Goedelian ontological arguments improved even more
September 11, 2009 — 12:13

Author: Alexander Pruss  Category: Existence of God  Tags:   Comments: 31

In a piece (based on a post I made on Prosblogion almost two years ago) that has just come out in Religious Studies (with a response by Graham Oppy), I prove a certain theorem.  Say that a property A is strongly positive iff, necessarily, having A essentially is a positive property.  Assume the following three axioms:

  • F1: If A is positive, ~A is not positive.
  • F2: If A is positive and A entails B, then B is positive.
  • N1: Necessary existence is positive.

Theorem T1: Given F1, F2 and N1, if A is a strongly positive property, then there exists a necessarily existing being that essentially has A.

Assume also:

  • N2: Essential omniscience, essential omnipotence and essential perfect goodness are positive properties.

Then we get the following result.

Corollary C1: Given F1, F2, N1 and N2, there exists a necessary being that is essentially omniscient, and a necessary being that is essentially omnipotent, and a necessary being that is perfectly good.

But I was unable to prove, without assuming further controversial axioms, that there is one being that is omniscient and omnipotent and perfectly good.  I can now do so as long as one grants the following axiom:

  • N3: There is at least one strongly positive property that, necessarily, is uniqualizing.

A property is said to be uniqualizing provided that it is impossible for there to exist in one world two distinct things that have the property.  For instance, being the tallest woman is uniqualizing.  Note that it is prima facie possible Janet to have a uniqualizing property in one world and for Patricia to have the same property–but in a different world.

Theorem T4: Given F1, F2, N1 and N3, there exists a unique necessary being that has all the strongly positive properties.

Corollary: Given F1, F2, N1, N2 and N3, there necessarily exists an essentially omniscient, omnipotent and perfectly good being.

Moreover, I think a good case can be made (see point 1 below) that N2 implies N3, so in fact, the controversial axioms are going to be F1, F2, N1 and N2, just as in T1.

First, two arguments for N3. 

1. It seems impossible for there to be two omnipotent beings in one world.  For then the exercise by each of omnipotence would have to be under the other’s control, and that would generate a vicious regress or circularity of control.  Hence, necessarily, omnipotence is uniqualizing, and by N2 (which the applications of Theorem T4 will anyway assume), omnipotence is strongly positive (note that axioms are supposed to hold necessarily).

2. It seems plausible that some necessarily uniqualizing property like being the wisest or being the creator of every being other than oneself or being the ground of being is strongly positive.

Other examples of plausibly strongly positive uniqualizing properties would be welcome. 

To prove T4, we need two little results:

Lemma L1: Given F1 and F2, any pair of positive properties is compossible.

(This is proved in the paper.  But the argument is easy.  If they aren’t compossible, then each entails the other’s negation.  Hence the negation of each is also positive by F2, and by F1 no positive property has a positive negation.)

Lemma L5: Given F1, if A is strongly positive, then having A essentially is also strongly positive.

(This is proved in the paper, using S4.)

Now, let’s prove T4.  Let U be a uniqualizing strongly positive property by N3.  By T1, there is a necessarily existing being that essentially has U.  Call this being “Umberto”.  Now, let A be any strongly positive property.  Let EA be the property of having A essentially.  By L5 and L1, there is a possible world w, and a being x in w that has both EA and U.  Since U is uniqualizing and Umberto exists in w and has U in w, it follows that x = Umberto.  Therefore, Umberto has EA.  But then Umberto has A in every possible world, since Umberto exists in every possible world.  Thus, we have shown that Umberto necessarily and essentially has every strongly positive property.  Moreover, clearly, nobody but Umberto can be like that, because one of these properties is U.  Q.E.D.

Comments:
  • Mike Almeida

    Alex,
    This is a neat argument. But I think the problem we talked about last time remains. Since F1 and F2 entail that the traditional attributes are compossible, and since the compossibility of the divine attributes is more or less the entire point of contention between theists and non-theists, no rational non-theist is going to concede that all of the traditional attributes are positive properties. They can agree that the properties are desirable and great-making, but they will (and should) deny that they have the logical properties of positive properties. Do you have an argument that the traditional attributes are positive properties in your sense of positive?

