Thursday, June 9, 2011

Necessity isn't so necessary

Following my retrospective on the ontological argument, I just remembered another thing I always thought was funny about it.  According to the argument, part of God's definition is that God necessarily exists.  But why?

I can just imagine, what if there were some powerful being which shared every property with God, except for the necessary existence.  That is, this being is exactly like God in every way, except that unlike God, it only exists in our world, not every possible world.  By definition, this being would not be God.  After all, it's not the greatest being imaginable.  We can imagine a being that is greater: one that exists in every possible world, not just ours.

But a fine distinction that would be if in our world, we're being subjugated/loved/ignored by an all-powerful and vengeful/benevolent/passive being!  I don't know about you, but I'd call that thing a god, even if it doesn't quite fit the definition in the ontological argument.  As for those other possible worlds where the being doesn't exist, who cares about 'em?  Depending on who you ask, there isn't even any metaphysical significance to the other possible worlds, they're just ideas.

That leaves the ontological argument in a funny position.  The god it argues for does not necessarily have any of the properties we normally assign to a god.  But it does have this extra property, necessary existence, which I do not think is necessary to qualify a being as a god.  Does it really have anything to do with gods, or is it just a logical game, as I've been treating it?  (Also applies to nearly every other philosophical argument for gods.)

And yes, I do have some idea of how ontological argument proponents would respond.  A transcendent being such as a god must also transcend all possible worlds, thus necessary existence is an inseparable quality of God.  But if we understand other possible worlds as mere ideas, this doesn't make any sense.


Magnus said...

I have come up with a reply to the ontological argument, and I'm interested in comments and criticism.

The ontological argument essentially tries to prove God's existence from God's possible existence. It does this by going through the step of necessary existence.

God is possible->
God exists necessarily->
God exists

The exact way in which the first step is argued or proven varies, but let us just accept that God has the property of "possible existence implies necessary existence." The theist then asserts that God is exemplified and therefore possible. The existence of God then follows.

One atheistic response is to try a similar argument for atheism. If God's non-existence is possible, then God's non-existence must follow.

The possible non-existence of God->
God does not necessarily exist->
God does not possibly exist->
God does not exist

I won't go into the exact logic of either argument, there are better sources than me, but both seem to be without any logical problems, so we are forced to deal with the definitions and premises.

Definition: God is a maximally great being.
Premise 1: God has a property which is logically equivalent to "possible existence implies necessary existence."

Theistic premise 2: God's existence is possible. (TP2)
Atheistic premise 2: God's non-existence is possible. (AP2)

Let us accept that premise 1 is obviously true. We are then dealing with the question of whether the definition is coherent, and whether premises TP2 and AP2 are true or false.

Obviously, the premises cannot both the false. We can also see that the premises cannot both be true. If either of these were the case God would both exist and not exist. In fact, God would both exist and not exist necessarily. Acceptance of AP2 and TP2 must therefore indicate that the definition of God is not coherent.

What if one premise is false, and the other is true?
Let us assume that TP2 is true and AP2 is false. Then God's necessary existence does follow, so we no longer have a contradiction. However, if you claim that AP2 is false then you are asserting "it is false that God's non-existence is possible," but this is the equivalent of saying that God is necessary. In order for the ontological argument to work without contradiction it must implicitly assume that God exists necessarily. The same is the case for the atheistic version. This seems to make the argument useless.

I have read a response from Dr.Craig on this problem (question 144). He says that we have to consider the epistemic probability of premises TP2 and AP2. He says that it is easy to imagine that God is coherent, but no-one has shown that he isn't. In other words the probability of God's existence being exemplified is high, but the probability of God's non-existence being exemplified is low. This approach does not save the argument.

Let us assume that the epistemic probabilities heavily favours theism:

P(TP2) = 0.98

But this implies that P(not AP2)=0.95, meaning that we know that there is a 95% chance of God's necessary existence. Can we use the ontological argument to increase this probability? No!

