Monthly Archives: July 2012

Lost and the Ontological Argument

Consider Gaunilo’s refutation of Anselm’s ontological argument:

…[I]t is said that somewhere in the ocean is an island, which, because of the difficulty, or rather the impossibility, of discovering what does not exist, is called the lost island. And they say that this island has an inestimable wealth of all manner of riches and delicacies in greater abundance than is told of the Islands of the Blest; and that having no owner or inhabitant, it is more excellent than all other countries, which are inhabited by mankind, in the abundance with which it is stored.

Now if some one should tell me that there is such an island, I should easily understand his words, in which there is no difficulty. But suppose that he went on to say, as if by a logical inference: “You can no longer doubt that this island which is more excellent than all lands exists somewhere, since you have no doubt that it is in your understanding. And since it is more excellent not to be in the understanding alone, but to exist both in the understanding and in reality, for this reason it must exist. For if it does not exist, any land which really exists will be more excellent than it; and so the island already understood by you to be more excellent will not be more excellent.”

If a man should try to prove to me by such reasoning that this island truly exists, and that its existence should no longer be doubted, either I should believe that he was jesting, or I know not which I ought to regard as the greater fool: myself, supposing that I should allow this proof; or him, if he should suppose that he had established with any certainty the existence of this island. For he ought to show first that the hypothetical excellence of this island exists as a real and indubitable fact, and in no wise as any unreal object, or one whose existence is uncertain, in my understanding” (Gaunilo of Marmoutiers, In Behalf of the Fool, 6).

To summarize, Gaunilo thinks that the ontological argument proves too much.  We should expect to be able to demonstrate the existence of a superlative within any class or species of a thing.  Not only would perfect islands exist, but perfect pineapples, perfect pencils, and perfect pizzas!

But Anselm is not without a response:

Now I promise confidently that if any man shall devise anything existing either in reality or in concept alone (except that than which a greater be conceived) to which he can adapt the sequence of my reasoning, I will discover that thing, and will give him his lost island, not to be lost again

But it now appears that this being than which a greater is inconceivable cannot be conceived not to be, because it exists on so assured a ground of truth; for otherwise it would not exist at all.

Hence, if any one says that he conceives this being not to exist, I say that at the time when he conceives of this either he conceives of a being than which a greater is inconceivable, or he does not conceive at all. If he does not conceive, he does not conceive of the non-existence of that of which he does not conceive. But if he does conceive, he certainly conceives of a being which cannot be even conceived not to exist. For if it could be conceived not to exist, it could be conceived to have a beginning and an end. But this is impossible (Anselm’s Apologetic In Reply to Gaunilo’s Answer In Behalf of the Fool, 3).

Admittedly, it is not very clear how Anselm’s response undercuts Gaunilo’s parody objection— at least at first blush.  The idea seems to be that whatever is “that than which a greater is inconceivable” cannot be thought to be contingent.  But islands, at least normal islands, are contingent.

Someone might decide to bite the bullet and insist that she has conceived of a necessary island.  Has she escaped Anselm’s criticism?  Interestingly enough, it seems that Anselm is willing to concede that if such a person truly has followed his line of reasoning, then such an “island” is no longer lost, but is never to be lost again.  Still, one might raise the question, “what kind of ‘island’ is it?”

The television show Lost offers an interesting perspective to this question.  Lost was well-known for referencing a wide variety of philosophic themes.  Many of the show’s characters are named after various philosophers, e.g. Locke, Rousseau, and Hume.  Recently, I stumbled across a snippet from Lostopedia that I thought was very interesting.  The author writes:

The underlying philosophy of the entire show is the 11th century discussion around what is called Anselm’s ontological argument for God and Gaunilo’s refutation using the “lost Island” argument…  And for television, a truly greater island would be one that moved in space, or in time, or even thought for itself. In fact this fallacious argument can be extended to prove the existence of anything, like tropical polar bears (Lostopedia, Philosophers/Theories).

Upon reflection, I think the show proffers insights into how one might respond to Gaunilo.  That is, if we start to imagine what kinds of attributes a perfect island should have,  the island begins to look less and less like an island, and more and more like God.  On the show, the island could cure John Locke and others, it had enormous power, it could travel through space and time, and seem to be self-aware and express intentionality.  Towards the end of the series the island seemed to be anything but an island at all!

