# A variant on the Anderson-Goedel ontological argument

The following is a version of the Anderson-Goedel ontological argument with somewhat weaker premises, and which is apparently not subject to Oppy's parody (Analysis, 2000):

Axiom 1. If a property is positive, its negation is not positive.
Axiom 2. If P is a positive property, and P entails Q, then Q is a positive property.
Axiom 3. Necessary existence is a positive property.
Definition 1. If P is a property, then EP is the property of having P essentially.
Definition 2. P is strongly positive iff it is necessary that EP is positive.
Definition 3. God-likeness is the property of having all strongly positive properties.
Axiom 4. God-likeness is a positive property.

Given an appropriate modal logic that includes S5 and assumes that properties are necessary beings, we get:

Theorem 1. If Axioms 1-4 hold, then there necessarily exists an essentially God-like being.

Note: Since EP entails P, by Axiom 2, any strongly positive property is positive.

The details of proof don't matter much but are below the fold for completeness.

The hard question is whether we can give an interpretation to the notion of a "positive property" that makes Axioms 1-4 plausible and the conclusion of Theorem 1 interesting.

The proof depends on several subsidiary results. (I start with Lemma 2, because Lemma 1 turned out to be unneeded and I don't want to renumber.)

Lemma 2. Given Axioms 2 and 3, necessary existence is strongly positive.

Proof of Lemma 2: Let N be necessary existence. If x exists necessarily, then it is necessarily true that x exists necessarily, by S4. Hence, N entails EN. By Axioms 2 and 3, EN is positive. Moreover, this proof works necessarily (we only made use of the axioms, and axioms are assumed to hold necessarily), so EN is necessarily positive.

Lemma 3. Given Axioms 1 and 2, every positive property is possibly exemplified.

Proof of Lemma 3: Suppose for a reductio that P is positive but not possibly exemplified. Hence, necessarily, nothing has P. Hence, necessarily and trivially, everything that has P has ~P. Hence, P entails ~P. Hence, ~P is positive as P is, by Axiom 2. But ~P is not positive, by Axiom 1. Hence absurdity ensues.

Lemma 4. If Axiom 2 holds, and P is strongly positive, then EP is strongly positive.

Proof of Lemma 4: Necessarily, EP is positive. EP entails EEP by S4. Hence, necessarily, EEP is positive, by Axiom 2. Hence EP is strongly positive.

Lemma 5. If P possibly is strongly positive, then P is strongly positive.

Proof of Lemma 5: Suppose possibly P is strongly positive. Then, possibly, necessarily EP is positive. By S5, necessarily, EP is positive. Hence, P is strongly positive.

Lemma 6. Given Axioms 1 and 2, if x has God-likeness, then x essentially has God-likeness.

Proof of Lemma 6: Suppose for a reductio that x possibly lacks God-likeness. Then, possibly, there is some strongly positive property that x lacks. Since properties are necessary beings, and given S5, existential quantification over properties and possibility can be interchanged. Hence, there is a property P such that possibly P is a strongly positive property that x lacks. Let P be such. Then, possibly P is strongly positive. Hence, P is strongly positive by Lemma 5. Hence, x essentially has P. But we said that possibly x lacks P. Hence a contradiction ensues.

Proof of Theorem 1: God-likeness is possibly exemplified, by Axiom 4 and Lemma 3. Thus, possibly, there is an x that has God-likeness. God-likeness entails necessary existence, by Lemma 2. Hence, possibly, there is a necessary being x that has God-likeness. But, by Lemma 6, anything that has God-likeness has God-likeness essentially, and this holds necessarily since it is a logical consequence of the axioms. Thus, possibly, there is a necessary being that essentially has God-likeness, since if a necessary being has an essential property, it is necessary that it have this essential property. Hence, possibly, necessarily there is an x that is essentially God-like. Hence, necessarily, there is an x that is essentially God-like, by S5.

