Holy Cyclops

Here, I argued that whether or not to accept the fatalistic argument really boiled down to a choice to how to view future events—as already fixed or not as already fixed. Propositions about future events might be taken as already having truth-values, or they might be taken as taking on truth-values only at the time of occurrence of those future events.

But if we assume the existence of a foreknowing God, that changes. Propositions about future events must then be taken as already having truth-values, which God already knows (although we don’t). The fatalistic argument I gave there, then, must go through:

1. p v ~p (Premiss, by the Law of the Excluded Middle)
2. p—>O(E) (Premiss: If it is true that E occurs at time t, then E has an occurrence-value)
3. ~p—>O(E) (Premiss: If it is true that E fails to occur at time t, then E has an occurrence-value)
4. O(E) (1, 2 ,3, Constructive Dilemma)

This applies to any future event E of which a foreknowing God has knowledge, whether it’s the result of human choice or not. And although one may still argue that God’s foreknowledge is like his looking through a time-telescope, so that he is not bringing about event E (or not-E) but is simply observing it or aware of it, one can no longer argue that event E’s occurrence-value isn’t yet fixed.  One can no longer argue that future contingent propositions are neither true nor false.

The background for this post appears here and here.

Chad McIntosh (of Doxazo Theos—see links) thinks that the problem with my second argument is that God’s foreknowledge isn’t necessary. He says that I’d be hard-pressed to find a theist who agreed that it was. But I was just reading Alvin Plantinga’s The Nature of Necessity, and he argues for just that point—he thinks that in order for a possibly existing entity to count as God, it can’t be omniscient in one possible world but not in another, or omnipotent in one possible world but not in another, or morally perfect in one possible world but not in another. He thinks that God must be maximally great—i.e., maximally excellent in all possible worlds—necessarily maximally excellent. If foreknowledge is part of omniscience, then at least one prominent theist thinks it’s necessary.

Still, I agree that if it’s not necessary, then the second argument, in its short form, fails, since N(Kx) is a premiss of the short form; and I assume that Chad would say that the third premiss of the second argument in its long form, N(Kx) v N(K(~x)), was false, so that it, too, would fail. Denying the necessity of God’s foreknowledge is indeed a way of rendering the arguments unsound.

But I suspect that Chad has in mind not that God isn’t necessarily foreknowing, but rather that his foreknowledge is contingent rather than necessary—that whether God foreknows x or foreknows not-x depends on the agent S’s choice, and is not “written in metaphysical stone” independent of S’s choosing. It is not God’s foreknowing that is contingent; it is what God foreknows that is contingent. And that’s how N(Kx) is false: it’s not N(K(something)) that’s false, but rather N(K(specifically x)) that’s false. Chad might then agree to the truth of N(Kx v ~Kx) but not to N(Kx) v N(K(~x)).

(Chad also notes that God needn’t be foreknowing because there are possible worlds in which, for example, God has not created time; but I am restricting myself to consideration of all possible worlds in which there are human beings making choices. The notion of necessity involved will then be one of relative necessity—necessity relative to a restricted class of possible worlds. Since the class of possible worlds in which there are human beings making choices is exactly the class in which freedom matters, necessity relative to this class seems strong enough to be opposed to freedom.)

I agree with Chad when he writes, “Were S to refrain from x and performed [sic] y instead, God’s foreknowledge would have been different.” I have, in fact, written a defense of that very view here.

What I don’t yet agree with is the necessity of deriving Nx (or Nx v N(~x)). Looking at the first argument I gave:

1. N(Kx—>x) (Premiss—to know that x will occur at t requires that x will occur at t)
2. Kx—>x (1, modal axiom M [or T], i.e., Np—>p)
3. Kx (Premiss—it is known that x occurs at t [since God has complete foreknowledge])
4. x (2, 3, modus ponens)

