Ontological argument

The ontological argument attempts to prove that a "maximally great being" must exist. The ontological argument was first proposed by St. Anselm in his book Proslogium in 1077. Since its inception, the ontological argument has been subject to many criticisms and continues to be debated about today. The ontological argument was created by Anselm as an attempt to supply Christians with some sort of arguable foundation for the belief in God which they already possessed.

Logical proofs
All versions of this argument boil down to:

P1: God is a maximally great being.

P2: It is better to exist in reality than to exist conceptually.

C1: God exists.

Anselm's
What then are You, Lord God, You than whom nothing greater can be thought? Anselm's reductio ad absurdum argument hence runs as follows:


 * 1) Assume God does not exist.
 * 2) 'God' is defined as "that than which no greater can be conceived".
 * 3) "That than which no greater can be conceived" must therefore not exist (from 1 & 2).
 * 4) "That than which no greater can be conceived" exists only in imagination, not in reality (from 2 & 3).
 * 5) If "that than which no greater can be conceived" were to exist in reality as well as in imagination, it would be even "greater".
 * 6) But that would mean "That than which no greater can be conceived" is not "that than which no greater can be conceived" (from 4 & 5).
 * 7) "That than which no greater can be conceived" must exist in imagination and also exist in reality for it to be the greatest thing conceivable.
 * 8) That means 'God' both does and does not exist (from 1 & 7).
 * 9) Premise 1 cannot be true (reductio ad absurdum).
 * 10) 'God' exists.

Like other medieval examples of its kind, the ontological argument was not intended to simply argue for the existence of 'God' in the usual sense — something that would have been taken for granted by most contemporaries — but rather to investigate the divine from a certain philosophical and theological point of view.

The film Monty Python's The Meaning of Life made light of this argument with this bit of dialogue: Chaplain: Let us praise God. O Lord,… Congregation: O Lord,… Chaplain: ...ooh, You are so big,… Congregation: ooh, You are so big,… Chaplain: so absolutely huge. Congregation: so absolutely huge. Chaplain: Gosh, we're all really impressed down here, I can tell You. Congregation: Gosh, we're all really impressed down here, I can tell You. Chaplain: Forgive us, O Lord, for this, our dreadful toadying, and… Congregation: And barefaced flattery. Chaplain: But You are so strong and, well, just so super. Congregation: Fantastic. Humphrey Williams: Amen.

Gödel's
Mathematician Kurt Gödel also advanced a version of this argument. There are ongoing efforts to formalise it, which have had some success. Thus, the logic is valid &mdash; but the objections to the axioms are the same as for Anselm's. His goes like this:
 * Definition 1: x is God-like if and only if x has as essential properties those and only those properties which are positive
 * Definition 2: A is an essence of x if and only if for every property B, x has B necessarily if and only if A entails B
 * Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified
 * Axiom 1: If a property is positive, then its negation is not positive
 * Axiom 2: Any property entailed by—i.e., strictly implied by—a positive property is positive
 * Axiom 3: The property of being God-like is positive
 * Axiom 4: If a property is positive, then it is necessarily positive
 * Axiom 5: Necessary existence is positive
 * Axiom 6: For any property P, if P is positive, then being necessarily P is positive
 * Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified
 * Corollary 1: The property of being God-like is consistent
 * Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing
 * Theorem 3: Necessarily, the property of being God-like is exemplified

Plantinga's
Alvin Plantinga has a currently-popular version of Godel's modal logic argument that uses a modal system of possible but not-actual worlds to give the appropriate semantics for statements about possibility & necessity.

To understand this argument, one must understand several concepts Plantinga uses, first, the concept of world-indexed properties. World indexed properties are properties that a being must have in order to exist, they're essential properties for said being. For example, if 'x' is the name of the actual world, and if I had the property of being fat in 'x', if I existed in some possible but not-actual world, in every world in which I exist, I have the property of being fat.

With this established Plantinga then brings out two more terms:
 * Maximal excellence: a being which has the property Maximal Excellence is omniscient, omnipotent and morally perfect
 * Maximal Greatness: a being which has the property Maximal Greatness is Maximally excellent in every world.