Alex, according to your Def. 3, we have,
Definition 3. God-likeness is the property of having all strongly positive properties.
But there is nothing in the axioms/defs. that ensures that all positive properties are compossible or that all positive properties can be coinstantiated in a single being. Therefore there is no reason at all to believe Axiom 4 is true,
Axiom 4. God-likeness is a positive property.
But then the proof can't be sound.

I wouldn't say that there is no reason to believe Axiom 4. It has some innate plausibility--it is plausible that conjunctions of strongly positive properties are positive.

Alex,

Maybe you're right. But among the central objections to ontological arguments, no doubt, is that nothing could instantiate all of the divine attributes or perfections. Making your property of "godlikeness" a positive property will seem to many people question-begging. After all, all positive properties are possibly exemplified, therefore godlikeness is possible exemplified. But then it is actually exemplified. Once Axiom 4 is granted, God's existence quickly falls out.

Mike:

Questions of how quickly something falls out are always tricky. The proof of Lemma 3 is not entirely trivial. :-)

As an autobiographical fact, it didn't occur to me when reading Axioms 1 and 2 that they entail that what is positive is possibly exemplified. Once I read the claim that the axioms (in the original Anderson-Goedel setting) entail that what is positive is possibly exemplified, I quickly managed to prove it. But that is how it sometimes is: One doesn't see that two propositions imply a third, but once one finds out (e.g., by testimony) that they do, it's easy to prove why. That's a non-trivial entailment.

That said, I do have a further argument for Axiom 4.

4a. If someone leads a flourishing human life centered primarily on loving a being under some description D, then, probably, the property of falling under D is positive.

4b. Some people live flourishing human lives centered primarily on loving the being that has all positive properties.

4c. Therefore, probably, having all positive properties (and, a fortiori, having all strongly positive properties) is positive.

Alex,

Very interesting! Can you give me a hint of what a positive property might be? By axiom 2 positive properties can be wildly disjunctive, conditional, or expressed with many negations. Also, take a property p and form the big disjunction of the all the rest of the properties. Property P is the same property as the negation of the big disjunction of the rest of the properties. But, intuitively, that description of property P doesn’t seem positive. (There are a lot of assumptions in this little argument which is meant only to gain some clarification on the notion of a positive property).

Alex,

What I had in mind is that once you assume Axiom 4, the philosophical work is done. All that is left is some logical work. Though I agree that the proof of Lemma 3 is interesting and not obvious, it does not help with the philosophical question of whether anything could instantiate all of the positive properties.I've been conceding that the positive properties include all of the traditional divine attributes. But that too is an open question. It might be, for instance, that the positive properties do not include omnipotence, since necessarily nothing is omnipotent. Or, more weakly, one could concede that all of the traditional divine attributes are positive properties, but disagree (on the basis of the modal argument from evil (MAE)) that anything instantiates them all. Some MAE's are quite powerful arguments.
Finally, on your proof of axiom 4, I admire your dialectical boldness! Doesn't premise (4b) entail that there is a being that has all of the positive properties!?

(4b') Sue loves the being with all of the positive properties.

If (4b') is true then,

(4c') :. (Ex)(Sue loves x)

Mike:

I don't find MAEs that powerful, because I think imaginability is merely a guide to possibility, and because on my view of all possibility as causal possibility, they don't work (but the ontological argument isn't all that valuable on that view either :-) ).

In (4b), I was using "loves" non-veridically, as in "Bob loves his imaginary friend." This is inconsistent with the usage in an earlier post of mine, and I apologize for the inconsistency. If you prefer "putatively loves" or something like that, that's fine. The point I want to make is that it just barely might be possible to lead a prima facie flourishing life centered on putative love for something unreal (e.g., take someone who leads a flourishing life centered on the overly charitable assumption that everybody else is a saint, and so he loves everybody else under descriptions that do not apply to them), but a flourishing life centered on putative love for something impossible is implausible.

Ted:

I confess to not having found a good account of positivity to make this argument go. I'm thinking a fair amount about it, though. :-) Let me share some of my thinking, and maybe you can help me.

Here are some options I'm not happy with.

