Two More Ontological Arguments
September 23, 2009 — 9:46

Author: Josh Rausmussen  Category: Uncategorized  Comments: 33

Pruss’s clever ontological argument inspired me to offer two ontological arguments of my own:


A Simplicity Ontological Argument
Axiom 1: The simplest determinate of an exemplifiable determinable is itself exemplifiable.
Clearly, if a determinable property can be exemplified, then so can at least one of its determinates. The idea behind axiom 1 is that if some determinates are simpler than others (easier to grasp or analyze, or built out of fewer elements), then the simplest determinates have the greatest epistemic probability of being exemplifiable. (Complexity might result in there being incompatible components.) So, if there is any question as to whether every determinate of an exemplifiable determinable is exemplifiable, the safest assumption is that the simplest of the determinates is exemplifiable.
Axiom 2: Greatness is an exemplifiable determinable
This axiom invites an entire essay devoted to discussing what “greatness” is. Here I simply I propose that we have a primitive, analyzable notion of greatness by which we can recognize instances of greatness of various flavors and degrees.
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. I’m also assuming that there is no such thing as minimal greatness, since for every degree of greatness near zero, one can conceive of there being something half that great (e.g., if something has 1 unit of power, one could conceive of there being something with .5 units). The final assumption is that there is no other specification of greatness that is as simple (e.g., as easy to grasp or analyze) as maximal greatness.
Therefore, maximal greatness is exemplifiable.
Axiom 4: Maximal greatness entails essential maximal power, essential maximal
knowledge, essential moral perfection, and necessary existence.
This is perhaps controversial, but I think it has some intuitive appeal.
From those axioms plus given S5, it follows that that there is an essentially maximally powerful, good, and perfect being. This argument draws on the apparent simplicity of maximal greatness.
A Causal Ontological Argument
(Disclaimer: This argument may be called ‘ontological’ because all its premises are supposed to be knowable a priori, but it may also be called ‘cosmological’ because it relies on a causal principle.)
Let a ‘causable property’ be a property such that there can be a causal explanation as to why that property has ever been exemplified at all. For example, being an armchair is a causable property because there can be (indeed has been) a causal explanation as to why that property has ever been exemplified.
Let a ‘contingent property’ be one that is exemplifiable, but not necessarily exemplified. I propose the following axiom:
Axiom 1: Every contingent property is causable.
One motivation for this axiom is that it appears to be the simplest axiom that accounts for all known cases of causable properties. There are many known cases of contingent properties that are causable and no known cases that are not. I propose that the simplest explanation for this is that being a contingent property entails being causable. (I suspect the entailment goes the other way, too.)
Consider next:
Axiom 2: being contingent is a contingent property.
This is plausible, though not wholly uncontroversial. It is plausible that there is a world containing no contingent things, for it is plausible that there can be contingent things whose non-existence doesn’t depend upon the existence of other contingent things.
From Axioms 1 and 2, it follows that being contingent is causable. But as you probably already realize, being contingent couldn’t be caused to be exemplified by any contingent thing or things without circularity: contingency would have to already be exemplified by the contingent cause(s) prior to the effect. That is to say:
Axiom 3: No instance(s) of a property P can be the causal explanation(s) as to why P is ever exemplified at all.
Therefore, if being contingent is causable, then it is possible for there to be a necessary being. From S5, there is a necessary being. Call it (or them), N.
Let’s continue:
Axiom 4: Every degreed determinate within a continuum of possible degrees is itself contingent.
For example, being 400.12 feet long is contingent–it need not be exemplified. The thought here is that for every degree of A among a continuous range of possible degrees, there could have been a slightly higher or lower degree of A exemplified instead.
From here, an argument for N’s being maximally great can go like this. Premise: every non-maximal degree of greatness is a degreed determinate within a continuum of possible degrees [any non-maximal degree can be slightly exceeded]. (Note: it is not clear that there are exemplifiable degrees of greatness infinitesimally close to maximal greatness.) Therefore, every non-maximal degree of greatness is a contingent property (axiom 4). Now if A and B are both contingent, then unless the negation of A entails B or the negation of B entails A, then the disjunction, A or B, is also contingent. Thus, more generally, every disjunction of contingent properties whose negations are compatible is itself contingent. From here, if the negation of every non-maximal degree of greatness is compatible with the negation of any other non-maximal degree [and surely they are: just because there isn’t anything great to degree D doesn’t thereby entail that a certain other contingent degree of greatness is exemplified], then it follows that the disjunction G of every non-maximal degree of greatness is itself contingent. So, G is causable [axiom 1]. G cannot be causally explained by a non-maximal thing given that every non-maximal thing would be an instance of G [axiom 3]. Therefore, G can be causally explained by a maximal being. Therefore, a maximal being is possible. This means that being maximally great is either contingent or necessary. If it were contingent, then it would be causable [axiom 1]. But anything that is dependent for its greatness on another being is not maximally great. Therefore, it’s not possible for a thing to be caused to be maximal great. Therefore, maximal greatness is not causable. Therefore, maximal greatness is not contingent. Therefore, maximal greatness is necessary. Therefore, in a world in which there are no contingent things (recall axiom 2), maximal greatness must be exemplified by N. Then, if maximal greatness entails essential maximal greatness [isn’t that plausible?], then N is maximally great, even today.
This argument draws on the apparent contingency and explicability of the exemplification of non-maximal degrees of greatness.

