Holy Cyclops

In The 70th Philosophers’ Carnival appears The Barefoot Bum’s analysis, here, of what goes wrong with Alvin Plantinga’s “Victorious” ontological argument for the existence of God. Since it’s something I’ve been looking at, I thought I’d take my own shot.

Plantinga’s argument takes differing forms.  For technical reasons, he puts it in terms of the exemplification of properties in possible worlds, rather than in terms of the existence of entities in possible worlds, and in its more detailed form, he puts it in terms of properties that entail other properties.  None of that will really affect my objections to the argument.  I’m going to present the simpler of the forms Plantinga presents in The Nature of Necessity.

Let maximal excellence (ME) be the property of being omniscient, omnipotent, and morally perfect—i.e., Godlike.

Let unsurpassable greatness (UG) be the property of necessary maximal excellence—of being maximally excellent in every possible world—of being Godlike in every possible world. 

Notice that in the widely accepted modal logic S5, which Plantinga uses, any statement that is necessarily true in one possible world is necessarily true in each possible world.  This is because if we had Np in world W[1] but ~Np in world W[2], we would have both P(Np) and P(~Np) (because truth in some possible world is what possibility means, in possible-worlds semantics)—but in S5, P(Np) collapses to Np and P(~Np)=P(P~p)=P~p=~Np, so we wind up with Np and ~Np, a contradiction.  In S5, a necessary truth in one possible world is a necessary truth in all other possible worlds, too.

Let a universal property be one which is instantiated in every possible world or in no possible world.  Note that UG is a universal property.  If UG is instantiated in any possible world, then N(ME) is instantiated in that possible world, so that N(ME) is instantiated in every possible world (because what is necessary is necessary in every possible world), so that UG is instantiated in every possible world.  Hence, either UG is instantiated in every possible world or in none of them.

1)  There is a possible world in which unsurpassable greatness is exemplified.     (Premiss)
2)  The proposition a thing has unsurpassable greatness if and only if it has maximal excellence in every possible world is necessarily true.    (Definition of UG)
3)  The proposition whatever has maximal excellence is omnipotent, omniscient, and morally perfect is necessarily true.      (Definition of ME)
3a)  Unsurpassable greatness is a universal property.  (As noted above)
4)  Possesses unsurpassable greatness is instantiated in every possible world.    (1,3a)
5)  Possesses unsurpassable greatness is instantiated in the actual world.  (4, universal instantiation)

In more compressed form:

1.  P(UG)      (Premiss)
2.  In some possible world, UG.   (Definition of possibility in possible-world semantics)
2a.  UG is a universal property.   (As noted above)
3.  In every possible world, UG.     (1,2a)
4.  N(UG).      (Definition of necessity in possible-world semantics)

And any being possessing unsurpassable greatness in the actual world is clearly an actually existing God.  Q.E.D. 

What is wrong with the argument?  Well, perhaps nothing is really wrong with it; but it certainly doesn’t give any reason to believe in God.  When one defines UG=N(ME), and then uses the premiss P(UG), he is using the premiss P(N(ME)).  But if he is working in S5, in which P(N(ME))=N(ME), it’s hardly surprising that the assumption of the possibility of the exemplification of universal greatness gets him the existence of God.  Defining UG as N(ME) guarantees, as Plantinga well realizes, that UG is a universal property:  Either UG is exemplified in all possible worlds or in none of them.  P(UG) seems like a tempting premiss, because it’s easy to confuse logical or metaphysical possibility with epistemic possibility.  One might think, “Gee, all I have to assume is that UG’s exemplification is possible?  That’s not much to ask!”  But it is a lot to ask when UG is defined as N(ME).  If one instead assumed the possibility of UG’s non-exemplification, a “proof” of God’s nonexistence would follow:

1)  There is a possible world in which unsurpassable greatness is not exemplified.     (Premiss)
2)  The proposition a thing has unsurpassable greatness if and only if it has maximal excellence in every possible world is necessarily true.    (Definition of UG)
3)  The proposition whatever has maximal excellence is omnipotent, omniscient, and morally perfect is necessarily true.      (Definition of ME)
3a)  Unsurpassable greatness is a universal property.  (As noted above)
4)  Possesses unsurpassable greatness is not instantiated in any possible world.    (1,3a)
5)  Possesses unsurpassable greatness is not instantiated in the actual world.  (4, universal instantiation)

