Debugging the ontological argument

I'm introducing a new series on the ontological arguments for the existence of God.  Ontological arguments are basically attempts to prove that God exists using pure reasoning alone.  For example:

God is defined to be the most perfect being.
It is more perfect for something to exist than to not exist.
Therefore, God exists.

However, this is just one ontological argument, and I will cover several more, culminating in Godel's ontological argument.

This is the masterpost for the series, containing a complete list of links, as well as an explanation of my purpose and perspective.

Outline of series:

• The definitional ontological argument
• Is existence a predicate? 
• These things do not exist
• The conceptual ontological argument
• Making sense of existence
• The uselessness of the ontological argument
• Interlude: Aesthetics in the ontological argument
• Plantinga's modal ontological argument
• Modal logic semantics
• Addressing the argument
• Gödel's ontological argument
• Gödel's workarounds to the ontological argument
• Consistency and the material conditional
• Gödel's positive predicates
• Improving on Gödel

Purpose:

The primary purpose is to talk about logic.  When we discuss apparent paradoxes, I find that many beautiful details about logic come out of the woodwork!

I also know there are lots of people out there who are just plain curious about the ontological argument, so my secondary purpose is to satisfy your curiosity.

I am not here to merely show that ontological arguments are wrong.  If I wanted to persuade people, I could simply assert the argument's absurdity--this is essentially the strategy adopted by Richard Dawkins in The God Delusion, for instance. The fact of the matter is that a vanishingly small number of people think ontological arguments are correct.  While I'm sure a few proponents will show up in the comments, most people look at ontological arguments and immediately find them absurd and unpersuasive.

My purpose here is to pinpoint errors in the ontological arguments.  I also call it debugging the argument, as if to go line by line in a computer program and find the missing semicolon. Although unlike debugging a computer program, I will not actually correct the errors, since ontological arguments are fundamentally unfixable.

There are definitely ways to refute ontological arguments without pinpointing the problems.  For example, an argument ad absurdum extrapolates the argument to reach an absurd conclusion:

Ted is defined to be the most excellent being.
It is more excellent for something to exist than to not exist.
Therefore, Ted exists.

But even if you accept the argument ad absurdum, that doesn't tell you what is wrong with the argument.  It's like executing a program and encountering an error--it tells you that there's a bug, but doesn't tell you where.  Thus, in this series, I will refer to, but not rely on argument ad absurdum.

Perspective:

I am a random person on the internet.  I am not here to explain historical refutations of the ontological argument, I am offering my own original vision.  For a more scholarly treatment, I recommend more formal resources such as Wikipedia or the Stanford Encyclopedia of Philosophy.

My perspective is more that of a mathematician than a philosopher.  Most ontological arguments can be framed in logical terms, and my writing will focus on these terms.  Some ontological arguments add rhetorical bells and whistles to the logic, and privately I view these ornaments as unhelpful at best and obscurantist at worst.

Expertise:

Unless you count my minor in math, my expertise here is basically self-taught. I taught myself philosophical concepts as necessary, sometimes consulting my boyfriend, who has a degree in philosophy.  Many years ago, I taught myself modal logic just so I could understand modal ontological arguments.  I wrote a bunch of blog posts about it (see references below), and argued endlessly with a few ontological argument proponents.

Basically, you shouldn't trust me based on my expertise, but based on my arguments and reasoning.

Comment Policy:

If you have comments that do not pertain to any specific post, please comment here on the masterpost.  I will not reply to all comments.  I do not promise to be polite.

Throughout this series I will use MathJax to render LaTeX equations.  This requires that you enable Javascript.  LaTeX will not work for comments, so in the comments I will adopt the following conventions:

E - existence quantifier
A - universal quantifier
~ - negation
^ - and
V - or
=> - implication (will be rendered as =>)
&lt;=&gt; - equivalence (will be rendered as <=>)
N - necessary
P - possible
-&gt; - entailment (will be rendered as ->)
|= - satisfaction relation

I encourage readers to adopt the same conventions to avoid confusion.

Complete list of old posts
This is all the stuff I've previously written on the ontological argument.  I'm afraid some of it is not very clear, and I may have changed my mind on some of the details.  This new series is intended to be my definitive word on the subject.

A simple ontological argument
A modal ontological argument
Godel's ontological argument
Godel's ontological argument, step by step
Modal ontological argument, revisited
Plantinga responds to me
An anecdote on obscurantism
Necessity isn't so necessary

Godel's workarounds to the ontological argument

This is part of my series on debugging the ontological argument.

In the previous post, I discussed Plantinga's modal ontological argument, which is about the simplest modal ontological argument you can get.  A simply-stated argument is quite valuable, but I believe that as a matter of technical feat, Plantinga's argument was surpassed by Gödel's Ontological argument (GOA).

