Two More Ontological Arguments

Josh, very cool. Questions about #1:

(1) It looks like the argument shows that every perfection is exemplified to the maximal degree. Congratulations, you seem to have reconstructed something like Aquinas's Fourth Way! But more seriously, does your argument show that all flavors of greatness of maximized in the same being? You say that maximal greatness just IS the greatest conceivable being, but you might think that it is the property of having all the great-making properties to the maximal degree. Your proof could show that this property is exemplified, but it looks like a set could have the property (a set containing a perfectly good being, another perfectly powerful being, another perfectly beautiful being, etc.). Sorry if I'm missing something.

(2) I am teaching Phil Religion this fall and I was trying to explain why some people deny S5. What's your defense of S5? My apologies if this generates a long answer!

(3) Post more!

Joshua, you write,

Axiom 3: Maximal greatness (being the greatest conceivable being) is the simplest determinate of greatness. . . I'm assuming that zero greatness is not a determinate of greatness, since if something had zero greatness it wouldn't even have the determinable, greatness.

It's hard to see why maximal greatness would be easier to grasp than, say, the level of greatness we see exemplified in human beings. Or, to make it even easier, the level of greatness exemplified by my lawn. Not too difficult.

Also,

Axiom 2: being contingent is a contingent property.

I don't think that's a principle you can endorse under S5, assuming you are contrasting contingent beings with necessary beings. If I'm possibly necessary, then (given S5) I'm necessary. So, it looks like being contingent entails being necessarily contingent. Similarly for properties. If the property of being left-handed is one that I instantiate contingently, then there is no world in which I instantiate that property necessarily. Since if there is any world in which I instantiate that property necessarily, then I instantiate it in every world in which I exist. Therefore, I do not instantiate it contingently.

Hi Josh.

Long time, hope Notre Dame is treating you well. I heard the conference last week was pretty interesting, wish I could have been there. Anyway, here's a thought. Return to Axiom 3.

Axiom 3: Maximal greatness (being the greatest conceivable being) is the simplest determinate of greatness.

Suppose this axiom is true. If it is, then we should accept Axiom 3*.

Axiom 3*: Slightly-less-than-maximal greatness (being the the-slightly-less-than-greatest conceivable being) is slightly less simple than the simplest determinate of greatness.

I can't think of any reason to accept Axiom 3, but not 3*. We can then keep adding 'slightly less than' into our definitions, repeatedly, to get axioms 3**, and 3***, etc. Now go back to axiom 1.

Axiom 1: The simplest determinate of an exemplifiable determinable is itself exemplifiable.

I think a similar thing to what I said above would be true here, namely,

Axiom 1*: A slightly-more-complex-than-the-simplest determinate of an exemplifiable determinable is itself exemplifiable.

And as with axiom 3*, so it goes with axiom 1*. Denying 1* would be like accepting that the color determinate scarlet is exemplifiable, but some color just slightly more intense than scarlet is not. Were your argument sound, these other arguments (it seems) would also be sound. We then get ontological arguments for the existence of infinitely many Gods! I guess I think this is a reductio on the initial argument.

Two other things: First, why the talk of the "simplest determinate" in the first place, do you need it? I would have thought that if a determinable D is exemplifiable, then necessarily, for any determinate d of D, d is also exemplifiable. No?

Also, I "think" I accept S5. But, nevertheless, the following speech sounds perfectly okay to me: Something could possess, that is, it is possible for it to possess, the property of being a necessarily existing thing, but actually, as a matter of fact, nothing has this property. So, actually, nothing has the property of existing necessarily even though this property might have been had by something.

The S5 inference at the end of your argument is incompatible with this speech, I "think". Or no? Anyway, I'm much more convinced of this speech than S5 if it entails this speech is wrong. Or am I missing something? I get the difference between metaphysical and epistemic possibility here. I'm saying that, metaphysically, just because the property of being necessarily G could be exemplified, it doesn't follow, logically, that it is in fact exemplified.

Best!

Not sure I understand your objection to axiom 2 [of my second argument]--are you saying that S5 is incompatible with the possibility that there be no contingent things (that being contingent is not exemplified)?