Comments:
  • Kevin

    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!

    September 23, 2009 — 11:32
  • Joshua Rasmussen

    Kevin,
    Thanks for those remarks. In reply: (1) The argument isn’t really aimed to show that all flavors of maximized greatness are in the same being; my goal was to show just that the greatest conceivable being is actual, and I understand the greatest conceivable being to include the classic divine attributes (though perhaps not every flavor of greatness…).
    (2) S5 seems intuitive when I think of it in terms of possible worlds–if there is a possible world in which x exists in all the other possible worlds, then x exists in all the possible worlds (or, if there is a possible world from the standpoint of our world, then that world should be possible from the standpoint of every world). This picture is based upon viewing possible worlds as necessarily existing abstracta (a view I’ve argued for in an APA paper using S4 plus some other principles, like serious actualism…)

    September 23, 2009 — 12:14
  • Mike Almeida

    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.

    September 23, 2009 — 14:32
  • Christian

    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!

    September 23, 2009 — 17:48
  • Joshua Rasmussen

    Mike,
    Would it make a difference if I said it is easier to fully grasp maximal greatness than to fully grasp the exact degree and flavor of greatness of a certain blade of grass? I admit that the notion of “complexity” in terms of grasping requires fleshing out…
    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)?
    Christian,
    Nice to hear from you! And nice question. I had that worry in the back of my mind, too, and now you’ve forced it to the front. 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. For example, I’m not sure a being could fail to know one proposition without thereby failing to know an infinite number, and I’m not sure there is slightly less than infinite power or moral perfection… I’m actually not sure one way or the other, which leaves me skeptical of your parallel argument (maybe the only necessary concrete being that is possible is a maximally great one). I’ll keep thinking about it… I actually think my second argument is stronger. πŸ™‚
    My sense about your S5 speech is that whatever motivation there is to think that there could (ontologically could) be a necessary being is canceled by an equally strong motivation to think that there could fail to be a necessary being combined with the realization that these two positions are incompatible if S5 is true. At least that’s how I perceive the situation.

    September 24, 2009 — 7:46
  • Mike Almeida

    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?

    September 24, 2009 — 8:50
  • Joshua Rasmussen

    Mike,
    Thanks for that clarification. I don’t think I’m saying what you think I am. I defined a ‘contingent property’ as “a property that is exemplifiable, but not necessarily exemplified.” By that I mean that there is some possible world in which that property is exemplified, but that it isn’t exemplified in every possible world. Given this, axiom 2 is equivalent to the proposition that there is a possible world in which there are no contingent things–that is, in which being contingent is not exemplified. That’s consistent with my being necessarily (essentially) contingent. Yes?

    September 24, 2009 — 10:16
  • Mike Almeida

    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.

    September 24, 2009 — 10:44
  • Joshua Rasmussen

    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.
    Well, axiom 2 only says that contingency (being contingent) need not be exemplified by anything, though it can be exemplified. I’m not meaning to say (or imply) that any particular contingent thing is contingently contingent. Another way to put axiom 2 is this: “contingent entity” can have an empty extension, though it doesn’t have to.