This series does not focus on history, but I will dryly summarize a few established facts.  Gödel formulated the argument in 1941, but only told people about it in 1970.  Gödel did not actually believe that the argument proved God, and was afraid that publishing it would send that message.  He was more interested in it from the perspective of logical investigation (which is my interest as well).  It was published posthumously in 1987, which as far as I know predates Plantinga's argument.

Long story short, I've gotten several people who said stuff like (actual quote), "The first premise you must accept is that Godel was almost certainly much, much, smarter than you."  And besides being the worst kind of argument from authority it's also historically wrong about Gödel's motivations.

But it is true, after all, that Gödel was very smart, and I find the GOA to be quite clever in ways that are not immediately obvious.  This post will focus on several things that the GOA got right.

1. Possibility is proven, not assumed

In the conceptual ontological argument, we had difficulty with the premise that God was "conceivable" in the sense required by the proof.  In the modal ontological argument, we had difficulty with the premise that God was "possible" in the sense required by the proof.  In the GOA, we don't need to assume that God is possible.  Instead, we prove it from more basic premises.

The basic structure of the GOA is as follows:
1. We define what is a positive property.
2. Using the definition of positive properties, we prove that positive properties are always consistent.
3. We prove that consistent properties are always possible.
4. The property of being God is taken to be positive.
5. Therefore God is possible.
6. Therefore God is necessary (this part of the proof is identical to Plantinga's argument).

If you're interested in the particulars, I wrote out the entire ontological argument years ago.  The goal in this series is to explain the steps in more depth rather than detail.

2. Existence is a predicate

Earlier in this series, I wasted much time with an in depth discussion of whether existence is a predicate. In a first analysis, it cannot be a predicate, or at least not a very useful one.  In later analysis, I note that existence can be a predicate after all, under certain conditions.  One possible condition is if we use second order predicate logic (SOPL), and that is what the GOA does.

In more detail, the GOA defines necessary existence as follows:
$$E(x) \Rightarrow \forall\phi~ (\phi~ess~x \Rightarrow \square \exists y~ \phi(y) )\tag{1}\label{ref1}$$
In plain language, necessary existence (E) means that in all (accessible) possible worlds, there exists an object with all exactly the same properties.  This requires SOPL since it applies a quantifier to a predicate ($\phi$).  In my opinion, this completely answers Kant's objection to the ontological argument.

You might notice that \ref{ref1} includes the phrase "$\phi~ess~x$" which means "$\phi$ is the essence of x".  This simply means that $\phi$ is the conjunction of all properties of x.

3. God's definition is no longer fluff

Earlier in the series, I complained that many version of the ontological argument have completely pointless augmentations.  They define God as a maximally great being, or a being with all the perfections, and this uses far more assumptions than is necessary.

In the GOA, God is defined as a being which has the conjunction of all positive properties, but this is actually a crucial component of the proof!  The entirety of the proof hinges on the definition of "positive properties", and there isn't an obvious way to reformulate the argument without it.

In fact, it almost seems too powerful.  One of the objections to the GOA is that God would even have positive properties that describe the world that God lives in, rather than God himself.  For instance, God would have the property that "trivialknot likes cake."  And since God necessarily exists, God would have this same property in all possible worlds, implying that I like cake in all possible worlds.

Because Gödel cleverly addressed so many of the standard problems with other ontological arguments, I will have to devote several more posts to the particular flaws in his argument.

Aesthetics in the ontological argument

This is part of my series on debugging the ontological argument.

Let's take a little break from ontological arguments, and switch to math.  Consider the following proof.

Theorem: Given a square, opposite sides are parallel.
Proof:
By definition, a square is a rectangle with all equal sides
By definition, all the corners of a rectangle are right angles.
In the following figure of a rectangle, line AB and CD are cut by transversal BC.
By the consecutive interior angles theorem, AB and CD are parallel.

There's something very wrong with this proof.  Logically speaking, it's completely valid and sound, and I can't really think of a better way to prove the theorem it intends to prove.  But think about it:  Why are we only proving the theorem for squares?

The entire proof rests on the fact that squares are a special kind of rectangle. It's obvious that if we just replaced "square" with "rectangle", we could prove a more general theorem, and use fewer steps.  "Given a rectangle, its opposite sides are parallel."

This isn't an issue of logic.  It's an issue of aesthetics.  My mathematical training taught me to make the fewest assumptions necessary.  My training taught me to prove more general cases when I am able.