1. A property is positive provided that it does not entail any imperfection or limitation. The problem with this is that Axiom 1 does not seem to be satisfied. Creating horses does not entail any imperfection or limitation, but neither does not creating horses.

2. We take some primitive set of "perfections" (e.g., properties that affirm in an unlimited way the presence of something unlimitedly valuable) and say that a property is positive provided it lies in the closure of that set under entailment. I don't like this. I want positivity to be closely related to some pretheoretic notion.

There are some others. I agree it's not much of an argument without some such account.

Minor update: We can get pretty far without Axiom 4, but with Axiom 4* instead:

Axiom 4*. If P and Q are strongly positive and mutually consistent (i.e., it's possible to have both), then their conjunction is positive.

Axioms 1-4 clearly entail Axiom 4*, so this is a weaker version.

Theorem 2. If Axioms 1-3 and 4* hold, and S is any finite set of strongly positive properties, then there necessarily exists an entity that essentially has every member of S.

Thus, if omniscience, omnipotence and moral perfection are strongly positive, it follows that there necessarily exists an essentially omniscient, omnipotent and morally perfect being.

Why does Theorem 2 hold?

Lemma 7. Given Axioms 1 and 2, if P and Q are positive properties, then P and Q are consistent.

Proof of Lemma 7: If they are not consistent, then P entails not-Q, and hence not-Q is positive, and hence not-Q is both positive and not positive.

Lemma 8. Given Axioms 1, 2 and 4*, any finite conjunction of strongly positive properties is strongly positive.

This is pretty easy to prove: just show that a conjunction of two strongly positive properties is strongly positive by using Lemma 7 and Axiom 4* (apply Lemma 7 not to the strongly positive properties P and Q, but to EP and EQ), and go on from there.

Given Lemma 8, we prove Theorem 3 pretty easily. Let S* be S plus necessary existence. Then, S* is a finite set of strongly positive properties, and hence the conjunction of all the members of S* is possibly instantiated by Lemma 8. It's not hard to go from that to the conclusion of Theorem 2.

Ted, fwiw

I believe there are accounts of positive properties in David Johnson's version of Godel's argument, 'A Modal Ontological Argument' and J. Howard Sobel's "GĂ¶del's Ontological Proof," in Being and Saying: Essays for Richard Cartwright, ed. J. J. Thomson, Cambridge, Mass., (1987) 241-61 (also in his Logic and Theism); Johnson's 'A Modal Ontological Argument'is chp. 4 in David Shatz (ed) Philosophy and Faith (MacGraw Hill, 2001)

Alex,

Axiom 4* does seem plausible, but how does it advance the argument? We still have the problem of showing that the traditional divine attributes are "strongly positive and mutually consistent". I incidentally do find Ted Guleserian's MAE
very powerful, and I don't think it depends on there being worlds that are not causally possible. That is, I don't see why he could not run the same argument on that restriction. All he really needs is some world in which there is gratuitous evil.

Hi Alex,

One other quick concern. There are properties P such that neither P not ~P is positive. For instance, let P be the property of weighing less than 10lbs. on Friday, November 30. Having the property is not positive and lacking it isn't either. It is nonetheless true that the disjunctive property of (weighing less than 10lbs. or not weighing less than 10lbs.) is a positive property. All necessary properties are positive, since they are all entailed by positive properties. That has to be mistaken, since a disjunction of non-positive properties cannot itself be a positive property.

Mike:

Discussion is advanced because an axiom is replaced by a more plausible one. :-)

Ah, but to argue that there is a causally possible world with gratuitous evil, you need to show that, given the things that existed in our world at the beginning, it was possible for evil to come about. But among the things that existed in our world at the beginning was the First Cause. Hence, one has to show that the First Cause would allow a gratuitous evil.

I don't have any problem with the idea that a disjunction of non-positive properties might be positive. Suppose P is such that neither P nor ~P is positive. (In the original Goedel setting, P is positive iff ~P is not, so that won't happen.)

Let Q be any positive property.