Maybe I'm not following you. It looks to me now like you're saying that since there is a world w in which I do not exist, I am not necessarily contingent. But that can't be right, since I'm necessarily contingent just in case I am contingent in every world that I happen to exist. So, I'm not necessarily contigent just in case there is some world in which I exist necessarily. But this is what I'm claiming is impossible. If there is any world in which I exist necessarily, then I exist necessarily in every world. That follows by S5. You can now generalize: for any contingent object O in any world w, it is true that O is necessarily contingent, or contingent in every world in which it exists. You can generalize to properties: for any contingent property P that x exemplifies, it is true that P is a necessarily contingent property of x. Suppose for reductio that there is some world in which P is a necessary (i.e. non-contingent) property of x. In that case, from S5, P is a necessary property of x in every world. But then there is no world in which P is a contingent property of x. But by hypothesis P is a contingent property of x, so our assumption for reductio is false.

Any better?

That's consistent with my being necessarily (essentially) contingent. Yes?

Indeed, it is. But it is not consistent with you being contingently contingent. I thought axiom 2 entailed that everything that has the property of being contingent (i.e., existing contingently) has the property of existing contingently contingently. If that were so, then there would be some worlds in which you have the property of existing contingently necessarily and some worlds in which you don't. But that isn't possible. If it is just a contingent fact that I exist contingently, then I do not have the property of existing contigently in every world in which I exist. But then in some worlds in which I exist I exist necessarily. You don't want to say that, right?

So maybe I need to get clearer on what you are after with the phrase having the property of existing contingently contingently.

Joshua,

This is sort of interesting. If the property 'being contingent' itself has the property 'being contingent', then isn't it true that anything that instantiates the property 'being contingent' has that property contingently? Or, no? You say,

Well, axiom 2 only says that contingency (being contingent) need not be exemplified by anything, though it can be exemplified.

So, let's take an example. My pencil is a contingent object, and so (trivially) instantiates the property of being contingent. So far, so good. But from axiom (2) it follows (unless I'm really lost) that the property of being contingent instantiated by my pencil itself has the property of being contingent. But that could only mean (as far as I can see--maybe this is where I've got you wrong) that there are some worlds in which my pencil possesses the property of being contingent necessarily. That is the shorthand way of saying my pencil instantiates the property of being contingent and the property of being contingent itself instantiates the property of being necessary. But if that is true, and we assume S5, then there is no world in which my pencil instantiates the property of being contingent and the property of being contingent is itself a contingent property. That is, Ax. 2 is false.

Maybe what you want to say is that contingent properties all have the property of being instantiated contingently. But I'm not sure that helps here.

When I say a property is contingent, all I mean is that it doesn't have to have any instances at all (though it could).

Right. So there is some world in which the property of being contingent is instantiated by (say) my pencil and a world in which it isn't. That's what I mean too by a property being contingent. So we agree here.

So the property of being contingent which is exemplified by your pencil is itself contingent if there is a world in which no contingent things exist, not even your pencil.

The property of being contingent that my pencil instantiates is itself a contingent property of my pencil, right? Just like the property of being red is a contingent property of whatever has it.

Do we disagree here?

Josh:

Very nice. Maybe Axiom 1 of the first argument should be weakened to be conditional: If there is a simplest determinate... Here, I don't mean: If the simplest determinate is possible. Then you just add an axiom that greatness has a simplest determinate.

The reason for this conditionality is that I am worried about cases where a determinable doesn't have a simplest determinate. For instance, is there really a simplest irrational number? (pi, sqrt(2), e?)

Well, being contingent is essential to my pencil (my pencil is contingent in every world in which it exists)...

Of course I agree, since this is the line I've been arguing on the assumption of S5. But now you're committed to saying that my pencil is a contingent being, and the property of being contingent is itself a contingent property, but not a contingent property of my pencil
Analogously, then, my pencil's being red is itself a contingent property, but it is not a contingent property of my pencil. I don't know what that means. It is pretty unconventional to assert that the property of my pencil of being red is a contingent property, but not a contingent property of my pencil.

I do understand how being contingent is an essential property of anything that is so (assuming S5). I just have no idea how these two claims are consistent.

1. being contingent is a contingent property. Ax. 2

2. being contingent is an essential property.

But, judging by the results so far, I'm probably not going to get it :)

1. There is a possible world in which there isn't anything that is contingent (exemplifies being contingent), and there is possible world in which something is contingent (Ax. 2).