And that's why looking at the definitional ontological argument, I boggle.
$$\text{God is defined to have all the perfections.}\tag{1a}\label{1a}$$ $$\text{Existence is a perfection.}\tag{1b}\label{1b}$$ $$\text{If something exists by definition, then it exists.}\tag{1c}\label{1c}$$ $$\text{Therefore, God exists.}\tag{1d}\label{1d}$$
Earlier, we placed all our focus on the inference \ref{1c}, because that's the most questionable step.  But let's also take a moment to focus on the abysmal aesthetics of \ref{1a} and \ref{1b}.

The entire proof is based on the fact that God exists by definition.  Why, then, is it necessary to assume that God has all the perfections?  It is sufficient to assume that God just has one perfection--existence.¹  If \ref{1c} were a correct inference, then we could prove a more general theorem, not just the theorem about God.

This is a problem for most ontological arguments.  They define God to have all the perfections, or to be the most supreme being imaginable.  In Alvin Plantinga's version, he goes so far as to define a "maximally great being" as omnipotent, omniscient, and morally perfect in every possible world.

Any decent mathematician would look at these words and say, "Why bother?"

Florid prose, or obscurantism?

I don't know what runs through the minds of philosophers who state ontological arguments with patently extraneous assumptions.  However, the florid definitions of God do serve a purpose, whether intentional or not.

The purpose is to prove only the special case of God, and draw attention away from the more general theorem.  Like with the theorem about squares, it's as if the mathematician wants you to only think about squares, and wants you to conveniently forget that the same theorem also holds for rectangles.

If people realized that the ontological argument works equally well for a large class of objects and not just God, then they might just realize that the ontological argument proves some things which are completely ridiculous.  For example, The Flying Spaghetti Monster is the most perfect cluster of noodles imaginable.  Therefore it exists.

I've made such reductio ad absurdum arguments before, and I find that ontological argument proponents dismiss them for various reasons.  For example, they would say the Flying Spaghetti Monster is inconceivable, or that it is conceptually inconsistent.

This is all missing the point.  When I look at the ontological arguments, it's clear that all the premises about God being the greatest and most perfect being are completely extraneous.  If ontological argument proponents truly think that the ontological argument only works to prove the existence of God and no other objects, then they clearly have some unstated premises.

In the next part of the series, I will be considering Plantinga's modal ontological argument.  However, the part of the argument where he talks about omnipotence and omniscience, that part will be dismissed and erased with contempt.

Maximization arguments

I should mention, for technical completeness, that sometimes the "florid" definition of God does some work after all, creating what I call the Maximization Ontological Argument.

In the Maximization Ontological Argument, God is specifically defined as the greatest thing that is conceivable.  Then it's asserted that a thing would be greater if it also exists.  I suppose this draws on the intuitive assumption that given any ordered set of things, there must be a maximum.  Of course, this assumption is just mathematically wrong (e.g. take the set of all integers, or the set of all irrational numbers less than one).

Even putting aside the fact that not all ordered sets have maximums, you can't simply assert things about a maximum on the basis that it would make it the maximum greater.  For example, consider the greatest even number less than 10.  Your first guess might be that the answer is 8, but consider that any number would be greater if you added 3 to it.  Therefore, 11 is the greatest even number less than 10.  I feel this argument is just flatly invalid, and doesn't really dignify in-depth discussion, so I'm going to move on.

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1. It's also worth noting that the extraneous assumption actively makes the proof weaker, because now we need the additional assumption that existence is a perfection.  

A simple Ontological argument

Today I will be considering a simple Ontological proof of God. Ontological arguments are basically attempts to prove God through concepts alone, without any real-world evidence. This particular variant has been fully discredited in modern times (and this is not just my atheist bias talking). This isn't so much an attempt to argue about God as it is an exercise in reasoning.

The Ontological proof is in three steps.

1. Definition: "God" is the greatest being imaginable.
2. Premise: It is greater to necessarily exist than to not necessarily exist.
3. Conclusion: God necessarily exists.

There is a very simple counterargument using what's called reductio ad absurdum. We show that if ontological argument works, various absurd results must also follow. Take the following example:

1. Definition: "Drod" is the greatest video game imaginable.
2. Premise: It is greater to necessarily exist than to not necessarily exist.
3. Conclusion: Drod necessarily exists.

Similarly, I could argue that the greatest sandwich exists, the greatest dad exists, and the greatest island exists. I do not even need to confine myself to objects of the form of "the greatest [blank] imaginable", nor do I have to assume any premises. Take the following:

1. Definition: The "IPU" is an invisible pink unicorn that necessarily exists.
2. Conclusion: the IPU necessarily exists.

I could continue to use this argument to prove the existence of anything, as long as I define it to necessarily exist. That is absurd. Unfortunately, while the reductio ad absurdum tells us that the ontological argument cannot possibly work, it does not tell us exactly where it goes wrong.