    September 24, 2009 — 11:27
  • Mike Almeida

    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.

    September 24, 2009 — 12:53
  • Joshua Rasmussen

    Mike,
    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). 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. That’s all I have in mind. I don’t mean that there are some worlds in which being contingent itself fails to be contingent. It’s not a matter of whether being contingent could fail to be contingent. It’s a matter of whether being contingent could fail to be possessed by anything at all. All I mean when I say that property x is contingent is that there is a possible world in which there are no instances of x and a possible world in which there is at least one instance of x: that is, x can be exemplified but doesn’t have to be. Does that make sense?

    September 24, 2009 — 13:54
  • Mike Almeida

    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?

    September 24, 2009 — 15:18
  • 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?)

    September 24, 2009 — 16:51
  • Joshua Rasmussen

    Alex,
    Nice suggestion. I’m curious what you might think of my argument for N’s being maximally great in my causal-onto arg. I had you in mind (for some reason) when I wrote it.

    September 24, 2009 — 17:53
  • Joshua Rasmussen

    Mike
    You wrote: 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.
    Well, being contingent is essential to my pencil (my pencil is contingent in every world in which it exists)… I wouldn’t say that being contingent is contingently exemplified by any particular thing. I just say it is contingently exemplified: there is a world in which it isn’t exemplified by anything (and one in which it is).

    September 24, 2009 — 18:13
  • Mike Almeida

    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.

    September 24, 2009 — 18:45
  • Joshua Rasmussen

    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 suspect being red is not essential to my pencil (though being contingent is). Still, being red is contingent because there is a possible world without red things. (Just making sure we’re on the same page.)
    I recall first hearing this terminology (of “contingent properties”) from Peter van Inwagen and assumed it was standard, though perhaps it is not. At least you now know what I mean. πŸ™‚

    September 25, 2009 — 6:55
  • Mike Almeida

    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 πŸ™‚

    September 25, 2009 — 7:03
  • Joshua Rasmussen

    Interesting… Can you see that these claims are consistent?:
    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).
    2. Necessarily, for all x, if x exemplifies being contingent, then in every possible world in which x exists, x exemplifies being contingent.
    If not, why not? If so, then we’re done, yes?

    September 25, 2009 — 7:52
  • Mike Almeida

    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.

    September 25, 2009 — 8:50
  • Joshua Rasmussen

    Mike,
    Nice comment–I think we’re making progress here. Thanks for your patience. πŸ™‚
    I see an ambiguity in the notion of “being contingent”. Something can be contingent in the sense that it exists in some but not all possible worlds. Or, it can be contingent in the sense that it is exemplified or obtains (in the case of states of affairs) in some but not all possible worlds. I think the moral of this whole discussion is that it may be better for me to avoid calling properties “contingent” because of this ambiguity. Better for me to use the terms “contingently exists” and “is contingently exemplified.” So, to be completely explicit, I mean this:
    1. There is a possible world in which there isn’t anything that contingently exists (exists in some but not all possible worlds), and there is a possible world in which something contingently exists (exists in some but not all possible worlds) (Ax. 2).
    2. Necessarily, for all x, if x exists in some but not all possible worlds, then in every possible world in which x exists, x exists in some but not all possible worlds.
    And you would accept that those are consistent, yes? If so, then we are done and I should like to thank you for bringing to light the importance of this clarification. Thanks!

    September 25, 2009 — 9:15
  • Mike Almeida

    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.

    September 25, 2009 — 10:42
  • Christian

    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?

    September 25, 2009 — 11:20
  • Joshua Rasmussen

    Christian,
    I do think greatness is a magnitude in the sense that it comes in various degrees, but I’m skeptical that all it’s determinates are along a continuum. An analogue would be power. I suspect power comes in greater and lesser degrees, but I’m skeptical that there is a degree of power slightly less than infinite power. Does that seem right to you? Perhaps all finite levels of greatness are exemplifiable (by your principle of continuity) and maximal greatness is too (by my principle of simplicity), but that no degree of greatness that entails necessary existence is continuous with maximal greatness… What do you think?

    September 25, 2009 — 12:53
  • Christian

    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.