With that established, the argument is put forth as such:


 * 1) A being has maximal excellence in a given possible world W if and only if it is omnipotent, omniscient and wholly good in W; and
 * 2) A being has maximal greatness if it has maximal excellence in every possible world.
 * 3) It is possible that there is a being that has maximal greatness. (Premise)
 * 4) Therefore, possibly, it is necessarily true that an omniscient, omnipotent, and perfectly good being exists.
 * 5) Therefore, it is necessarily true that an omniscient, omnipotent and perfectly good being exists (axiom S5).
 * 6) Therefore, an omniscient, omnipotent and perfectly good being exists.

Craig's
William Lane Craig defends Plantinga's ontological argument for the existence of God, rendering it as:


 * 1) It is possible that a maximally great being (God) exists.
 * 2) If it is possible that a maximally great being exists, then a maximally great being exists in some possible world.
 * 3) If a maximally great being exists in some possible world, then it exists in every possible world.
 * 4) If a maximally great being exists in every possible world, then it exists in the actual world.
 * 5) Therefore, a maximally great being exists in the actual world.
 * 6) Therefore, a maximally great being exists.
 * 7) Therefore, God exists.

Responders
Gaunilo of Marmoutier was the first to respond to the ontological argument. Gaunilo argued that, following Anselm's absurd logic, it is impossible to imagine an island of unrivaled beauty without such an island existing in reality. Gaunilo's writing forced Anselm to admit that his argument depended on the ambiguity of its terms, and began a philosophical tradition of responding to the ontological argument which would be continued by Immanuel Kant and Gottlob Frege.

Problems
The argument is fallacious due to several flawed assumptions.

"Great"-ness
The most noticeable of these is the assumption that that which exists in reality and imagination is somehow "greater" than that which exists only in imagination. "Greater" and "greatness" as a quality is not at all defined in this context, and it is only the far overreaching manner in which the term is applied that allows this argument some semblance of logical appeal. The first counter to the argument was developed by Gaunilo of Marmoutier in the eleventh century.

Anselm performed a bit of sloppy reasoning by assuming that there was a difference between our concept of a God and a God which exists in fact, so that he could elevate the latter case as supreme. But if God can be shown to exist through means other than pure reason (such as by direct observation, or historical veracity), then his existence is automatically incorporated into the true concept of God. We can have false concepts of God all we want, but the true concept of God always tracks the status of God in reality, whether he exists or does not. So it is never possible to demonstrate the existence of God purely by juggling our definitions of God and making a word salad, which is what the ontological argument is all about. So, while it may be assumed that Anselm merely intended to show how the concept of the ultimate entity (God) includes necessary existence, this is not clear from the form of the argument. In any case, it appears that Anselm meant to ask how, if something (as opposed to nothing) exists necessarily, can existence be deduced not to be a property even of that which exists contingently.

The weakness of "greatness" opens up the argument to further refutations. What happens if two people disagree on what makes something "great"? If a xenophobe comes up and says, "maximally great includes maximal hatred for conscious life", what argument can be presented for that not being an actual quality of greatness? In fact, what argument at all is put forward for how we determine what is greatness? Consider the claim: "something that is maximally great cannot be denied." The argument suddenly becomes a reductio ad absurdum with the simple addition of that and "I deny god."

If this is supposed to follow from the definition of "maximally great being," then that definition needs substantial defense. Otherwise, it is question-begging. It suffers from the same problem as St. Anselm's: existence is not a real predicate. A being that exists in every possible world is not greater than a being who does not exist in every possible world.

Strong atheism
One has no reason to accept the possibility of premise (3). Further arguments must be given to make the possibility premise plausible, and therefore the argument is essentially useless from a natural theological perspective (Plantinga took note of this fact, and admitted that it was not a successful piece of natural theology).

Counter-proof
Another objection to the argument is also quite simple: one could change the possibility premise, and flip the argument on its head:


 * 1) A being has maximal excellence in a given possible world W if and only if it is omnipotent, omniscient and wholly good in W; and
 * 2) A being has maximal greatness if it has maximal excellence in every possible world.
 * 3) It is possible that there isn’t a being that has maximal greatness. (Premise)
 * 4) Therefore, possibly, it is necessarily true that an omniscient, omnipotent, and perfectly good being does not exist.
 * 5) Therefore, it is necessarily true that an omniscient, omnipotent and perfectly good being does not exist. (axiom S5)
 * 6) Therefore, an omniscient, omnipotent and perfectly good being does not exist.