In my opinion, the central question of this argument is this: what sort of control do you have over objects when you define them? Normally, when I define something, I have full control over its properties. If I define a triangle to be a polygon with three sides, these properties must be true. There is no triangle with more than three sides, because if an object has more than three sides, it cannot be a triangle.

Why can't I do the same with the property "necessarily existing"? Let's try the same argument I tried with the triangle. "There is no God that doesn't necessarily exist, because if an object doesn't necessarily exist, it cannot be God." Aha! The argument with the triangle only proves that if there is a triangle, then it has three sides. Similarly, this argument only shows that if God exists, then God must necessarily exist. The Ontological Argument basically assumes what it is trying to prove. We fully control the properties of an object that we define, but we cannot control whether there is an instance of the object in the real world. (Some philosophers go on to say that existence cannot be a property of an object, but that distinction is irrelevant for now.)

In the spirit of critical-thinking fun, I shall attempt a similar argument going in the other direction. This isn't a serious argument, just what passes for humor in my mind. Any refutations?

1. Definition: "God" is the greatest being imaginable.
2. Premise: A being would be greater if its existence could be proven with the Ontological Argument.
3. Premise: The Ontological Argument is impossible.
4. Conclusion: God is impossible.
   

Gödel's modal ontological argument

Update 2015: I wrote a new series on ontological arguments.

Previously, I've explained a simple ontological argument, and also a modal ontological argument. However, there's another modal ontological argument which interests me: That which was proposed by Kurt Gödel. It's Gödel's ontological argument!

It's all so clear now! There must exist an x such that G(x)! (That is, God exists!)

What's that, you weren't convinced? Indeed, most skeptics would be unimpressed, since it looks so obscurantist. The argument seems to work solely on the principle of "blinded by Science!", or in this case, "blinded by Symbolic Logic!" Obviously, the proof doesn't really prove what it purports to prove, or the argument would be backed by philosophers everywhere. I certainly don't expect everyone to have the knowledge to be able to read the proof, much less pinpoint the precise location where it might go wrong.

However, as it happens, I had already taught myself a bit of modal logic, so I thought, why not try reading the proof, you know, for fun? I'm not going to go through it step by step (unless my readers really want me to), but instead, I'll outline the basic idea, and explain where I think a flaw is. [Update: By request, I went through the proof, step by step.]

The basic idea of Gödel's argument is based on the previous modal ontological argument. According to the modal ontological argument, God, by definition, necessarily exists. Therefore, if it's even possible that God exists, then God must exist in all possible worlds, including the one which we happen to live in. The problem is that we still have yet to show that God is possible.

And this is where Gödel comes in. Gödel supplies a proof of the fact that God is possible.

But first, we must define a "property". A property is an attribute of an object. For example, we might let "x" denote a particular apple, and "R" denote the property of being red. Thus R(x) is the statement "This apple is red." Gödel's central insight was to categorize some properties as "positive" and other properties as "not positive". We use P(R) to denote the statement "The property of being red is a positive property."

Gödel never tells us exactly what he means by "positive". That's because it doesn't really matter what exactly it means. The proof works, regardless. All that matters is that the positive properties obey a few axioms.

• Axiom 1: If property A is positive, and if property A entails property B, then B is positive.
  
• Axiom 2: If property A is positive, then the property not-A is not positive.
  
• Axiom 3: The property G is a positive property. (G is the property of being "God-like"; an object with property G has all positive properties)
  
• Axiom 4: If a property is positive, then it is positive in all possible worlds.
  
• Axiom 5: Necessary existence is a positive property.

The question is, is it even possible to construct a concept of "positive" such that it obeys all these axioms? Maybe, maybe not. The power in these axioms is that they seem so intuitive. If they were not intuitive, then I would simply disagree with the axioms, and be done with it. But since they are intuitive, I must pinpoint exactly why our intuition goes wrong. And then I can disagree with the axioms, and be done with it.

Let's focus on axiom 1. What does it mean for property A to entail property B? That means that for every possible object in every possible world, if it has property A, then it also has property B. Of course, if there is not a single object in any of the possible worlds with property A, then A automatically entails B.

For example, let's consider the property of being an invisible pink unicorn, which we will call U. Let us presume that U is an impossible property. There is not a single object in any possible world which has the property U. Therefore, U entails O, which is the property of being omnipotent. I mean, have you met any invisible pink unicorns which are not omnipotent? I doubt it. Similarly, U entails every property. All of them, every single one. U entails the property of being solid dark blue, the property of being a flying spaghetti monster, and the property of not being a flying spaghetti monster.