Joshua,

No, those are not consistent. Even on that reading of Ax. 2, it's necessarily false. There isn't any such world. Proof: Let S be any contingent state of affairs. If S is contingent, then so is ~S. It is necessaruly true that S v ~S. Therefore, for any world w, some contingent state of affairs obtains, S obtains or ~S obtains. Therefore, for any possible world w there is something that is contingent (exemplifies being contingent), viz., the state of affairs S or the state of affairs ~S.

That's an interesting idea! Maybe it's true that there's a world in which nothing contingently exists or nothing is contingently exemplified. I don't know. Maybe you're not so interested in following this out (that's of course fine), but here's a consideration. Uncreated essences are contingent things. In some worlds there contingently exist only uncreated essences, just as in some worlds there contingently exist only skinny squirrels. But now take a world W in which we believe nothing contingently exists. There are nonetheless essences in W (if Plantinga, and probably PvI, are to be believed on this topic). Let E be one such essence. Suppose E is a created essence, then there contingently exists a created essence E in W (i.e. it is true in W that there might have existed no created essences). Suppose E is an uncreated essence, then there contingently exists an uncreated essence in W (i.e., it is true in W that there might have existed no uncreated essences). So every world contains some contingently existing things.

Josh,

"Well, I accept your axiom 1*, but I question whether it makes sense for there to be a degree of greatness that is slightly less than maximally great."

This makes me wonder whether you think greatness is a magnitude. I think that greatness is a magnitude, and, in particular, that it is a continuous magnitude. Greatness is a property like mass or volume or height, for example; one that allows us to order objects that have it. Do you doubt that greatness is like this, and if so, could you say why?

Hey Josh,

I guess I'm note sure what I think. But here's a thought. Consider lines A and B that both begin at some point C. Suppose that both A and B are infinitely long, so that both A and B have an infinite magnitude, namely, infinite extension in one direction from C. Now, suppose that we annihilate one point on A, namely, the point that exactly overlaps C. Thus, A does not have this point as a part, whereas B does have this point as a part.

It will be true that A and B still have infinite extension, though they will still differ since A lacks a part that B has. Thus, A will be distinct from B. A may not have slightly less extension than B since we are supposing their extension remains the same, but we can still distinguish between these two lines.

Now, run my argument this way. There could be two distinct magnitudes of greatness, both infinite, while one being could have a power, for example, that the other lacks. Both magnitudes of greatness are determinates of greatness, both are exemplifiable, both are infinite, but yet they are distinct magnitudes.

With this understanding in mind we can then state an analog of axiom 1 as follows:

Axiom 1*: Any determinate of an exemplifiable determinable is itself exemplifiable.

I think the magnitude of greatness described above is a determinate of greatness, it is distinct from maximal greatness, it is also an infinite magnitude, but then we reach the conclusion that if your argument is sound, so is the analogous argument. So we wind up with infinitely many Gods.

Anyway, the idea is that we can get the same conclusion I suggested without appealing to the claim that there is a magnitude of greatness that is slightly less than infinite greatness.

"I'm not sure it makes sense (is coherent) for there to be a determinate of greatness that entails that a single power is missing (analogous to a single point missing off a ray)."

I'm not sure it makes sense either. But, on the assumption that it does make sense to say something could be infinitely great, I'm inclined to think it also makes sense to say that something could lack some power, or knowledge, or some amount of goodness and yet still be infinitely great. This is all I need, right?

Does it follow that if something lacks the single power to move a speck of dust while being infinitely powerful, then it is infinitely less powerful than a being that is infinitely powerful and has this ability? As you put it, would it follow that there is an "infinite gap" between their powers? I guess I don't see why. This would be false for the line case. Line B is not infinitely longer than A and yet B has a point that A lacks. I'm inclined to say there would not be an infinite gap, but rather, there would be a finite gap, namely, the difference between having one power.

"One difficulty with this proposal is that it doesn't account for determinates that might entail necessary existence plus an essential limited non-maximal degree of greatness. But in reply, it may be plausible that if something has a finite degree of something (among a continuum of possible degrees), then it could have it to a slightly higher or lower degree. Then, if I'm right that non-maximal degrees of power (and other great-making features) are all finite (because they are infinitely distant from the maximal degree), then there are no essential non-maximal degrees of greatness."