It seems clear to me that in every possible world in which God foreknows that agent S will choose to perform x at t, agent S will in fact choose to perform x at t, and in every possible world in which God foreknows that agent S will choose to perform not-x at t, agent S will in fact choose to perform not-x at t; agent S’s choice of whether or not to perform x at t is fixed once God’s foreknowledge of which he will perform is fixed. The argument works just as well the other way, of course: Once S’s choice is fixed, so is God’s foreknowledge; and since the relevance of the argument is usually with respect to human freedom of choice, we naturally want to say that as a causal or compulsory matter, that is the way it really works—that God’s foreknowledge is like his looking through a time-telescope and seeing what will happen, rather than like his reaching out and forcing events to occur as they do; and I agree with that; but I do observe that as a strictly logical matter, God’s foreknowledge seems clearly to fix S’s choice. In no possible world can we have both Kx and ~x, and in no possible world can we have both K(~x) and x; what we have, for any possible world in which freedom is a live issue, is N([Kx^x] v [K(~x)^(~x)]). The two—God’s foreknowledge and S’s choice—logically fix each other. This isn’t causation or compulsion, but once you have one, you also have the other. If freedom is supposed to mean that at any time before the choice, the choice is not yet fixed, then this seems to defeat human freedom.

But the real problem with the view that it is S’s temporally later choice of x at time t that causes God’s temporally earlier foreknowledge that S will choose x at t comes about if one tries to combine it with the view that God can interact with the universe in any way whatsoever. If one allows God, who foreknows that agent S will choose to do x at t, to tell agent S beforehand of his future choice, there seems to be a problem: Why can’t agent S, having been informed of his future choice, now decide to behave contrarily and to do not-x at t? Obviously, he can’t so choose, for to do so would violate God’s foreknowledge; but how is he free if he can’t so choose? This argument has force even though Nx would normally be thought of as false. If choice x is the choice to put on a green shirt, we would normally think of it as entirely within S’s power to put on a green shirt and also entirely within his power to put on a blue shirt instead. Even if God foreknows that S will put on a green shirt, it may still be that what God foreknows is really that S will freely choose to put on a green shirt. But if God foreknows S’s choice and also tells S about it, why can’t S choose instead to put on a blue shirt, if he is still free? It seems that the sort of interaction that might lead to S’s choosing differently than foreknown is denied to a foreknowing God, if S is thought to remain free; and it seems that S loses his freedom if a foreknowing God does take part in such an interaction.

Perhaps I need to look around for some characterizations of freedom.

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).

By way of introduction (possibly at too-great length):

Here, I wrote, in part,

“The simple argument for the incompatibility of God’s foreknowledge with human freedom of the will, using ‘Np’ to represent ‘metaphysically necessarily p,’ ‘Pp’ to represent ‘metaphysically possibly p,’ ‘x’ to represent ‘x occurs at time t,’ and ‘Kx’ to represent ‘it is known that x occurs at time t,’ is

1.  N(Kx—>x)    (Premiss—to know that x will occur at t requires that x will occur at t)
2.  Kx—>x          (1, modal axiom M [or T], i.e., Np—>p)
3.  Kx                (Premiss—it is known that x occurs at t [since God has complete foreknowledge])
4.  x                   (2, 3, modus ponens)

Hence, if we assume that event x at t is foreknown, we know that event x at t occurs—and, therefore, not event not-x at t.  But where is human freedom if what is foreknown dictates what occurs?”  I then proceeded to argue that while this argument did not actually imperil human freedom of the will, God’s foreknowledge combined with God’s telling a purportedly free agent what he would do—or, more broadly, God’s initiating any chain of events leading a free agent to choose to do otherwise than God foreknew—would pose a problem:  If God says to person S, “You, person S, will choose to do x at t,” why couldn’t S be contrary and choose to do not-x at t instead?  If he can’t, where’s his freedom?

Chad McIntosh replied, “The simple solution for the defender of divine foreknowledge to the argument as you outlined it is to point out how, even if sound, the argument doesn’t negate creaturely freedom. This is because the necessity in (1) does not carry over to the conclusion, (4). But carrying necessity to x, at least in the argument as you’ve outlined, would be guilty of an invalid modal operator shift. As it stands, all that follows is x, not Nx. In other words, (4) still allows for possibly ~x, which is entirely consistent with creaturely freedom. What you need is an argument that establishes Nx.”

To which I replied, “It seems to me that although Nx is not established, it is also not needed. In every possible world, we have Kx—>x and K(~x)—>~x, so whether x or ~x is foreknown (in particular, which is foreknown by God) fixes whether it is x or ~x that occurs.