So if U were a positive property, then every property would be positive. Obviously, this is not the case (see axiom 2). Therefore, U is not a positive property.

But this contradicts our intuition of what makes something positive (and recall that these axioms were based on intuition in the first place). Surely, there are some things which are positive, but sadly impossible. For instance, the invisible pink unicorn may be impossible, but if she existed, she would surely be a force for good. Wouldn't I be justified in saying that U is a positive property? No, because if I assume U is "positive", then I am essentially assuming that U is possible.

You might notice that in axiom 3, we assume G, the property of being God-like, is a positive property. Here, we're basically assuming that G is possible. Oops!

[An aside: In discussions of Gödel's modal ontological argument, people will inevitably start talking about Gödel himself. I haven't really seen any evidence that Gödel necessarily saw this argument as anything more than an interesting exercise in modal logic. Which is, of course, how I see it.]

The uselessness of ontological arguments

This is part of my series on debugging the ontological argument.

Previously, I rebutted the conceptual ontological argument (COA).  But I want to make one more point about its soundness and validity.

Soundness and Validity

In logical/deductive arguments, we have the concept of validity, and soundness.  An argument is valid if its conclusions follow from its premises.  An argument is sound if its conclusions follow from its premises, and the premises are true. Consider the following arguments:

Socrates is a man.
All men are mortal.
Therefore, Socrates is mortal
(Valid and sound)

Socrates is a snake.
All snakes have eight legs.
Therefore, Socrates has eight legs.
(Valid but not sound, and conclusion is false)

Socrates is a flying submarine.
All flying submarines are mortal
Therefore, Socrates is mortal.
(Valid but not sound, and conclusion is true)

Socrates is a talking plant.
All mortals are talking plants.
Therefore, Socrates is mortal.
(Neither sound nor valid, but the conclusion is true)

Note that if the argument is unsound, the conclusion can either be true or false. Soundness, however, always implies validity, and always implies that the conclusion is true.

Is the conceptual ontological argument sound?

So what about the COA?

The COA takes a single premise, that "God is conceivable".  From there it concludes that God exists in the real world.  The objection I raised before is that the proof works only if we take a particular definition of "God is conceivable."  Namely, if we conceive of an object, we can call it God only if the object exists in the real world.  This makes it rather difficult to verify that God is conceivable!

But I would say that the COA is valid, and it's only the soundness I question.

Now suppose for a moment that God really does exist.  We know that all things that exist are also conceivable.  So you'd have to conclude that the premise is true, and COA is sound.  If God exists, then the COA is sound.  Equivalently: if the COA is unsound, then God does not exist.

As an opponent of ontological arguments, people often think my role is to argue that ontological arguments are unsound.  However, if I did successfully argue that COA is unsound, I would not just refute the COA, I would be disproving God entirely!  While I am unapologetically atheist, such a conclusion remains outside the scope of this series.

I do not wish to argue that ontological arguments are unsound, instead I wish to argue that they are "useless".

Useful arguments

My concept of "usefulness" is my own idea, but it's inspired by something written by Chris Hallquist.  He says:

An argument can be sound and still not be any good. Imagine arguing with someone who believes that the Sun orbits the Earth rather than the other way around. Now imagine giving them the following argument: “Premise: the Earth orbits the Sun. Conclusion: the Earth orbits the Sun.” If the premise of this argument is true, the conclusion must be true, and the premise is true. Thus the argument is sound. 

Yet you couldn’t blame anyone for not being persuaded by that argument.

With that in mind, I say that an argument is useful if its conclusions are more difficult to demonstrate directly than its premises are.

Note that usefulness is not a logical concept.  Within the formal logic, there is no notion of "direct demonstration" and no notion of "difficult".  Usefulness is a pragmatic concept.  Consider the following sound but useless arguments:

The Earth orbits the sun.
Therefore, the Earth orbits the sun.

If Mars is a giant egg, then the moon is made of cheese.¹
The moon is not made of cheese.
Therefore, Mars is not a giant egg.

All four cards in my hand are aces.
Therefore, at least one card in my hand is an ace.²

Since I do not believe in gods, I also believe that the COA is unsound.  However, even if you disagree on this point, you must agree that the COA is useless.

It is only possible to demonstrate that "God is conceivable" if you first demonstrate that God exists.  Since the conclusion of the argument is that God exists, that means that the premise is at least as hard to demonstrate as the conclusion.

This is an important point moving forward, because I will argue that all the remaining ontological arguments in this series fail by being useless.³

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1. This premise is true if we're using the material conditional.  The material conditional will appear much later in this series, but for now I refer you to Google.