    September 11, 2009 — 14:32
  • Mike,
    Well, how about this rational process. We have the concept of a perfection and the concept of a positive property. We reflect on the concept of a positive property (in the paper, I have three different characterizations of this concept), and this reflection leads us to F1 and F2. We reflect also on the concept of a perfection, and we come to an axiom that relates pefections and positive properties: all perfections are positive. We also have paradigm cases of perfections, paradigm cases which we cannot easily deny because it was by abstracting from these that we arrived at the concept of a perfection. These cases are the traditional attributes.
    So, the argument form is this:
    1. The traditional attributes are perfections. (Paradigm cases are presumed to be genuine cases, barring argument to the contrary.)
    2. All perfections are positive. (This we get on reflection on the connection between positiveness and perfection.)
    3. Therefore, the traditional attributes are positive.
    4. Therefore, God exists by T4.

    September 11, 2009 — 22:17
  • Mathis

    Hi Mike,
    consider the same argument, just with omniscience and perfect goodness left out and have omnipotence defined as “the greatest possible power”. Would that be an improvement?
    As for me, I have problems with F2 and N1, where F2 seems of greater importance.
    Consider the property S:
    Sx = x is so that for every P that is a possible
    – horrible thing
    – instance of great suffering
    – morally wrong action
    – etc.
    P is possible.
    S is entailed by every property but it doesn’t seem positive.

    September 12, 2009 — 3:31
  • Mathis

    P.S.
    Since I started reading the Blackwell Companion to Natural Theology that Craig and Moreland edited, I finally had the time for the chapter on the ontological argument.
    There, Maydole offers a solution to the above problem. Make the following adjustments:
    F2′: Nontautological properties which are entailed by a positive property are positive
    N1′: Necessary existence is positive and nontautological.
    N2′: Essential omniscience, essential omnipotence and essential perfect goodness are positive and nontautological properties.
    And he writes that “the resulting ontological argument would still be valid, but its premises would not imply that tautological properties are positive”.
    But is it valid? I don’t see it.

    September 12, 2009 — 4:26
  • Mike Almeida

    We reflect also on the concept of a perfection, and we come to an axiom that relates pefections and positive properties: all perfections are positive.
    I don’t think it follows from the concept of being a perfection that all perfections are compossible. It would have to be a priori necessary that P is a perfection only if P is positive. But I don’t see the a priori necessity. In any case, this is not the sort of thing that objectors to the traditional ontological argument are going to find persuasive.

    September 12, 2009 — 9:57
  • Clayton Littlejohn

    “F2: If A is positive and A entails B, then B is positive.”
    So there are no positive properties that entail the possession of any negative property? That might be right, but it’s not sufficiently self-evident to some of us to see that it’s right. I suppose if you were sceptical of the very existence of negative properties, this would be easy to swallow. F2 would be trivially true. If, however, you are not sceptical of the existence of negative properties, could you spell out the rationale for F2 for us here?

    September 12, 2009 — 16:57
  • Christian

    “F2: If A is positive and A entails B, then B is positive.”
    I share Clayton’s worry.
    P entails not-(not-P).
    But P is positive and not-(not-P) is negative. So that would be a counterexample, unless you think both expressions express the very same property, namely, P. But why would that be? The first involves no function-application, while the second does.

    September 12, 2009 — 18:11
  • CliveStaples

    I understand that any necessary being exists by definition. But how do you show that the set of necessary beings is not empty?

    September 12, 2009 — 21:11
  • Mathis

    Clayton,
    it doesn’t seem like “entails the possesion of any negative property” is positive. It seems very negatve.

    September 13, 2009 — 2:09
  • Let me quote from the beginning of the paper:
    The basic primitive notion in a Gödelian argument is that of a positive property. We can understand that in several different ways, each one giving rise to a different interpretation of the argument. For instance, one might take a positive property to be one that in no respect detracts from any respect of the excellence (or greatness or value, depending on how we prefer to phrase it) of the entity that has the property but whose negation does detract from some respect of the excellence (or greatness or value) of the possessor. Or one might take a positive property to be one that does not entail any limitation but whose negation does. Or one might start with a somewhat Leibnizian structure of the space of properties, on which there are some basic properties that are mutually compatible (e.g., because they are logically independent of each other), and then count a property as positive provided that it is entailed by at least one of the basic properties. Or, following Robert Maydole’s discussion of “perfections”, one might take a positive property to be one that it is better to have than not to have (Maydole 2003, 302).
    Each of these interpretations makes prima facie plausible (and in the case of some of the interpretations, entails) the following two “formal” axioms:
    Axiom F1. If A is positive, then ~A is not positive.
    Axiom F2. If A is positive and A entails B, then B is positive.
    The correctness of F1 on the excellence, goodness, greatness and no-limitation readings is clear, and on the Leibnizian interpretation F1 follows from the compatibility of the basic properties. Moreover, if a property doesn’t detract from the excellence (or goodness or greatness) of an entity, then anything it entails had better not detract from it either. On the other hand, if a property detracts from the excellence (or goodness or greatness) of an entity, so does any property that entails that property. Hence if ~A detracts from excellence (etc.), and A entails B, then ~B detracts from excellence (etc.), since ~B entails ~A by contraposition. This yields Axiom F2 on the excellence, goodness and greatness interpretations. Exactly the same reasoning shows that if a property does not entail any limitation but its negation does, the same holds for any property that it entails. And closure under entailment is trivial on the Leibnizian interpretation, so F2 follows once again. Maydole gives arguments for F1 and F2 on his interpretation (Maydole 2003, 302).