Suppose we are able to imagine an even greater island, one that not only travels through space and time, but somehow manages to transcend it.  Perhaps this island would be morally perfect—the island on the TV certainly wasn’t.   Would the island be pure actuality?  Would it be omnipotent and omniscient, and omnipresent?  At some point our greatest “island” just happens to have an ill-selected pseudonym.  It would be more appropriate to consider it divine.  And as we reflect on its nature, we’re no longer wondering how many coconuts, beautiful hula-girls, palm trees, or secluded beaches the island should have.  When we reflect upon the phrase “that than which none greater can be conceived” we realize that it is a description that cannot be grafted onto just any other term.  When attached to terms referencing contingent things, we’re either uttering nonsense, as we do when we speak of round-squares, or we are no longer talking about a contingent thing at all.  If we  loosen up the concept sufficiently to accommodate the Anselmian phrase, we’ve traded our initial concept for the divine concept.  The very meaning of this phrase is that which blocks Anselm’s argument from proving too much.

Revised Meta Modal Ontological Argument in QML

Several weeks ago I posted an ontological argument that I had hoped would overcome many of the powerful parody objections employed against the various Hartshorne-Malcolm-Plantinga modal arguments.  Unfortunately, when I tried to symbolize it, an error became quite obvious.  The lesson is that I should symbolize the arguments before posting them.  This new version essentially reaches the same conclusion I had sought, but it makes use of a two-place predicate.  It’s a simple solution, but the results are quite compelling, at least as far as I can tell (I look forward to potential criticisms).

But before I delve into the argument, I would like to address a worry recently raised on an excellent blog, Third Millennial Templar (the post is linked here).  The concern is whether this argument somehow confuses or conflates epistemic possibility with the relevant modal possibility (metaphysical or broad-logical) needed in a modal ontological argument.  And this worry is well-founded.  After all, the motivation behind (1) is based upon Graham Oppy’s admission that it is an open question whether there is a sound ontological argument that is yet to be found.  In other words, it is at least an epistemic possibility that a sound ontological argument exists.

Nonetheless, I don’t think that this meta modal argument conflates epistemic possibility with other modal possibilities.  Rather, it makes use of epistemic possibility as only part of the background justification for (1).  The rest of the justification comes from the fact that while assertions bear an onus, an epistemic possibility receives the benefit of the doubt for being logically possible.  It would be ruinous to our modal epistemology if we demand positive proof for every logical possibility.  Therefore, we default to the position that something is logically possible unless and until it can be shown to be impossible.  The onus is on the one claiming impossibility, as that is the stronger claim.  The problem with earlier versions of the ontological argument is that, while granting logical possibility to God leads to the conclusion that God exists, maintaining the proof commits one to special pleading so as to not permit the logical possibility that God does not exist.  The benefit of the doubt must be granted to both the existence and non-existence of something, unless there are independent reasons to think that thing is necessary or impossible.  Granting both possibilities to God results in a symmetrical anti-ontological argument that just as easily disproves God’s existence.  Consequently, there is no reason to prefer the conclusion of the ontological argument to that of the anti-ontological argument from reason alone.  So to claim that these earlier ontological arguments serve as disproof for their anti-ontological counterparts is unfounded.  Had one started from the assumptions of the anti-ontological arguments, they would have found equal disproof for the premises of the ontological argument.  But which argument holds logical priority?  Neither!

This argument is different.  While it still grounds the logical possibility of the first premise on the benefit of the doubt, it is not disrupted by symmetrically granting the possibility of its contradictory.  Fair is fair!  To push the argument further two varieties of parody arguments are considered, possibility is granted, and they are shown to be consistent with the conclusions of the meta modal ontological argument. So while older versions of the ontological arguments are blocked by symmetry, this argument is not.  Again, we might be tempted to restrict the benefit of the doubt, or presumption of possibility, in light of this proof, but such a restriction seems a bit ad hoc and potentially destructive to modal reasoning generally.  We’d have to propose a principle that prevents this argument, but still permits other varieties of modal proofs that rely upon benefit of the doubt based modal reasoning.  But struggling to manufacture just the right restrictions is merely to assume a conclusion and to work backwards to prevent an argument that conflicts with that conclusion!  This would be reminiscent of the old logical positivist project of developing a working verification principle that restricts meaning to only those propositions that positivists thought should be meaningful.  Any restrictive principle they developed was either too restrictive, not restrictive enough, or self-defeating.  All of this is to say that it is legitimate to assume the logical possibility of something that is epistemically possible, so long as one is also willing to grant the logical possibility of the contradictory.  That is, one should presume possibility of something not known to exist actually or necessarily only if the possibility is granted to both the existence and non-existence of the possibility in question and there are no known reasons to think it impossible.