2. The idea here is that proving that all four cards in my hand are aces is at least as hard as proving that at least one is an ace.  Actually this argument isn't sound because I'm not actually holding any cards right now, but please grant the hypothetical.

3. There's an interesting philosophical question: Even if all known ontological arguments fail, how can we be sure that we won't find one which succeeds?  In my opinion, to address this question, we need a good way to define "success".  We already have ontological arguments which succeed, in the sense of being sound.  For example, I define my keyboard as a keyboard that exists.  I can conceive of this keyboard (I'm typing with it!), therefore it exists.  The premise is correct, the reasoning is correct, and the conclusion is correct.  However, the argument still fails in the sense of being useless.

Modal ontological argument, revisited

Update 2015: I wrote a new series on ontological arguments.

Let's revisit the modal ontological arguments for God.  Though I have covered these arguments and variations at length, and have not changed my opinion about them, I didn't necessarily cover them in the clearest manner possible.  It's a tough balance, because for most people, symbolic logic is like math.  Math = scary!  Other readers understand the symbolic logic, but clearly have difficulties translating between the symbols and their underlying meaning.  My goal is to explain it so both groups understand what I'm saying.

Modal ontological argument, reviewed

Definition: If God exists, then God necessarily exists.
Premise: It is possible that God exists.
[Insert mess of logical reasoning here]
Conclusion: God exists.

I omitted the mess of logical reasoning because I don't want to scare away my readers. Unfortunately, this makes it difficult to convey the amount of respect the logical arguments deserve.  Richard Dawkins and other atheists often seem to think it's all a bunch of high-sounding gibberish.  And perhaps it is.  But that particular gibberish is absolutely solid, as solid as 2+2=4.

There are basically two reasonable objections to the argument.  Either we must object to the premise, or object to the definition.

Objection to the premise

To explain the problem with the premise, I must distinguish between two kinds of possibility.

Epistemological possibility: For all we know, God exists.
Modal possibility: Among the set of "possible worlds", there exists a world in which God exists.

The conclusion of the proof follows only from modal possibility, not from epistemological possibility.  However, the epistemological statement is the one that is intuitively true, while the modal statement could be true or false.  Modal possibility is intended to be a translation from epistemological possibility to logic, but the translation is not perfect.

In particular, the translation fails when we talk about the set of possible worlds itself.

Epistemological statement: For all we know, the universe is deterministic (ie there is only one possible world).
Modal translation: Among the set of "possible worlds", there exists a world in which it is true that there is only one possible world.

The epistemological statement is sensible, but if we naively copy it as a modal statement, it's a mess.  The first statement only says that the universe may be deterministic, while the second statement can be used to prove that the universe must be deterministic.  Clearly, we need to be careful with our translations when discussing determinism.

Similar to determinism, the existence of God also says something about the set of possible worlds.  If God exists, then s/he exists in all possible worlds.  If an object does not exist in all possible worlds, then we cannot call it God.  So we need to be careful with our translations when discussing God too.

Here is a much better translation to logic:

Epistemological statement: For all I know, God exists
Logical translation: God exists, or God does not exist.

The conclusion of the ontological argument does not follow from this proper translation.

Even proponents of the modal ontological argument must accept that there are problems with translation.  If we take the statement, "For all I know, God doesn't exist" and naively translate it to modal logic, then we would conclude that God doesn't exist.

Alternate premise: Logical consistency

Some ontological arguments have a cleverer premise to replace the old one.

Old Premise: It is possible that God exists.
New Premise: There is no contradiction in the existence of God.  In other words, God is "consistent."

The old premise logically follows from the new one.  If you're familiar with symbolic logic, I hope you already knew this.  If not, here is a simple proof.

Suppose that the old premise is false; it is not possible that God exists.  The statement "If P, then Q" is always true if P is false.  Likewise, the statement "If God exists, then Q" is always true, because God doesn't exist.  It's true even if Q is a contradictory statement (ie "God is blue and not blue").  Therefore, the new premise is false; the existence of God implies a contradiction.

If the old premise is false, then the new premise is false.  Equivalently, if the new premise is true, then the old premise is true.

But now we will run into another problem of translation.  I will distinguish between two kinds of consistency.

Self-consistency:  The object has no contradicting properties in its definition.
Logical consistency: The object implies no contradictions.

Proponents of the ontological argument often expect me to disprove the self-consistency of God.  Perhaps they expect me to argue that God's omnipotence contradicts its omniscience, or something like that.  But they fail to realize that I don't have to.  The proof requires logical consistency, not self-consistency.