    September 13, 2009 — 19:56
  • CliveStaples: The proof of T1 in the paper proves that if F1, F2 and N1 hold, then there are necessary beings.

    September 13, 2009 — 19:58
  • Clayton Littlejohn

    “Moreover, if a property doesn’t detract from the excellence (or goodness or greatness) of an entity, then anything it entails had better not detract from it either. On the other hand, if a property detracts from the excellence (or goodness or greatness) of an entity, so does any property that entails that property. Hence if ~A detracts from excellence (etc.), and A entails B, then ~B detracts from excellence (etc.), since ~B entails ~A by contraposition. This yields Axiom F2 on the excellence, goodness and greatness interpretations.”
    I haven’t had my morning coffee yet, but this seems really questionable. Suppose A entails B but B does not entail A. It seems that B might be negative and it might be that something is better off being B & A than being ~A & ~B and better off being B & A than ~A & B. B should count as negative if B is a flaw (right?). It is not the possession of A that would detract from the bearer of A’s excellence if the bearer is better of being A & B than ~A & ~B and ~A & B (right?). It still would be the possession of B that would detract from the bearer’s excellence. B is the flaw and A is the corrector, say. Suppose all humans sin and that’s not a contingent fact about us. Suppose that some humans seek forgiveness for their sins. Having that property entails that you have the property of being a sinner, but the property of seeking forgiveness for one’s sins is not what detracts from the excellence of a human.

    September 14, 2009 — 8:14
  • Mike Almeida

    I haven’t had my morning coffee yet, but this seems really questionable. Suppose A entails B but B does not entail A. It seems that B might be negative and it might be that something is better off being B & A than being ~A & ~B and better off being B & A than ~A & B.
    Clayton seems right here, but maybe for reasons that he won’t like. Plantinga (see his ‘Felix Culpa’) defends the view that among the greatest goods is the Incarnation and Atonement. Let W be a world that has the property P = being a world in which there occurs the Incarnation and Atonement. Any world that has property P also has the property E of being a world in which profound evil occurs. P is positive, but P entails E, and E is clearly negative. It is the rare theist (I don’t know one) that wants to deny that P is positive. I don’t know one who denies that P & E is better than ~P & ~E.

    September 14, 2009 — 8:43
  • Clayton Littlejohn

    I’m happy to be right for Plantingian reasons, Mike, except for those cases where I think Plantinga is wrong (which, admittedly, I thought was always). This morning after the comment, I was wondering about cases of the sort that someone like Plantinga might like (e.g., cases where there is the creation of significantly free creatures (+) where it cannot be that this creative act takes place without the creation of conditions that always involve some sin or other (-)).

    September 14, 2009 — 10:24
  • Mike:
    I don’t think P is positive, because P entails the property of being a world, and being a world rules out all sorts of perfections, such as perhaps being concrete, conscious, a person, etc. P is no more positive than being essentially a perfect dog is. Being perfect is nice, but being essentially a dog is quite limiting.
    Let’s see if your example can be transposed to a property of God. Let P* = becoming incarnate and atoning. Then, P* is a positive property. And P* entails being such that there is evil. But while being evil is plainly negative, I deny that being such that there is evil is negative. However, I do not think P* is strongly positive.
    Clayton:
    Minor point. Seeking forgiveness for one’s sins might not entail being a sinner, since “one’s sins” might be in the scope of an intentional operator. So let’s say “Seeking forgiveness for one’s actual sins”. Well, let’s try to put that property in a clearer logical form.
    “x seeks forgiveness for his* actual sins” (“his*” being Castaneda’s quasi-indicator) may be something like: “x is such that (exists S)(S is a sin and x has done S) and (S)(S is a sin and x has done S and x seeks forgiveness for S).” Or else, if the forgiveness-seeking is not distributed over sins, it’s “x is such that (exists S)(S is a sin and x has done S) and x seeks forgiveness for all of his* sins”. In both cases, we have a conjunctive property, one conjunct of which is that x has sinned. I think the defender of Goedel’s argument and the “detracts” reading can bite the bullet quite easily and say that, because of that conjunct, the property is negative.
    I think this highlights a deeper issue. In English, we have a way of highlighting the salient aspect of a property. Thus, we can say that
    (1) x seeks forgiveness for the sins he actually committed
    and we can say that
    (2) x has committed sins and he seeks forgiveness for them.
    It is plausible that (1) says something purely good about x, while (2) doesn’t. But unless we have a very abundant theory of properties, we’re not going to have properties so finely cut that they distinguish between (1) and (2). We’ll just have the conjunctive property of having sins and seeking forgiveness for them. This does mean that (1) should not be analyzed as of the form:
    (1*) x has P.
    There are deep and interesting issues here, aren’t there?