In more compressed form:

1.  P(~UG)      (Premiss)
2.  In some possible world, ~UG.   (Definition of possibility in possible-world semantics)
2a.  UG is a universal property.   (As noted above)
3.  In every possible world, ~UG.     (1,2a)
4.  N(~UG).      (Definition of necessity in possible-world semantics)

Therefore, an unsurpassably great being does not exist in any possible world, so there is no God.  (The conclusion that there is no God requires the ascription of unsurpassable greatness to God.  Without it, one simply has P(~N(ME))=P(P(~ME))=P(~ME), so that in some possible world there is no God, but might be one in the actual world.)

The use of unsurpassable greatness, defined as necessary maximal excellence, is a trick.  One might use it to “prove” the existence of unicorns.  Let maximal unicornness (MU) be the property of being one-horned, white, equine, and so on; let unsurpassable unicornness (UU) be the property of necessary maximal unicornness (UU=N(MU)).  Notice that UU, just like UG, is a universal property.  Then

1)  There is a possible world in which unsurpassable unicornness is exemplified.     (Premiss)
2)  The proposition a thing has unsurpassable unicornness if and only if it has maximal unicornness in every possible world is necessarily true.    (Definition of UU)
3)  The proposition whatever has maximal unicornness is one-horned, white, equine (and so on) is necessarily true.      (Definition of MU)
3a)  Unsurpassable unicornness is a universal property.  (As noted above)
4)  Possesses unsurpassable unicornness is instantiated in every possible world.    (1,3a)
5)  Possesses unsurpassable unicornness is instantiated in the actual world.  (4, universal instantiation)

In more compressed form:

1.  P(UU)      (Premiss)
2.  In some possible world, UU.   (Definition of possibility in possible-world semantics)
2a.  UU is a universal property.   (As noted above)
3.  In every possible world, UU.     (1,2a)
4.  N(UU).      (Definition of necessity in possible-world semantics)

And, therefore, unicorns exist. 

Well, obviously not.  The point is that one must have some reason, in Plantinga’s “proof,” to prefer P(UG) to P(~UG).  The two are jointly inconsistent, so you can’t have both.  But one cannot give any reason to prefer P(UG) that is independent of the conclusion that God exists.   So, even if the argument is valid—and the making of St. Anselm’s argument into a valid one is the reason for Plantinga’s labeling it “victorious”—we have no reason to think it is sound.  But more than that, we have no reason to accept its crucial premiss:  P(UG).  Plantinga seems to think that it is rational to accept that premiss, and therefore rational to accept the conclusion that God exists.  But since P(UG)<—>N(UG), it is precisely as rational to accept P(UG) as it is to accept N(UG); how rational can it be to accept N(UG) without reason?  I am not claiming that it is more rational to accept P(~UG); only that I can see no rational reason for accepting either P(UG) or P(~UG).

I was reading Alvin Plantinga’s The Nature of Necessity yesterday, and he quoted and analyzed, in great detail, a couple of passages, one from William Kneale and one from W.V.O. Quine, whose quotation and analysis reminded me once again that professional philosophers sometimes make silly mistakes.

The discussion was about essential and accidental (necessary and contingent) properties.  Kneale’s anti-essentialist argument was, in my reconstruction from memory, that one couldn’t say that the number twelve was essentially composite, because surely it is only a contingent fact that the number of apostles was twelve, so the number of apostles couldn’t be essentially composite; but since the number of apostles and the number twelve are the same number, twelve can’t be essentially composite.