The argument is as follows…


Ax =df x is a sound modal ontological argument concluding to the existence of a sufficiently defined divine being.
Cx =df x is the conclusion of a sound modal ontological argument that asserts the existence of a sufficiently defined divine being.
Θxy =df x is sufficiently defined as a divine being by proposition y.

T1. ◊(∃x)Fx → (∃x)◊Fx (Barcan Formula)
T2. ◊□(∃x)Fx → □(∃x)Fx (S5 axiom)

1.  ◊(∃x)Ax (premise)
2.  ◊(∃x)Ax → ◊(∃y)Cy (premise)
3.  (∀y)(◊Cy → ◊□(∃z)Θzy) (premise)
4.  ◊(∃y)Cy (1,2 MP)
5.  ◊(∃y)Cy → (∃y)◊Cy (T1)
6.  (∃y)◊Cy (4,5 MP)
7.  ◊Cu (6 EI)
8.  ◊Cu → ◊□(∃z)Θzu (3 UI)
9.  ◊□(∃z)Θzu (7,8 MP)
10.  ◊□(∃z)Θzu → □(∃x)Θzu (T2)
11.  □(∃x)Θzu (9,10 MP)
12.  (∃z)Θzu (11, NE)

To prove no disruption on first level symmetry, we grant

13.  ◊¬(∃x)Ax (premise)

And (13) is shown to be consistent insofar as one might defend this premise,

14.  ◊¬(∃x)Ax → (∀y)(◊¬(∃z)Θzy ∨ ◊(∃z)Θzy) (premise)

To prove there is no disruption what might be called a second level symmetry, consider the following additions:

Bx =df x is a sound modal anti-ontological argument concluding to the non-existence of a sufficiently defined divine being.
Dx =df x is the conclusion of a sound modal anti-ontological argument that asserts the non-existence of a sufficiently defined divine being.

15.  ◊(∃x)Bx (premise)
16.  ◊(∃x)Bx → ◊(∃y)Dy (premise)
17.  (∀y)(◊Dy → ◊□¬(∃z)Θzy) (premise)
18.  ◊(∃y)Dy (15,16 MP)
19.  ◊(∃y)Dy → (∃y)◊Dy (T1)
20.  (∃y)◊Dy (18,19 MP)
21.  ◊Dv (20 EI)
22.  ◊Dv → ◊□¬(∃z)Θzv (17 UI)
23.  ◊□¬(∃z)Θzv (21,22 MP)
24.  ◊□¬(∃z)Θzv → □¬(∃x)Θzv (T2)
25.  □¬(∃x)Θzv(23,24 MP)
26.  ¬(∃z)Θzv (25, NE)
27.  ¬(∃z)Θzv · (∃z)Θzu (12,26 Conj)

It is not evident that (27) is a direct contradiction without the further premise,

28*.  v = u (premise)

But there is no reason to presume this identity, at least there is no more reason to assume (28*) than there is to presume,

28′. v ≠ u (premise)

Further, we should take the fact that a contradiction would be generated by claiming identity between v and u as prima facie evidence that they are not identical.  And so, this version of the meta modal ontological argument successfully demonstrates that there exists a being that is sufficiently defined as divine by some proposition.  In other words, a divine being exists, which is shown by these last two steps,

29.  (∃z)Θzu (27, Simp)
30.  (∃x)(∃z)Θzx (29, EG)

That is, by (30) we have proved that there exist some x and some z such that z is sufficiently defined as a divine being by proposition x.  Or, to put it simply, there exists some such being that can be sufficiently defined as a divine being.  As far as I can tell, that means that God exists.

In future posts, I hope to defend the key premises in a bit more detail, along with my use of the S5 axiom and Barcan Formula.  In the meantime, I would appreciate any challenges, criticisms, questions, or suggestions.