    September 14, 2009 — 10:27
  • Mike Almeida

    Let’s see if your example can be transposed to a property of God. Let P* = becoming incarnate and atoning. Then, P* is a positive property. And P* entails being such that there is evil. But while being evil is plainly negative, I deny that being such that there is evil is negative.
    Your initial objection looks question begging. Why wouldn’t I respond that what’s shown rather is that positive properties might entail properties that are not positive?
    Anyway, P* entails much more than what you suggest. It entails that there is a world which instantiates the property of containing profound evil. That is, necessarily, if God has the property of atoning for the deep sins of the world, then then there is some world whose sins are deep and atoned for. But the entailed property of containing deep sins is obviously not positive. So, P* is not positive, either, or positive properties might entail negative ones.

    September 14, 2009 — 12:00
  • Mathis

    The “O Felix Culpa” counterexample looks powerful.
    I don’t think that P* is positive (or negative) because it’s not something that adds up to the quality of his being, it’s an expression of love.
    As for “necessary existence”, “omnipotence”, “omniscience” and “moral perfection”, they seem to add up to the quality of the being that posesses them.
    I think we should revise our concept of “positive”.

    September 14, 2009 — 12:18
  • Mathis

    Here’s a different idea:
    Assume “O Felix Culpa” is a good theodicy – it gives a plausible (and true) reason for why a morally perfect being would allow evil. It does so by showing that evil is entailed by incarnation and atonement. Necessarily, if God becomes and incarnate and uses atonement, he allows evil to enter the world.
    If the theodicy works, then allowing evil to enter the world does not make God morally evil. Why? Because IA makes God morally good. It seems that theodicies rest on:
    (*) If performing A is morally good and performing A entails performing B, then performing B is not morally evil.
    In fact, the whole point of “O Felix Culpa” seems to be:
    Allowing evil was actually [in some sense] good, because it allows incarnation and atonement.
    To me, this indicates again that our concept of “positive” needs revision.
    How about this:
    (1) IA is good*
    (2) IA entails E(IA)
    (3) E(IA) is good* (from 1,2 – see what I did here?)
    Where E(IA) is the evil that entered the world because of incarnation and atonement – good* is to be understood in context of greater-good-theodicies.
    If anyone here thinks that E(IA) is not good*, then it seems he must deny that IA is good*.

    September 14, 2009 — 12:37
  • Mike:
    When we say that a property is positive, we mean that it is positive to its bearer. It might still imply negative things about other beings. Thus, that x is essentially omnipotent entails that nobody else is, and that x is smarter than everybody else entails that everybody else is less smart than x.
    So I bite the bullet and say that God’s allowing profound evil is not a negative property of God.
    A property that contains a limitation isn’t going to be positive. Thus, “being a world such that …” is not going to be positive, just as “being a number such that …” is not going to be positive, because (at least on standard views of worlds and numbers) being a world and being a number are very limiting.
    But what the examples do show is that the axioms may well fail if we replace “positive” by “positive for Ks” where K is some kind. (Thus, being a fast runner is positive for dogs, and, maybe, being consistent is positive for worlds.) This enables a Geachian criticism: There is no such thing as a property that is positive simpliciter, just as there is no such thing as being good simpliciter–positiveness and goodness are kind-relative. I guess the defender of the argument just has to deny that.