We can use the two ontological arguments to prove that the probability of God's existence is 98% and that the probability of God's non-existence is 5%, but any overlap renders God incoherent, so the actual probability of existence must be 95%. This is the same as before, so the ontological argument fails to increase the probability of God beyond what was implied in our premises.

miller said...

I agree with everything you said up to this point.

P(TP2) = 0.98

If someone gave me these epistemic probabilities, I would simply conclude that they have estimated the probabilities incorrectly. TP2 & AP2 doesn't render god incoherent, it renders your premises inconsistent.

There was at least one person who in arguing with me claimed that the ontological argument at least increased the epistemic probability of God. I think it does no such thing, not even a little (and this is in contrast to most other arguments). If proponents want to argue about probabilities, I would like to see a Bayesian argument, not just hand-waving.

BTW, you should know I have more detailed analysis of the modal argument.

magnus said...

I have struggled somewhat with the problem of having both premises. The only reason why I included the possibility of having AP2 and TP2 both being true is for thoroughness, not because I personally support it. The only way I can imagine justifying it is by saying that "the possibility of A" and "the possibility of not A" should be true as a matter of principle. If we were debating the existence of Bigfoot or any other empirical claim, then they should both be true. It actually seems like they must be simultaneously as long as Bigfoot is logically coherent idea. If we were debating the conclusion of a mathematical proof we would not accept the notion that the proof was valid in one possible world, but invalid in another. Logic and mathematics are often used as analogies to God by people using the ontological argument, but God would be an actual being not an abstraction. If it were the case that for all potential actual beings premises AP2 and TP2 must be true, then we are forced to conclude that God is incoherent.

Until I see a very good reason to believe that the God claim is in a category where premises AP2 and TP2 must be true as a matter of principle I see no reason to accept the possibility. It's a confusing point, so I am inclined to agree with you and simply leave the possibility behind along with the option of both premises being simultaneously false.

miller said...

In my view, the set of all possible worlds is just an abstraction. If we wanted, we could define this set to include just this world. If we did so, then ALL claims would be in the same situation as god. That is, either bigfoot possibly exists or possibly does not exist, but not both.

But people usually define the set of all possible worlds as a much larger set (the details of which are left ambiguous).

The problem is that God, as defined in the modal argument, has properties which depend on all possible worlds, rather than just the world in which God happens to be in. Even if there are many possible worlds, there is only one set of possible worlds. And so God either possibly exists or possibly doesn't exist, but not both.

If it is intuitive that both AP2 and TP2 are true, I think it is because we are imagining not just one set of multiple possible worlds, but multiple sets of possible worlds. Unfortunately, that's not how modal logic works. So even if our intuitions are correct, they can't be translated into modal logic.

Larry Hamelin said...

we are imagining not just one set of multiple possible worlds, but multiple sets of possible worlds. Unfortunately, that's not how modal logic works.

Not exactly. We can apply modal logic separately to each meta-level.

I don't know of any analysis of how multi-level modal logic would work out. I suspect that there's not much interesting to say about it.

miller said...

Yeah, you could probably modify the axioms of modal logic to model multiple sets of possible worlds. Maybe it's just a matter of rejecting axioms 4 and 5, or maybe there's more to it than that. The ontological argument depends on those axioms though.

Larry Hamelin said...

I don't know you have to modify any axioms. If you have a set of sets of possible worlds, then you can make ordinary modal statements about each element in the set of sets.

We might, for example, define a set of sets of possible worlds, such that each element in the larger set is the set of all possible worlds where some particular statement was true (but all other statements could vary). It would seem that all of the axioms of ordinary modal logic would apply mutatis mutandis to this larger set.

If each "possible world" within each individual element were all disjoint, then I don't think there are any problems. If, however, there are overlaps, then there might be interesting subtleties.

I dunno. In any event, I'm not really clear that there's any substantive difference between modal logic and set theory.