The argument is completely valid, and suffers from the same flaw as Plantinga’s. There really is no reason to accept the formulation of the possibility premise (3), either – without additional arguments, at the very least.

"Maximal excellence"
The late Philosopher of Religion from Boston University Michael Martin objected to the concept of “maximal excellence” altogether. He claims that the concept is contradictory, and therefore logically impossible. The way to formulate this type of objection is to form multiple property disproofs (or maybe even single property disproofs) of the orthodox theistic god-concept. There are a few of these types of arguments in earlier posts on this blog.

Begging the question
Richard M. Gale, a metaphysician from the University of Pittsburgh, claims that the possibility premise begs the question. Basically, one is not justified in an epistemic sense to accept the possibility premise unless one also understands the nested modal operators in system S5. Within the modal system S5, “possibly necessary” means the same as “necessarily”. Since the concept of a being with “maximal excellence” entails this being’s necessary existence in a possible world, the possibility premise (3) contains nested modal operator “possibly necessary”. Since “possibly necessary” is equivalent to “necessarily” (within the system S5 that Plantinga needs for his argument to even get off the ground), the argument begs the question in the possibility premise (3), since the premise contains the conclusion within itself.

Metaphysical vs epistemic possibility
The modal ontological argument, in some presentations, relies on an equivocation between metaphysical and epistemic possibility. It may very well be that the existence of a maximally great being is epistemically possible (i.e. we don't know that it's false) but not metaphysically possible (i.e. non-contradictory). If the concept of a maximally great being is not self-consistent, then it is not metaphysically possible for such a being to exist. Compare: we don't know whether the twin prime conjecture is true or not. Suppose it is false but we don't yet know it; it follows that it is (metaphysically) necessarily false. We might nevertheless agree that it might be true because we don't know its truth value.

The issue with the metaphysical possibility as it relates to the first three premises can be clearly shown with a competing version of the argument:
 * 1) It is possible that a maximally great being (God) does not exist.
 * 2) If it is possible that a maximally great being does not exist, then there is some possible world where a maximally great being does not exist.
 * 3) If a maximally great being exists in some possible world, then it exists in every possible world.
 * 4) A maximally great being does not exist in every possible world (from 2).
 * 5) Therefore, a maximally great being (God) does not exist.

This further highlights that the argument has two likely sources of error: with the construction of the argument in general (in which case the argument is not useful for proving anything) or a problem specific to the first premise (in which case the possibility of the existence or non-existence of the character God must be defended with further arguments). Of course it is also entirely possibly the problem lies in both areas, and it is neither possible to prove an actuality from a mere possibility or accept a possibility without supporting empirical evidence.

Specificity
Perhaps the simplest objection to this argument, which works when it is used to justify a particular monotheistic religion, is that, even ignoring any problems with the axiomatic system required for its soundness, it proves nothing whatsoever about any properties of God beyond existence and "maximal greatness" - whether the one true god is YHWH, Allah, Satan, Ahura Mazda, Mahavishnu, Sithrak the Blind Gibberer or J. R. "Bob" Dobbs is as open a question as it has ever been. This makes it rather useless in apologetics specific to any particular religion.

Like many other arguments for God's existence, this reasoning is nonsensical because it could also be used for just about anything. The argument can be completely made laughable simply by changing "God" to "The Most Perfect Island" (or something similar). The argument remains structurally valid (that is, nothing in the symbolic formulation of the argument is incorrect), however, we come to the laughable conclusion that "The Most Perfect Island" must exist.

Similarly, you could replace "God" with "Unicorns" and define "Unicorns" as "that than which no greater horse can be conceived". We now have an argument for the existence of unicorns, another mythological creature.

Alternately, as SMBC does, you could prove that you totally have a girlfriend from Canada.