There's a lot here. I agree that if some degree of a finite magnitude can be exemplified, then any finite degree of this magnitude could be exemplified. And I think I agree that all non-maximal degrees of power, for example, are finite. This follows if we simply mean by 'maximal' that the degree exemplified is infinite. But what's the reason for thinking that there couldn't be two magnitudes of power, both infinite (and thus, maximal) and yet they be distinct magnitudes? You seem to say that if they are distinct, then one of these magnitudes cannot be infinite. The idea behind the line case was that isn't so. Another example, the set of evens is infinite. The set of naturals is too. They are distinct sets, but they have the same cardinality, that is, both sets are infinite. This feature of infinite magnitudes seems to me to be a condition on a magnitude being infinite. Maybe I'm missing your point though, am I?

I think it's important to note that the argument can be run on the assumption that only one of God's great-making attributes is a continuous magnitude with an upper bound. Thus, all I need to say is that either power, knowledge, or goodness works this way. And, of course, there's the huge worry that such magnitudes do not have upper bounds at all! I know Mike A. has a paper on this. And in fact, I think there is no upper bound on greatness. For any magnitude of greatness there is a greater magnitude of greatness that could be possessed. Similarly for length: for any possible length, there is a greater possible length. I'm working under the assumption that this is wrong, but I actually think it is very plausible.

Sorry if I was being dense Josh. I think I'm getting your point now. And you're reasoning makes sense for sure, although I do have a few worries.

So, we are considering a single power P of N that W lacks. We are also supposing W and N are maximally powerful beings, say, they are intrinsic duplicates save for the power P. You then ask us to suppose that either P is finite or infinite, and then raise a worry for both options.

You say that, if P is infinite, then it's at least unclear that W's power remains infinite. For how can an infinitely powerful being lack an infinite power. Or, differently, how can a maximally powerful being be such that another being could be infinitely more powerful than it, namely, by having P?

So here's a thought on this horn. I'm not sure what we are to be supposing when we consider power P. This makes it unclear to me exactly what kinds of conclusions we are licensed to draw from the supposition. The general question: What is an infinite power? For example, I have the power to partly occupy infinitely many temporal and spatial regions. Is this an infinite power? Does having an infinite power require the ability to bring about (cause) infinitely many distinct states of affairs, of a certain kind, to obtain? I can do this. For any state of affairs I can bring about, I can also bring about that infinitely many negative states of affairs fail to obtain, namely, all of those that are incompatible with the state of affairs that I can bring about.

The idea is just that I'm not sure what an infinite power is (or a finite power) and so I'm unsure what I'm supposing when considering the argument.

You say: Then, it's not at all clear that W's degree of power would still be infinite given that there is a certain infinite degree of power that it lacks.

But I would turn to the case of sets, or lengths, or volumes... to challenge this inference. A set can have an infinite cardinality while lacking infinitely many members that a different set, of the same cardinality, has. If this is correct, and if sets help us to see the application of infinite magnitudes more clearly, then I would say we have a reason to doubt the inference above. Of course, if there is an independent reason to treat powers differently than sets, then this won't work. But I'm not getting the independent reason yet.

Then there is the other horn. We can suppose that W lacks the power, a finite power, to move a piece of lint. You say, But then W surely lacks a host of other powers--like the power to lift two pieces of lint or to create a kitten. Indeed, W would seem to only have a finite degree of power.

Can't we suppose that W only lacks the power to lift a particular piece of lint? So, he can lift any other pieces of lint, and he can create a kitten. I'm unsure why we should think that powers are interdependent in the way you suggest. Or, more accurately, I don't see why we should think that every power that admits of a continuous magnitude and an upper bound is dependent on other powers. Just consider a maximally perfect being that's evil. He is able to kill an innocent person, although he has maximal power. Now consider a duplicate of this being that is morally perfect. This being lacks the ability to kill an innocent person. In this case, the relevant power is finite. But, it's also true that both beings have maximal power, and so, the fact that one being has this power and the other lacks it does not entail that one of these beings has only finite power.

Or so it seems to me.

I think some powers are interdependent. I don't think all powers are interdependent. And we only need one power that is a continuous magnitude and not interdependent to generate the worry. And this seems right to me. My ability to make a sandwich does not depend upon my lacking the ability to fly, or my having the ability to write about the ontological argument. Presumably God has both of these powers, and so God has powers that are not interdependent.

Josh,

That's a good example. Maybe the literature on supertasks would be helpful, I don't know?