Consider: Q1=(Q and P). This is not positive if P is not. Likewise Q2=(Q and ~P) is not positive. But the disjunction of Q1 and Q2 is uncontroversially positive, since it's equivalent to Q. That assumes equivalent properties are identical in positivity.

Mike:

Actually, given Axioms 1-3 and 4*, we only need to show that the traditional attributes of God are strongly positive. We don't need to show that they are compatible--that follows from Lemma 7.

Alex,

Take any positive property P. P entails P v Q (where Q is a negative property). It follows that in worlds where x's nontrivial properties include only negative properties including Q (and does not include P) it has the non-trivial positive property P v Q. So there are no worlds in which anything has no non-trivial positive properties. But certainly something in some world has no non-trivial positive properties! Therefore, the logical properties of positive properties must be mistaken.

Regarding the following,

Ah, but to argue that there is a causally possible world with gratuitous evil, you need to show that, given the things that existed in our world at the beginning, it was possible for evil to come about.

Not so! What you have to show is that there is a world that includes the same laws as ours in which we have gratuitous evil. It need not contain any of the contingent facts of this world.

Mike;

I think we're disagreeing on what "causally possible" means. By "causally possible" I mean that there is a causal history, starting at the same starting point as actually occurred, leading to the events. On my causal account of modality, the origination is necessary.

I don't see why it is a problem that nothing can lack a positive property. This seems exactly right. A positive property is a positive way of being, without admixture of limitation, or something like that. Everything should have some positive property to exist.

Mike & Alex,

I'd certainly benefit from a discussion on the MAE if either of you are willing to post something on it.

Alex,

I like the proof of lemma 7. But reconsider the larger proof in light of lemma 7. You've got Godlikeness defined this way:

Def. 3: God-likeness is the property of having all strongly positive properties.

And you take the instantiation of Godlikeness to be a proof that God exists. But it is not true that Godlikeness is instantiated only if God exists. Suppose, as you do, that necessary existence is a positive property. Now by lemma 7 any proeprty that is inconsistent with necessary existence is not a positive property (rather it's negation is positive). So suppose we find a good argument that Omniscience, Omnipotence, Omnibenevolence are all inconsistent with necessary existence. Does it matter to the proof that God exists? No, it doesn't matter at all to this proof. In that case the set S of positive properties will be following:

S={necessary existence, non-omnipotence, non-omnisience, non-omnibenevolence, etc., etc. . . .}

And Godlikeness G will be the property formed from the conjunction of the properties in S. You conclude from G's instantiation that God exists. But it is false that whatever instantiates G is God. So, it is not necessary that (x)(x instantiates Godlikeness only if x is God).

Christian,

I'm not a contributor, so I cannot post on MAE. Maybe Alex will post on it. Otherwise, you might want to look at Ted Guleserian's, 'God and Possible Worlds: The Modal Problem of Evil' Nous ('83).
To my mind, it's the best version of MAE.

Mike:

That's a nice objection, and one rarely made, I think. It's analogous to saying that, maybe, that than which nothing greater can be conceived is not omniscient, etc. (e.g., because omniscience is impossible or something like that).

That said, it seems highly plausible that essential omniscience, essential goodness and essential omnipotence are positive.

Alex,

All this argument shows is that there is something that has the property of Godlikeness. Nothing in the proof guarantees that anything that instantiates Godlikeness is God. But that is just to say that the argument is invalid.

Maybe this is a better way to put it. To make the argument valid you need to add the premise that the traditional divine attributes are positive properties. Since positive properties are mutually consistent, this is a premise that MAE's (and other arguments against the consistency of divine attributes) dispute.

Mike:

"I'm not a contributor, so I cannot post on MAE."

Wow! You think someone would have invited you after all of your wonderful contributions.

(HINT...HINT...HINT)

I trust Mike knows that he has an open invitation to return to posting as a contributor.

Thanks Christian. I was a contributor a while back, and since have, without much (some would reasonably say without any) success, tried to make better use of time. Thanks Matt.

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