However, if you try to substitute God in this way, the apologist will typically say that a perfect, and thus metaphysically necessary unicorn cannot exist, because it needs space to exist. The apologist here is simply finding possible worlds where the unicorn could not exist. In a similar way, you can say that there are possible worlds where God cannot exist. An example of a possible world where all there is, are innocent people being tortured, and God could not exist in this world because he would've stopped it. If the apologist says that this world cannot exist simply because God exists, because he would've stopped it, then what would stop someone from saying that there is a metaphysically necessary unicorn, because there are no possible worlds without space and time, because the metaphysically necessary nature of the unicorn would "shape" the other possible worlds to allow for its existence (by having it so every possible world had a space where the unicorn would exist.), in the same way that God existing would "shape" possible worlds to allow for his existence (by not allowing all of those innocent people to be tortured).

Treating existence as a predicate
Kant objects that existence is not a predicate. A hundred thalers that I merely imagine, he says, have all the same predicates as a hundred real thalers.

A more formal objection is that the ontological argument implicitly jumps between using existence as a quantifier (∃x) over the domain of real objects, and existence as a predicate (Exists(x)) over the domain of imaginable objects, that is, a property that an object in the domain may or may not have. When you combine these in the same argument, weird things happen that can lead to invalid proofs.

As an example whose flaw is more evident, consider this argument. We want to prove that there exists a unicorn. It suffices to prove the stronger claim that there exists an existing unicorn: after all, an existing unicorn is a unicorn, therefore, if an existing unicorn exists, then a unicorn exists. There are two possibilities:


 * 1) An existing unicorn exists.
 * 2) An existing unicorn does not exist.

Possibility 2 is clearly contradictory: how can an existing unicorn not exist? Therefore, possibility 1 is true, so an existing unicorn exists. Therefore, a unicorn exists.

What's wrong with this argument?

There are two problems with it. One is the substitution of the domain of discourse. "There exists a unicorn" is, implicitly, a statement in the domain of real objects, where existence is a quantifier, ∃x. "An existing unicorn is a unicorn" is, implicitly, a statement in a domain that includes all imaginable unicorns, where existence is a predicate that may be true or false for a given object: ∀x(Exists(x) & Unicorn(x) ⇒ Unicorn(x)), which is trivially true.

The second problem is that the natural language statement "An existing unicorn exists" is ambiguous. There are two possible interpretations of it:


 * 1) ∀x(Exists(x) & Unicorn(x) ⇒ Exists(x)) — "anything that is an existing unicorn exists".
 * 2) ∃x(Exists(x) & Unicorn(x)) — "there exists a unicorn with the property of existence".

Interpretation 1 is trivially true, and its negation indeed leads to a contradiction, but it is entirely possible for the statement to be vacuously true — that is, true simply because the set of objects satisfying the antecedent is empty.

Interpretation 2 is the "interesting" one in that it's not trivially true. However, its negation, ~∃x(Exists(x) & Unicorn(x)), does not result in a contradiction: it is entirely possible that there does not, in fact, exist a unicorn with the property of existence.

In a similar vein, we can now point the flaw in the following argument:


 * 1) I define God as an entity that possesses all desirable properties.
 * 2) Existence is a desirable property.
 * 3) Therefore, God has the property of existence.
 * 4) Therefore, God exists.

The problem with this argument is that even if we quantify over all imaginable objects and treat existence as a property, and even if we can meaningfully define which properties are "desirable", and even if you agree with this definition of God, the argument is still logically invalid. Statement 3 actually means "anything satisfying the definition of God has the property of existence", while statement 4 means "there exists an entity satisfying the definition of God". Since statement 3 can be vacuously true, statement 4 does not follow from it.

Or, rewritten formally in second-order logic,


 * 1) ∀x(God(x) ⇔ ∀P(Desirable(P) ⇒ P(x))) (axiom)
 * 2) Desirable(Exists) (axiom)
 * 3) ∀x(God(x) ⇒ Exists(x))
 * 4) ∃x(God(x))

Here, inferring 3 from 1 and 2 is logically valid, but inferring 4 from 3 is not; 3 can be vacuously true.

A similar rebuttal of the ontological argument was provided by Immanuel Kant in his .