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.