Tuesday, February 27, 2018

A problem for Goedelian ontological arguments

Goedelian ontological arguments (e.g., mine) depend on axioms of positivity. Crucially to the argument, these axioms entail that any two positive properties are compatible (i.e., something can have both).

But I now worry whether it is true that any two positive properties are compatible. Let w0 be our world—where worlds encompass all contingent reality. Then, plausibly, actualizing w0 is a positive property that God actually has. But now consider another world, w1, which is no worse than ours. Then actualizing w1 is a positive property, albeit one that God does not actually have. But it is impossible that a being actualize both w0 and w1, since worlds encompass all contingent reality and hence it is impossible for two of them to be actual. (Of course, God can create two or more universes, but then a universe won’t encompass all contingent reality.) Thus, we have two positive properties that are incompatible.

Another example. Let E be the ovum and S1 the sperm from which Socrates originated. There is another possible world, w2, at which E and a different sperm, S2, results in Kassandra, a philosopher every bit as good and virtuous as Socrates. Plausibly, being friends with Socrates is a positive property. And being friends with Kassandra is a positive property. But also plausibly there is no possible world where both Socrates and Kassandra exist, and you can’t be friends with someone who doesn’t exist (we can make that stipulative). So, being friends with Socrates and being friends with Kassandra are incompatible and yet positive.

I am not completely confident of the counterexamples. But if they do work, then the best fix I know for the Goedelian arguments is to restrict the relevant axioms to strongly positive properties, where a property is strongly positive just in case having the property essentially is positive. (One may need some further tweaks.) Essentially actualizing w0 limits one from being able to actualize anything else, and hence isn’t positive. Likewise, essentially being friends with Socrates limits one to existing only in worlds where Socrates does, and hence isn’t positive. But, alas, the argument becomes more complicated and hence less plausible with the modification.

Another fix might be to restrict attention to positive non-relational properties, but I am less confident that that will work.


Tom said...

Dr. Pruss,

I do not think the arguments here are terribly concerning, for the Goedelian proof, at least not the first one. Your comment at the end about relational properties is probably more useful than you seem to think, at least in the case of God.

In my view, God 'actualizing W0' is not even a "real" property of God. Or to put it another way, it is not an intrinsic property, essential or accidental, of God. This seems to be your view too given your work on divine simplicity. And I think restricting the attention to positive, intrinsic, properties is completely appropriate and not ad-hoc.

As for your second argument, this is a little trickier but I have two thoughts:
(1) To the extent that "being friends with" involves Cambridge properties, what I say above about God is applicable. It will be more complicated since obviously "being friends with" is not entirely a Cambridge property, nevertheless, I have a feeling that this might be work-able.

(2) I am not confident that it is impossible that Kassandra and Socrates exist in the same world. I do not know your views on essentiality of origins and the specifics of what makes E, E as opposed to E'. That said, it is not completely absurd to suppose that if there was an egg that had the exact same DNA content as E (call it E**), then whether or not S2 fertilizes E or E** is irrelevant, and either are sufficient for the existence of Kassandra.

Now, the simultaneous existence of E and E** in the real world is manifestly extremely unlikely. But when talking about possibility as opposed to probability, there is no reason to say that E and E** cannot coexist (it is even biologically possible, although obviously extremely unlikely)

Regardless of specifics, I think the complexity of the issue at very least shows that there are good reasons to suppose the premise of the Goedelian argument can be resurrected despite these apparent objections

VeritasSeeker said...

Dear Dr. Pruss,

As per request, I am posting the relevant portions of my response here.

My intuition is that ^the property of being friends with Kassandra is positive^ is false at w0, at least because it implies (but does not strictly imply!) the impossible property of being friends with a non-existent person.

Moreover, I just don't think that being friends with Socrates or Kassandra is a positive property! I think that these properties are neutral properties, which are neither positive nor negative. Recall that on your definition a positive property is one which in no way detracts from the greatness of a being which has it, but whose complement does. I don't think that not being friends with Socrates or possible Socrates-like entities detracts from our greatness. Indeed, it seems strange to believe that the properties of lacking friendships with some great possible persons x1....xn are negative properties (i.e., complements of positive properties), since that would mean that God has an infinite amount of negative properties!

Wielka Miska said...

Does every property of God need to be positive? If not, then creating w0 can be just neither positive nor negative.

This way we could treat a Godelian argument as a framework for proving the existence of things, one instantiation of the argument would prove an omniscient being, another one an omnipotent thing, and so on. Instead of trying to find the widest possible definition of positivity, we could focus on more tangible, narrower positivities, where it’s easier to prove that they are internally compatible. For example, if we said that being able to cause things that are possible in principle is positive, and also existence is positive, and that’s it, nothing else is positive (unless we need to add some things due to entailment, but I’m not sure what that would be in that case) - then we can prove the existence of an omnipotent being. That’s an instance of the strongly positive properties you mention, and if a wider definition is self-conflicting, then it’s fine to me - I don’t expect the omnipotent God to be able to create square circles either.

Similarily with omniscience, but - because positive properites need to be positive necessarily - I wouldn’t focus on keeping facts in memory, because that is backed by the material brain, whose existence relies on the structure of the actual world. Instead, I’d rather define having fact X (where X is any necessary fact) in consciousness as positive (assuming consciousness isn’t material, hence could be instantiated in any possible world, which means that this is a better candidate for necessarily positive properties). That even seems to work well with the immutable, simple God from cosmological arguments - He shouldn't have to use a “storage” for memories. (This one might be a bit bold and just assumes too much.)