“As long as neither Nx nor N(~x) has been established, one may say, as I suggested, that the foreknower’s knowledge is the result of the agent’s freely choosing x or, alternatively, freely choosing ~x, even though there is no escaping the complete correlation between what is foreknown and what is chosen; one may say that although the agent’s choice is temporally fixed before he makes it, it is ultimately metaphysically fixed by his own choice, which in turn makes the foreknower foreknow what he foreknows. But a problem does arise if the foreknower (in particular, God) tells the agent which choice he is going to make. Why can’t the agent, upon being told which choice he’ll make, simply be contrary and choose the opposite? That’s where the problem arises.”

End of introduction.

While I still agree with what I wrote, it has occurred to me that one can rather easily get the conclusion that Chad McIntosh claims I need, with only slightly different premisses:

1.  N(Kx—>x)    (Premiss—to know that x will occur at t requires that x will occur at t)
2.  N(Kx)          (Premiss—it is necessarily foreknown that x will occur at t)
3.  Nx               (1, 2, modal modus ponens)

(Modal modus ponens states that from N[p—>q] and Np, one can conclude Nq.)

Anyone who thinks that God necessarily foreknows all events, including the outcomes of all human choices, will have to endorse the argument.  Of course, we, not knowing whether x or not-x will occur at t, would have to fill in the second premiss as “N(Kx) or N(K(~x)),” but a foreknowing God would know which one he foreknew.  If he foreknew not-x, then we’d simply rewrite the argument with “~x” replacing “x.”  Anyone not happy with thinking of it that way would instead write

1.  N(Kx—>x)             (Premiss—to know that x will occur at t requires that x will occur at t)
2.  N(K(~x)—> ~x)       (Premiss—to know that not-x will occur at t requires that not-x will occur at t)
3.  N(Kx) v N(K(~x))     (Premiss—either it is necessarily foreknown that x will occur at t or it is necessarily foreknown that not-x will occur at t (a consequence of God’s necessary foreknowledge))
4.  N(Kx)—>Nx             (1, Modal Distribution)
5.  N(K(~x))—>N(~x)   (2, Modal Distribution)
6.  Nx v N(~x)             (4, 5, 3, Constructive Dilemma)

Whether Nx or N(~x) is the case will, of course, depend on whether God foreknows that x will occur at t or that not-x will occur at t. 

Naturally, I, not being a believer in God, do not endorse the claim of God’s necessary foreknowledge; but it seems to me that those who do will have to live with human beings’ lack of metaphysical freedom (even though we may very well choose how to act on the basis of conscious deliberation and of evaluation of possible consequences of our actions).

God is sometimes characterized as omniscient, and his omniscience is sometimes taken to include knowledge of the future–of all future events.  But his foreknowledge of events, it is sometimes argued, is logically inconsistent with human beings’ having freedom of the will.  After all, if God already knows what will happen, how can we have any real choice in the matter?  We must choose exactly in accordance with God’s foreknowledge.   

The simple argument for the incompatibility of God’s foreknowledge with human freedom of the will, using “Np” to represent “metaphysically necessarily p,” “Pp” to represent “metaphysically possibly p,” “x” to represent “x occurs at time t,” and “Kx” to represent “it is known that x occurs at time t,” is

1.  N(Kx—>x)  (Premiss—to know that x will occur at t requires that x will occur at t)
2.  Kx—>x        (1, modal axiom M [or T], i.e., Np—>p)
3.  Kx                  (Premiss—it is known that x occurs at t [since God has complete foreknowledge])
4.  x                     (2, 3, modus ponens)

Hence, if we assume that event x at t is foreknown, we know that event x at t occurs—and, therefore, not event not-x at t.  But where is human freedom if what is foreknown dictates what occurs?

The problem with this can be shown if we simply take “Kx” to denote postknowledge instead of foreknowledge.  The same argument works just as well, and yet we don’t think that postknowledge dictates the event x’s occurrence at t.  Rather, we think that event x’s occurrence at t dictates which of “x occurs at t” and “not-x occurs at t” is known.  For an omniscient postknowing being, x dictates Kx, and not-x dictates K(not-x).  We might similarly say that for an omniscient foreknowing being, x dictates Kx, and not-x dictates K(not-x)—i.e., that whether x or not-x happens at time t determines whether x is known or not-x is known, even if the knowledge is foreknowledge instead of postknowledge.

