rejected.


  1. Stanford Encyclopedia of Philosophy [1]
  2. 2.0 2.1 Philosophy of Religion [2]
  3. Internet Encyclopedia of Philosophy [3]
  4. Notice that "exists" is a broad property, as it pertains to how x is spread in possibility space - it is part of the Real World. Hence the idea that broad properties are not properties (i.e. that they require special treatment) implies that existence is not a property, which is Kant's famous objection to Anselm's ontological argument.
  5. To accommodate such speak technically requires extending our model. For each broad property we can consider it and its negation as a meta-element, and construct meta-possible worlds out of their combinations. Within each meta-possible world there is a host of possible worlds. Only one meta-possible world is really possible; it is just that we don't know which, so we need to use the meta-possibility as a means of representing this lack of knowledge. Instead of complicating things in this way, I've simply noted the difference between the two types of "possible".