Tuesday, January 23, 2018

An asymmetry between the theistic and atheistic modal ontological arguments

A simple version of the modal ontological argument goes as follows:

  1. Necessarily: If there is a God, then necessarily there is a God. (Premise)

  2. Possibly, there is a God. (Premise)

  3. So, possibly necessarily there is a God. (By 1 and 2)

  4. So, there is a God. (By 3)

It is well-known that there is a very similar argument for the opposite conclusion. Just replace (2) by the premise that possibly there is no God, and you can change the conclusion to read that there is no God. So it seems we have a symmetric stalement. Though, perhaps, as Plantinga has noted, we get to conclude from the arguments that the probability that God exists is 1/2, which when combined with other arguments for theism (or maybe with a particularly plausible version of Pascal’s Wager?) it could be useful.

However, interestingly, the symmetry is imperfect in a way that I haven’t seen mentioned in the literature. Consider this atheistic ontological argument:

  1. If there is a God, then necessarily there is a God. (Premise)

  2. Possible, there is no God. (Premise)

  3. So, it’s not necessary that there is a God. (By 6)

  4. So, there is no God. (By 5 and 7)

This argument differs from the theistic one in two ways. First, the atheistic argument can get away with premise (5) which is formally weaker (given Axiom T) than (1). This is not a big difference, since (5) is no more plausible than (1).

But there is a bigger difference. In the theistic argument, to derive (4) from (3) requires the somewhat controversial Brouwer Axiom of modal logic (which follows from S5). But the atheistic argument does not need any axioms of modal logic, besides the uncontroversial modal De Morgan equivalences behind the inference of (7) from (6).

My first thought on noticing this asymmetry was that the atheistic argument is somewhat superior to the theistic, at least when the audience isn’t sure of S5 (or Brouwer).

My second thought was that the atheistic argument is more subject to the objection that its possibility premise begs the question. For the conclusion follows more directly from the possibility premise, and that makes a question-beggingness objection a little bit more plausible.

I don’t know exactly what to think now.

Anyway, nothing earthshaking here. For those of us who think S5 is true, the differences are pretty small. But it’s worth remembering that the symmetry is imperfect.

21 comments:

Walter Van den Acker said...

Dr Pruss

The atheistic modal ontological argument is, in most cases, not used to argue against God's existence. What it instead wants to show is that the conclusion of the theistic modal ontological argument is unjustified, for the simple reason that, considering Brouwer (or S5), possibly necessary is equivalent to necessary and, as a result, the claim that God is possible can only be proven if His necessity is also proven.
Proving that X is simply possible, in and out of itself, is relatively easy. Proving that a necessary being is possible carries an incredibly high burden, just as high as the burden to prove that a necessary being exists.
If the latter can't be done, neither can the former, so the modal "shortcut" doesn't add anything interesting to the problem at hand, which is that the theist makes a necessity claim he can't back up.
Most atheists, on the other hand, do not make a necessity claim.

Christopher Michael said...

What stops the theist from running this argument to remove the asymmetry?:


9. If there is no God, then necessarily there is no God. (Premise)
10. Possibly, there is a God. (Premise)
11. So, it’s not necessary that there is no God. (By 10)
12. So, there is a God. (By 9 and 11)

(9) is different from your (5), but I don't see why (5) should be given any more credence than (9).

Alexander R Pruss said...

Christopher:

Premise 5 is an immediate consequence, with no modal axioms used, of the facts that:
(a) God (if he exists) is a necessary being
(b) God (if he exists) is essentially God.
And these facts are a consequence of the concept of God.

But Premise 9 only follows from these facts if one assumes something like the Brouwer axiom. For instance, suppose that there are two possible worlds @ (actual) and w, with each world accessible from itself, and with w accessible from @ but @ not accessible from w. Suppose that God exists and is divine at w but not at @. Then:
i. There is no God. (Since there is no God at @.)
ii. Possibly, there is a God. (Since there is a God at w.)
iii. At every possible world where God exists, God is a necessary being and essentially divine. (Since the only possible world where God exists is w, and God exists and is divine at all worlds accessible from w, namely at w.)
iv. Premise 9 is false, because its antecedent is true but its consequent is false.

Alexander R Pruss said...

Walter (call me Alex):

The atheist makes a claim that about as quickly implies a necessity claim, if Brouwer is true. For the atheist says that there is no God. But this implies that necessarily there is no God, if Brouwer is true.

The fact that P entails Q does not mean that we have to have independent evidence for Q in order to have evidence for P.

Walter Van den Acker said...

Alex

The atheist only makes an implicit necessary claim if he agrees that God, if He exists, is necessary. That is not the only God concept around. Swinburne's God, e.g. isn't necessary.
So there is no God does not in and out of itself imply that necessarily there is no God.
And even in the cases the atheist claim that there is no God does imply God's impossibility, my point is that most atheists do not use the atheistic modal ontological argument to argue that God doesn't exist, instead they use it to argue that the theistic modal argument does not show that God exists.

Alexander R Pruss said...

Walter:

Swinburne is just wrong about that. The perfection that is essential to divinity entails necessary existence. :-)

I agree that few people use the atheistic modal ontological argument to argue that God doesn't exist. Though few use the theistic modal ontological argument *by itself* to argue that God does exist.

Walter Van den Acker said...

Alex

The "atheistic modal ontological argument" is a reaction to the theistic one, which, as is obvious from your version, concludes ... "So, there is a God."
I have discussed the TMOA with quite a few theists who maintain that it does show God's existence. On the other hand, I don't know of any atheist who uses the AMOA as an argument to show that God doesn't exist. I think therein lies the true asymmetry.
Maybe there are a few that I don't know of, but it's certainly not one of the main arguments for atheism.
I, for one, don't use it that way, although I most certainly don't qualify as a prominent atheist.