Self-consistency is not sufficient to establish logical consistency.  For an object to be logically consistent, not only must its definition be properly formed, but the world must cooperate.  (More precisely, the set of "possible worlds" must cooperate.)  Suppose that the world does not cooperate, and the object does not exist.  If the object does not exist, then the existence of the object implies a contradiction.  Namely, it implies that the object both exists and does not exist.  I didn't even have to look at the definition of the object.

Of course, I don't know whether the world cooperates with the ontological proof or not.  The proponents have no idea either, but think they do.

Philosophers ought to teach themselves some mathematics. In geometry, there is a famous axiom called the Parallel postulate.  It is famous because many mathematicians thought they could prove it.  Modern mathematicians know that it is impossible to prove, because there is no contradiction in assuming it false.  Likewise, it is impossible to disprove.  The Parallel postulate is self-consistent.  The negation of the Parallel postulate is also self-consistent.  But in any given geometrical system, only one can be true.  Thus, only one can be logically consistent.

Objection to the Definition of God

Definition: If God exists, then God necessarily exists.

If we define a fork to be an object with a handle and prongs, then we can give the following statement: "If a fork exists, then it has prongs".  If a fork does not exist, we can't even talk about "it", much less ascribe it properties.  If an object does not have prongs, then it is not a fork.  That is the rationale behind the definition.

How can you disagree with a definition?  Can't we define any object we like?  If the definition makes no sense, can't we just say that the object doesn't exist?

I don't know about philosophy, but in mathematics, you can't just define any object you like.  Consider the set of all sets that do not include themselves (like the male barber who shaves all men who do not shave themselves).  Call this Russell's Set.  In "naive" set theory, you are allowed to define any set you like, including Russell's set.  But Russell's set leads to a paradox.  Therefore, "naive" set theory is inconsistent.

Naturally, mathematicians want a set theory that doesn't have paradoxes.  So they formulated Zermelo-Fraenkel set theory, which has specific rules dictating what sets you are allowed to define.  These rules do not allow us to define Russell's Set, and thus avoid its paradoxes.

Is modal logic more like "naive" set theory, or like Zermelo-Fraenkel set theory?  Can you define anything you like, or do you have to follow specific rules?  I suspect it depends on your choice of axioms.  It's a difficult question that I don't feel qualified to answer, which is why I prefer objections to the premise.

For what it's worth, Immanuel Kant's original objection to the ontological argument might fit in this category.  Kant argued that existence is not a property that you can include in the definition of an object.

Alternate definition: The greatest being

Many proponents of ontological arguments like to have it both ways.  On the one hand, we are allowed to define God.  On the other hand, we are not allowed to define "the unicorn which necessarily exists."

But to be fair, they're not exactly parallel.  In most ontological arguments, God is not defined as "the deity which necessarily exists".  Rather, God has a much more specific definition.

Definition: If God exists, then s/he is the greatest being conceivable.
Additional premise: We can conceive of a being as greater by conceiving it as necessarily existing.

I think that this new definition hurts the ontological argument.  For one thing, we have a whole new premise.  I don't have any particular problem with the premise, but it just seems so extraneous and unnecessary.  I refuse to argue with the additional premise, because it seems like a tactic to draw attention away from the real flaws of the ontological argument.  In my naive optimism, I expected this tactic from conspiracy theorists, not philosophers.

And the new definition does not help in the slightest.

Let's say that Russell is the name of "the male barber who shaves all men who do not shave themselves.  As I said before, in Zermelo-Fraenkel set theory, there are rules against defining Russell.  If we are not allowed to define Russell, then obviously, we are also not allowed to define Russell's wife!  Russell's wife may not have any self-referential paradoxes, but she requires the existence of Russell, who does have self-referential paradoxes.

Let's say we have a rule against defining necessarily existing beings.  Obviously, we are also not allowed to define the wives of any of those beings.  We are not allowed to make any definition which implies necessary existence.  If we accept the additional premise, then the definition of God implies necessary existence.  Therefore we are not allowed to construct the definition of God.

Concluding remarks

I've written over 1600 words now, so I think I'm done.  Dear readers, you are lucky that I'm on spring break (or unlucky, if you see it that way).

Multiple past experiences tell me that someone trained in philosophy will start lecturing me about implication, contingency, and other things.  You're welcome!  But a few notes: 1) I'm not an idiot when it comes to logic.  2) Philosophers who do not know how to communicate are useless.  3) I do not necessarily agree with the objections to the definition. 4) I view the ontological argument the same way I view a fun proof that 0=1.

A modal ontological argument

Update 2015: I wrote a new series on ontological arguments.

If you wish to understand this post, you should probably read my exposition on modal logic first. You may also wish to read my analysis on a much simpler ontological argument.