    September 14, 2009 — 12:43
  • Mathis

    Alex,
    I don’t think becoming incarnate and atoning is positive for God.
    1) As I said before: it doesn’t improve his quality – it’s merely an expression of his love
    2) It entails that he suffers

    September 14, 2009 — 12:51
  • Mathis

    So how about this:
    A property is positive if and only if having it makes the being that has it more worthy of worship in terms of the beings quality.
    Alex argument seems to work with this account, but the “O Felix Culpa” counterexample doesn’t work with it, because:
    – Improving the quality of a world doesn’t make the world more worthy of worship – even pantheists agree that a world isn’t worthy of worship .
    – Becoming incarnate and atoning makes a being more worthy of worship in terms of the beings actions, not of the beings quality.
    I think this works fine.

    September 14, 2009 — 13:54
  • Mike Almeida

    When we say that a property is positive, we mean that it is positive to its bearer. It might still imply negative things about other beings
    But isn’t it negative to God that he has the property of having beloved creatures that suffer profoundly in W? How could that not be negative to God?

    September 14, 2009 — 17:30
  • Mike:
    That he has them suffer is compatible with the proposition that he has them suffer for a very good reason, so maybe it’s not negative.
    That said, probably a better way of responding is this. What is positive is so unequivocally. In particular, where there are two incompatible and incommensurable goods, G1 and G2, then neither G1 nor G2 will be positive, because each rules out the other, and to that extent is non-positive. Now, incarnation and atonement are a great good, but there is an incompatible good–and probably incommensurable with this one–namely the good of incarnation and community with sinless human beings. Though both goods are very good indeed, neither good is positive in the relevant sense, since each rules out the other.

    September 14, 2009 — 21:25
  • I think (I haven’t checked all the details) that I can get away with weakening the conjunction of F1 and F2 to the conjunction of:
    F1*: If A is strongly positive, then ~A is not strongly positive.
    F2*: If A is strongly positive and A entails B, then B is positive.
    F3*: If A is strongly positive, then necessarily EA (having A essentially) is strongly positive.
    Since the incarnation and atonement are not strongly positive–it wouldn’t be good if God essentially had to atone, since then he necessarily would have to create sinners–the counterexample disappears.

    September 14, 2009 — 21:42
  • Mathis

    That said, probably a better way of responding is this. What is positive is so unequivocally. In particular, where there are two incompatible and incommensurable goods, G1 and G2, then neither G1 nor G2 will be positive, because each rules out the other, and to that extent is non-positive.
    I’ve been trying to avoid formulations like this. Now your premises say that the traditional attributes don’t rule out each other. Not so good.

    September 14, 2009 — 23:53
  • Mathis

    I think (I haven’t checked all the details) that I can get away with weakening the conjunction of F1 and F2 to the conjunction of:
    F1*: If A is strongly positive, then ~A is not strongly positive.
    F2*: If A is strongly positive and A entails B, then B is positive.
    F3*: If A is strongly positive, then necessarily EA (having A essentially) is strongly positive.

    This sounds very good. Maydole’s MPA might also work if he would use:
    (M1) The negation of a perfection is not a perfection
    (M2*) I having EP is a perfection and P entails Q then Q is a perfection.
    (M3) Supremity is a perfection
    Where supremity is defined as the property of being so that no greater being is possible and nothing else could possibly be equally great.
    Since “supremity” is identical to “essential supremity”, it is possibly exemplified which, together with the controversial Barcan Formula, is all he needs.
    I think the new F2 is exactly what Gödel-style arguments need.

    September 15, 2009 — 13:28
  • I definitely want to avoid the Barcan Formula!

    September 15, 2009 — 16:22
  • Mathis

    I definitely want to avoid the Barcan Formula!
    This might be a good idea, but I think Maydole’s idea that an ontological argument doesn’t need to rely on things like your N1 is interesting.

    September 16, 2009 — 11:13
  • I prefer to rely on a controversial but true premise to relying on a false premise. 🙂

    September 17, 2009 — 8:40
  • Here’s a fun argument without N1. Start with F1, F2, N2 and:
    N4. If p is a possibly true proposition, then MKp, the property of possibly knowing p, is positive.
    Let p be the proposition that there are no essentially omnipotent beings. If possibly p, then MKp is positive. By N4 and L1, essential omnipotence is compatible with MKp, which is absurd. Hence, necessarily, ~p. Hence, necessarily, there is an essentially omnipotent being.
    Suppose we add the claim that the conjunction of compossible strong positives is positive. Then we can get an essentially omnipotent, omniscient and perfectly good being.
    I bet there are other fun things we can do with N4, but I need to get ready to go to class.

    September 17, 2009 — 9:09
  • Mathis

    I prefer to rely on a controversial but true premise to relying on a false premise. 🙂
    Actualist bias ;-P

    September 17, 2009 — 9:43