Wielka Miska said...

If I were to use an ontological argument by itself, I'd rather choose a variant of the Godel's one.

The traditional ontological argument seems to work well with some kind of a cosmological argument. When the cosmological one gets you to the point where you accept either a necessary cause or brutal facts, you have given cause for the possibility of the existence of a necessary being, which is the fuel that the ontological argument needs.

Alexander R Pruss said...

I prefer the Goedelian ones, too.

There is a semblance of the atheistic modal ontological argument in Findlay's classic piece:
1. God is a necessary being.
2. Every being has the possibility of non-existence.
3. So, God doesn't exist.
Findlay's premise 2 is stronger than the claim that possibly God doesn't exist.

One other place where the pair of arguments comes up is in a remark by Plantinga that given the two arguments, it may seem reasonable to assign probability 1/2 to the thesis that possibly God exists. (And that would be useful. It would mean that any evidence for or against the existence of God tips the scales into making the existence of God more or less likely than not.) If the atheistic argument is a bit stronger than the theistic one, then you don't get 1/2, but maybe something like 0.45.

Wielka Miska said...

Wouldn't assigning a probability to the possibility of God's non-existence entail assigning probability to the existence of brute facts?

1. God is a necessary being.
2. Possibly, God doesn't exist.
3. So, God doesn't exist.

My points:
4. God is the necessary cause (it's somewhat different from 1).
5. Since God doesn't exist (3), the necessary cause doesn't exist.
6. The world exists.
7. Brute facts are possible (5 and 6).

Wielka Miska said...

"assigning probability to the existence of brute facts" => "assigning probability to the possibility of existence of brute facts"

Wielka Miska said...

Another comment, but I've just had a thought about what I wrote above: if we were thinking about the probability of the possibility of brute facts (I know I'm in a dubious area right now), then we could say that e.g. there's a 0.5 probability that brute facts are possible, and 0.5 that they aren't. But if we assign a non-0 probability to brute facts not being possible, it means that we say that it is possible for a necessary cause to exist, thereby launching the ontological argument.

Walter Van den Acker said...

Alex

Doesn't "Every being has the possibility of non-existence" imply that the PSR is false?
While I am not convinced the PSR is true, I wouldn't make the claim that it is false.
God (possibly) doesn't exist dos not entail there are no necessary beings.

The 1/2 probability also has its problems. Let's say we have proof for the resurrection.
Resurrecting someone takes a lot of power, even supernatural power, but it odesn't require omnipotence. Hence, the resurrection would be evidence for a powerful supernatural being, but it doesn't tip the scales into making the existence of God (defined as a necessary being) more or less likely than not.
I really think people who make the claim that X is necssary do not realize how extreme this claim really is. It is a claim that there cannot possibly be any alternative for X.

Wielka Miska said...

Walter, I also don't see how resurrection would immediately entail that a necessary being is possible. It might be so, but it would need further argumentation. However, cosmological proofs in general point to a necessary being as a "root" explaination - whatever it might be. In fact, they point to it as strongly as any evidence in the real world could point to something. This is a pretty solid argument for the possibility of the existence of such a thing.

Walter Van den Acker said...

Wielka

A necessary "thing" is not the same as God. Various things could be necssary.
Moreover, cosmological proofs only point to a necessary being if the PSR is true.

Alexander R Pruss said...

Walter:

I do think that if there is no necessary being, the PSR is false. But not everyone agrees. Quentin Smith has argued that one can have the PSR without a necessary being, in a naturalistic setting, by having circular causation, say. I think he may have some other options he considers.

As for resurrection, P(resurrection|no omnipotent being) is less than P(resurrection|omnipotent being). Why? Well, if there is no omnipotent being, there may or may not be a being powerful enough to resurrect someone. But if there is an omnipotent being, then there is a being powerful enough to resurrect someone (at least given some background assumptions about personal identity, which will affect the "no omnipotent being" at least as much). So that there is a resurrection is some evidence for an omnipotent being.

Heath White said...

What about this version? No Brouwer needed, I think:

1. Either (necessarily, there is a God) or (necessarily, there is no God) (df. God)
2. Possibly, there is a God (premise)
3. It is not necessary that there is no God (from 2)
4. Necessarily, there is a God. (1,3 DS)

Note that 1 is equivalent to "If God is possible then God is necessary" and then 4 follows from 1,2 by MP.

Walter Van den Acker said...

Alex

Well, if Qunetin Smith is correct, cosmological arguments do not even point to a necessary being even if the PSR holds.

And of course P(resurrection|omnipotent being) is higher than P(resurrection|no omnipotent being), but in fact, the probability of just about anything is higher if there is an omnipotent being, but that doesn't make the probability of an omnipotent being's existence signifcantly higher.
The probability of a stage magician pulling a rabbit out of his hat is higher if the magician truly has magic powers that allow him to create rabbits out of thin air, but that does not entail that the rabbit is good evicence for such magic powers.

Alexander R Pruss said...

Heath:

But why think (1) is true if one doesn't have Brouwer? Without Brouwer, there is the third option that possibly there is a necessarily existent God, but actually there is no necessarily existent God.

Heath White said...

I guess you are right. I suppose I don't need convincing about S5.

micah said...

Structurally, it seems that if (6) is question-begging, then so is (2). So which one is weaker or more plausible is going to depend on what else one brings into it, as others have above (PSR, evidence of the resurrection, and so forth).