To briefly review the simpler ontological argument: The argument says that we can define God as a necessarily existing being. Therefore, by definition, God exists. However, this takes the power of definition too far. The most we can say is that if there exists an object which we can properly call God, then that object, by definition, exists. If God exists, then God exists.

The problem with the argument is that it's rather useless to define God as a being which exists. But hold on! I never defined God as a being which exists. I defined God as a being which necessarily exists. This definition is not quite as useless. We can say that if there exists an object which we can properly call God, then that object, by definition, necessarily exists. If God exists, then God necessarily exists.

By definition of God: g g

The letter "g" represents the specific statement "God exists". Note, however, we did not completely define God. In fact, we could replace God with all kinds of absurd objects, and the proof would still hold. This is actually quite a problem for the ontological argument (as well as many other proofs of God). The proof is valid for any statement p as long as pp is necessarily true. For instance, we could replace the object God with "the invisible pink unicorn which necessarily exists" or "a left shoe which necessarily exists".

Nonetheless, let's continue onward. I'm going to go through the proof, step by step.

• Theorem 1: ¬g ¬g (The contrapositive of the definition of g)
  
• Theorem 2: ¬g ¬g (Using the definition of to substitute into Theorem 1)
  
• Theorem 3: (¬g ¬g) (Application of Axiom N to Theorem 2)
  
• Theorem 4: ¬g ¬g (Application of Axiom K to Theorem 3)
  
• Theorem 5: ¬g¬g (Axiom 5 applied to ¬g)
  
• Theorem 6: ¬g ¬g (Combining Theorems 4 and 5)
  
• Theorem 7: ¬¬g ¬¬g (The contrapositive of theorem 6)
  
• Theorem 8: gg (Using the definition of to substitute into Theorem 7 twice)
  
• Theorem 9: g g (Combining Theorem 8 with Axiom T)

You may have noticed that I did not prove what I set out to prove. I only proved g g, which says that if g is possible, then it is true. That's not quite the same as saying that g is true, but it's still something. Now, all we have to do is prove that g is at least possibly true.

And here is where the problems begin. All the previous work was purely logical manipulation, and is necessarily true if you accept the axioms of modal logic. I thought the axioms were pretty reasonable, and rejecting them would be too high a price to pay. However, it seems we need another premise, g. To support this premise, lots of arguments have been offered by various people, but I don't think they're nearly as fun or rigorous as the modal logic section.

One common argument for g is that g is self-consistent. I can conceive of a God without having any contradictions. Based on my current knowledge, it is entirely possible that God exists. It's possible that God exists, therefore God exists.

The problem is that this same argument seems to apply to the statement ¬g. ¬g is a self-consistent statement. I can conceive of a world in which God does not exist without having any contradictions. Indeed, I can conceive of a world where nothing at all exists, where there could not possibly be any contradictions since there is nothing around to contradict. Based on my current knowledge, it is entirely possible that God does not exist. Therefore we can take the premise ¬g. By Theorem 2, we conclude ¬g: God does not exist.

Obviously, the premises g and ¬g cannot both be true. Either God exists in all possible worlds, or God exists in none of them. Since we used the same argument for both both g and ¬g, that argument must be fallacious. But where exactly did it go wrong?

The error, I think, is in the concept of . g does not quite mean "g is possibly true." In fact, it means "Among all possible worlds, there exists at least one in which g is true." The concept of "All possible worlds" was never exactly defined. In fact, the definition is arbitrary. I could have declared "all possible worlds" to be our world and our world alone, and you never would have been able to prove me wrong from the axioms.

I also could have declared "all possible worlds" to be the set of worlds which are metaphysically possible (a rather complex philosophical concept). Under this definition, we would not be able to prove g or ¬g.

However, if I had declared "all possible worlds" to mean the set of all worlds which are self-consistent, then we would run into problems. Because under this definition, both g and ¬g appear to be true (unless the concept of God is inconsistent). And they contradict each other.

Similarly, if I had declared "all possible worlds" to mean the set of all worlds which are epistemologically possible (meaning, it may be true, for all we know), then we would have the same contradiction.

And what happens if we construct a pathological definition of "all possible worlds" such that g is true and ¬g is not true? Then I might question Axiom T, since it is no longer obvious that our world is included among this so-called set of "all possible worlds."

And it goes on and on. Well, wasn't it clever, at least at first? I think so. It seemed for a moment that we arrived at a paradox, a sort of 1=2 moment. Ontological arguments tend to be that way. Most people immediately recognize that it is a little too clever, that it proves a statement which is a little too strong to be true. Similarly, most philosophers think that ontological arguments fail, though they may disagree on exactly why they fail.

In my opinion, ontological arguments are merely interesting philosophical curiosities. It's rather silly when an apologist actually tries to use one as a serious argument.