Monday, April 2, 2012

Goedelian ontological argument in negative form


  1. Necessarily, if a property B is limiting, so is any property A that entails B.
  2. Necessarily, if a property B is limiting, its negation is not limiting.
  3. Possibly lacking existence is limiting.
  4. Possibly lacking omniscience is limiting.
  5. Possibly lacking omnipotence is limiting.
  6. Possibly lacking perfect goodness is limiting.
  7. Possibly not being creator of everything else is limiting.
  8. It is not possible that x is a creator of y while y is a creator of x.

In a forthcoming paper, I prove using S5 that (1)-(8) entails:

  1. There exists a necessary being that is essentially omniscient, omnipotent, perfectly good and creator of everything else. This being has every property that it would be limiting to possibly-lack.


Dan Johnson said...

Alex, does this form of the argument have a possibility premise? I don't see one. How do you run the proof without one?

Alexander R Pruss said...


It follows from 1 and 2 that the negation of a limiting property is always possibly instantiated.

For if A is a limiting property, suppose ~A is not possibly instantiated. Then, necessarily, everything has A. But then A is entailed by every property. But then A is entailed, inter alia, by ~A. Thus ~A is limiting by 1. But ~A is not limiting by 2. So we have absurdity.

Say that a property is positive iff its negation is negative. Then, in fact, for any pair of positive properties A and B, they are compossible. This follows from 1 and 2.

Johnny-Dee said...

I have a quick worry about premise (1). Suppose there is a property like, being a finite creation of an infinite Being. My concern is that this property is limiting, and furthermore that this property entails the property of being an infinite Creator. While being a finite creation of an infinite being is limiting, it appears to entail a property (being an inifinite Creator), which is not limiting.

Do you have a way of defining "limiting" or properties that staves off this kind of objection?

Alexander R Pruss said...

I should have said: Property A entails property B iff necessarily(if x has A, then x has B).

Your case isn't a case of property entailment.

Moreover, a limiting property can entail a non-limiting one. It's the opposite that can't happen.

Johnny-Dee said...

I see. On your view of property entailment, X's being a child does not entail some Y's being a parent. The properties in question must belong to the same being for property entailment to occur.

Dan Johnson said...

Hi Alex,

This is really cool. One worry, though. Isn't the entailment you are talking about just trivial entailment? If a non-limiting property (~A) is not possibly instantiated, then it entails every property, right? Because there aren't any possible worlds where it is instantiated by an individual and any other property isn't.

But if your first premise is supposed to include trivial entailments, doesn't it lose a great deal of its plausibility? At the least, it threatens circularity (which, as you know, I do not regard as a fatal flaw in an argument).

Maybe I'm just saying that the circularity that often accompanies the possibility premise in ontological arguments is present here too. My initial excitement that you avoided needing that element may have been premature.

Alexander R Pruss said...


Well, it seems there's got to be something like a possibility premise somewhere. :-) I think the possibility premise here is encapsulated in 3-6.

I do see your point about the inclusion of trivial entailments making 1 less plausible, though.

But, still, is there an uncontentious counterexample to 1?

Thomas Henry Larsen said...

"Possibly lacking existence is limiting."

Does this argument rely on existence being a property?

Dan Johnson said...

Alex, maybe there isn't an uncontentious counterexample to (1), but the contention about such counterexamples might just run along theistic/atheistic lines, which would mean that the premise makes the argument beg the question (or be circular in some other way). In other words, it might be the atheists who accept the counterexample (because of their atheism) and the theists who don't accept it (because of their theism).

Sorry, I'm struggling to come up with an example to illustrate this now. (Maybe the properties in 3-6 are examples, however.) Of course, even if all this is true, it doesn't mean that the argument is useless. Even if it is circular, it might be useful (as you probably expected me to say), and you may find some people for whom it isn't circular -- who accept premise 1 even if they aren't theists.

All this implies that the possibility premise is really an implication of premise 1 or the combination of 1 with 2 -- as your argument for the possible instantiation of the negations of all limiting properties indicates.

Alexander R Pruss said...

At this point, it gets hard to see what is question-begging and what simply leads to a conclusion contrary to one's views. Obviously, any valid argument for the existence of God leads to a conclusion contrary to the atheist's views.

The atheist can accept the formal premises 1 and 2, as long as she rejects what I call the non-formal premises--the ones that say that possibly being non-existent, possibly being non-omniscient, etc. are limiting. Or she can accept the non-formal premises and deny 1 or 2.

One difference between this and the standard S5 argument is that in the standard S5 argument it is pretty clear which premise the typical atheist will deny. Not so here, I think.

If I were an atheist, I think I would want to deny 2: I might think that it's necessary that every possible entity is limited, and hence every property is limiting. But notice that this is prima facie more than atheism by itself requires of one. Atheism at most requires that one hold that every entity is limited (and even that's not completely clear--some atheists believe in a multiverse, and a multiverse perhaps could count as an unlimited entity).

Alexander R Pruss said...

Mr. Larsen:

The argument requires that contingency, or contingent existence, be a property. That's less problematic than assuming that existence is a property.

In any case, there isn't a particularly good argument against existence being a property, apart from an argument from the authority of Kant.

WilliamM said...

this is an interesting take on the argument. Here's a problem: usually, the properties that are used to run this argument ("positive", "perfection") are chosen in such a way that the corresponding variation of (2) is true by definition (and (1) is not as much). In your case, (1) is true by definition, but it seems to me that (2) is problematic.

More precisely, problematic depending on one's idea of "limiting". Isn't every property in some sense limiting that it limits you from having its negation?

At this point, it gets hard to see what is question-begging and what simply leads to a conclusion contrary to one's views. Obviously, any valid argument for the existence of God leads to a conclusion contrary to the atheist's views.

I'm glad I'm not the only one who has reached this conclusion.

Alexander R Pruss said...

I don't think being non-cruel or believing that 2+2=4 are limiting.

Sam Calvin said...
This comment has been removed by the author.
Sam Calvin said...

Alex, your first comment sounds like Robert Maydole's defense of the possibility premise in his Modal Perfection Argument.

Sam Calvin said...

Alex, you mention that a forthcoming paper will discuss this argument (which I find fascinating) in more detail.

Any word on when it might be available?

Alexander R Pruss said...

I think it's in this collection. Which is very expensive.

You can email me for a copy of my contribution.