    September 25, 2009 — 16:11
  • Joshua Rasmussen

    Interesting thought, Christian.
    Your example of the lines reveals that there can be two things that are infinite with respect to a property (length), though they exemplify distinct determinates of that property. Furthermore, the determinates are “really close” to each other (one point difference). I’m inclined to think that in that case, every determinate is exemplifiable if any are (because of a principle of modal continuity). You then got me wondering whether greatness might me like that. Here are a couple thoughts:
    1. 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). Consider a really minimal power–like the power to move a speck of dust. Now imagine that a being would be maximally powerful except for the fact that it lacks this one power to move a speck of dust. Well, if it can’t even move a single speck of dust, wouldn’t it thereby be infinitely less powerful than a maximally powerful being? Wouldn’t there be an infinite gap between its power and the power of a maximally powerful being? My worry here is that when there is an infinite gap, modal continuity may break down, in which case I lose motivation to think that every determinate of power can be exemplified. What do you think?
    2. It also occurred to me that the result that every determinate of greatness is exemplifiable may not actually be inconsistent with classical Christian theism. Allow me to explain. On classical Christian theism, God consists of three centers of consciousness (Father, Son, and Spirit). This allows for God to essentially consist of and enjoy the good of loving relationship (including the category of relationship in which a plural jointly loves a single), making him completely self-sufficient. Now, on classical Christian theism, the Son (at least) is capable of voluntarily limiting his power, knowledge, and perhaps even moral fortitude (making him morally reliant on the Father to various degrees). On this conception, perhaps every determinate of maximal greatness that entails necessary existence is exemplifiable by the Son in various possible worlds.
    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. If that’s correct, then perhaps your Axiom 1* doesn’t actually pose a reductio. In fact, if it is true, it would seem to clarify the nature of God. πŸ™‚
    These are all tentative thoughts, as I surely might be missing something.

    September 26, 2009 — 8:01
  • Christian

    “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.

    September 26, 2009 — 12:59
  • Joshua Rasmussen

    Christian,
    Thanks for those further observations. What do you think of this reasoning?:
    Let N be a maximally powerful being (assuming for the sake of argument that this is a coherent notion…) Now let P be a power. And let W be a being that has all the powers of N except for P. P is either infinite or finite. Suppose P is infinite. 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. So then, suppose that P is finite. For example, P might be the power to move a piece of lint. In that case, if W lacks P, then W lacks the power to lift a piece of lint (or some other finitely difficult activity). 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. So, either way it’s not clear that there can be an infinitely powerful being that lacks a power that a maximally powerful being would have.
    The idea is that powers are not analogous to points on a line or elements of a set. The disanalogy arises by virtue of the fact that powers are dependent upon one another–one cannot lack one power without lacking many others.
    Does this reasoning make sense, or do you see a fly in the ointment?

    September 26, 2009 — 20:16
  • Christian

    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.

    September 26, 2009 — 23:07
  • Joshua Rasmussen

    I had a further thought:
    Let Bob be a monster with an infinite number of arms. It can lift all of them at once–it has infinite power. Now Sue, let’s say, is also a monster with an infinite number of arms. But one of Sue’s arms breaks. She can still lift an infinite number of arms at once, so she has infinite power, though she isn’t as great as Bob…
    What can I say about this case? There are distinctions I’d like to make here…but this deserves more thought on my part first–so I will keep thinking about it… πŸ™‚

    September 27, 2009 — 7:57
  • Christian

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

    September 27, 2009 — 12:43
  • Joshua Rasmussen

    I guess my question returns to whether there is a degree of greatness continuous with maximal greatness. I can admit that there are different degrees of infinite greatness continuous with each other. But is there an infinite degree that entails necessary existence continuous with what I take to be the simplest degree-namely maximal greatness?
    I guess it strikes me as incoherent for there to be a being that has just one less power than a maximally great being. Take maximal power–a being that can cause any state of existence as is possible for any being to cause. If we take away just one power (the power to move a speck) from such a being, don’t we thereby take away an infinity of powers? It presently seems so to me.

    September 27, 2009 — 13:17
  • Joshua Rasmussen

    Oops! My last two posts were written without me even seeing your reply post that just precedes them! So, realize the two comments above weren’t in response to your last comment. They were actually independent thoughts I had this morning. I will like to think about your last comment carefully. Thanks for it!

    September 27, 2009 — 13:22