I’ve rendered it in more detail than he did in the quoted passage.  But one can easily see the mistake:  An equivocation on the meaning of “The number of apostles.”  Does “the number of apostles” mean “the actual number of apostles (i.e., twelve),” or does “the number of apostles” mean “the possible number of apostles (i.e., twelve or eleven or thirteen or…)”?  “The number of apostles” and “twelve” denote the same number only if “the number of apostles” is intended as “the actual number of apostles (i.e., twelve)”; if one intends “the number of apostles” as “the possible number of apostles (i.e., twelve or eleven or thirteen or…),” then one can no longer equate twelve with the number of apostles.  One may either say

1. Twelve is a composite number.
2. The (actual) number of apostles is twelve.
3. Therefore, the (actual) number of apostles is composite.

or

1. Twelve is a composite number.
2. The (possible) number of apostles might be twelve but might be some other number, like eleven or thirteen.
3. Therefore, the (possible) number of apostles might be composite but might not be.

In the first case, “the (actual) number of apostles” is a Kripkean “rigid designator,” if I’m remembering his terminology correctly, always equalling twelve and therefore always composite, just like twelve—rendering the argument against essentialism toothless.  In the second case, “the (possible) number of apostles” is a non-rigid designator, not always composite but also not always equalling twelve—again rendering the argument against essentialism toothless.  Only if one could argue that the number twelve had the kind of fluidity of designation that “the (possible) number of apostles” has could one go on to argue that twelve is not necessarily composite—but, of course, that can’t be done.

Quine’s argument, again in my reconstruction of it, was that whether or not a property is thought to be necessary depends on how we describe the property-bearer—that properties of objects are not essentially necessary or non-necessary but are, rather, only necessary or non-necessary relative to our descriptions of those objects.  His example is as follows:  We might normally say that, in some sense, mathematicians are necessarily rational but are not necessarily bipedal, and that cyclists are not necessarily rational but are necessarily bipedal.  (Let’s set aside any question about either the rationality of all mathematicians or the bipedality of all cyclists.)  But suppose a mathematician is also a cyclist.  Then are we to say that he is both necessarily rational and not necessarily bipedal and also not necessarily rational but necessarily bipedal—a contradiction (a pair of them, really)?  Our assessment changes with our change in description:  We say that the mathematician-cyclist is both necessarily rational and necessarily bipedal.

I’m sure I’m not rendering his argument as persuasively as he did, but its main point is the contradiction given.  Two points can be made about this:  First, it may be that saying that mathematicians are not necessarily bipedal, and that cyclists are not necessarily rational, is saying something too strong.  We do not know of all mathematicians that they are not necessarily bipedal; perhaps some of them (like the mathematician-cyclist) are necessarily bipedal, while others aren’t.  (Similarly for cyclists and the non-necessity of their being rational.)  What we can justifiably say is that mathematicians are necessarily rational and possibly bipedal (and similarly that cyclists are possibly rational and necessarily bipedal).  But then the contradiction vanishes:  The mathematician-cyclist is necessarily rational and possibly bipedal, and he is possibly rational and necessarily bipedal, and that doesn’t contradict his being necessarily rational and necessarily bipedal.  Second—and, in my view, the more important point, and the one that qualifies Quine’s argument for inclusion in a post entitled “Silly Philosophical Mistakes”—is that while it is true that we make our descriptions on the basis of what we know, or on the basis of our present interests or purposes, rather than on the basis of what is true, and that for that reason our ascriptions of modal status with respect to different properties will seem to vary according to our present interests and purposes—in contradiction to the essentialist view that necessary properties, at least, are essential to entities—it is nevertheless also true, in opposition to Quine, that while what we know of an object may change (so that we realize, when we learn that a mathematician is a cyclist, that he is not only necessarily rational, as we had thought, but that he is also necessarily bipedal), and that while our choice of description may vary according to our interests or purposes (so that we may know very well that a mathematician is also a cyclist but may or may not choose to ignore it for the moment, resulting in our sometimes describing the mathematician-cyclist as necessarily rational and possibly bipedal and in our sometimes describing the mathematician-cyclist as both necessarily rational and necessarily bipedal), neither our knowledge change nor our choice of description implies that what is true of the object changes or is somehow malleable.  At best, Quine’s is an argument for description-relativism; it isn’t, as he appears to want it to be, an argument for fact-relativism (or for relativism of modal facts to description).  (Even if Quine thinks that we know the facts about objects, that simply means that a description of the object that expresses those facts must be complete in order to capture all of those facts—an incomplete description, chosen for our own reasons, might fail to capture all of those facts.)