Yet, there’s a clear asymmetry between the cases of foreknowledge and postknowledge.  If a person A makes the choice between x and not-x at time t, and if, at some later time, a postknower tells person A which choice he made, we see nothing strange about it, and it certainly doesn’t affect how we think of person A’s freedom to choose.  On the other hand, if a foreknower tells person A which choice he will make, there is a problem:  Why can’t person A just be contrary and choose to act otherwise than he has been told he will?  If he can do so, then he has the power to render foreknowledge false, which surely can’t be done; if he cannot do so, then he doesn’t seem to be choosing freely. 

It seems that if foreknowledge is to be compatible with freedom of the will, any foreknower is limited in what he can do between the moment of his first attaining foreknowledge and the time t when x is chosen.  In particular, he cannot initiate a chain of events that might lead person A to do what he otherwise would not do.  The simplest way to ensure this would be for the foreknower to remain passive.

Do we, then, limit God, so that although his omniscience extends to complete foreknowledge of events, he does not exercise any power he might have over the course of human events?  (While this might be a stronger restriction than really necessary, it seems like an aesthetically pleasing choice.)  Do we instead say that God’s omniscience does not extend to knowledge of future events? 

Modal logics dealing with possibility and necessity have sometimes been used to provide purported proofs of God’s existence, starting with some definition or characterization of God.  These, along with arguments like St. Anselm’s and Descartes’s, are classified as ontological arguments, as they attempt to argue from what it would mean to be God to the existence of God.  They use the modal operators “necessarily” and “possibly,” which are usually symbolized using a square and a diamond, respectively; but I’m going to use “N” and “P,” since they’re easier to type and since they have more mnemonic value for those of you who are not already very familiar with modal logic.  Thus, “Np” will mean, “necessarily p,” or, “p is necessarily true,” and “Pp” will mean, “possibly p,” or, “p is possibly true.” 

There are various versions of modal logic dealing with metaphysical necessity and metaphysical possibility—other modal logics have to do with other modalities, like moral obligation and moral permissibility, temporal necessity and temporal possibility, or epistemic necessity and epistemic possibility, but modal ontological arguments for the existence of God are concerned with metaphysical necessity and metaphysical possibility—and not everyone agrees on which version is the appropriate one for discussing necessity and possibility; but while there might be some question as to which modal logic is the appropriate one for discussing necessity and possibility, and while I have seen modal ontological arguments for God’s existence formulated using different modal logics (in particular, I’ve seen them using the widely accepted modal logic S5 and have also seen at least one using the modal logic KB), I will not be concerned with the somewhat technical question of whether to accept or reject any particular modal logic.  I can make my point no matter which version of modal logic is used.  (I will note that what is true in K or in KB is also true in S5.) 

The modal arguments I’m addressing in this post make these two assumptions:  First, if God exists, then his existence is not merely metaphysically contingent (possible but not necessary) but is metaphysically necessary; second, it is possible that God exists.  The idea behind the first premiss is that the world around us is merely contingent—its existence was not necessary; it could have failed to exist—and that the merely contingently existing—that which could have failed to exist—requires a necessarily existing entity either (i) to explain why it exists instead of failing to exist or (ii) to bring it into existence and to maintain its existence.  Personally, I don’t see the need for (ii)—I don’t see why, if it’s possible for something to exist, it nevertheless needs something more to make  it exist; it could exist, and it does, so what’s the problem?  Metaphysical possibility doesn’t say, “Possibly p only if something else exists”; it simply says, “Possibly p.”  As for (i), I’m not convinced that existence is the sort of thing that has an explanation; why the state of the universe is what it is at a particular time may have an explanation in terms of its state at an earlier time, for any time after time t=0, but asking for an explanation of why the universe exists at all might be pushing the notion of explanation beyond its appropriate limits.  However, such doubts are irrelevant to the point I want to make here, so let me simply grant the first premiss—i.e., let me grant that if God exists, he exists necessarily.

The problem—you knew I’d get to it eventually, didn’t you?—is that for non-contingent entities, the assumption of possible existence is tantamount to the assumption of necessary existence, while the assumption of possible nonexistence is tantamount to the assumption of necessary nonexistence.  It seems innocuous to assume that God possibly exists; assuming mere possibility isn’t assuming very much, is it?  But for an entity defined as non-contingent, assuming possibility is  assuming a lot—just as assuming the possibility of its nonexistence would be assuming a lot.  If one accepts the premiss of God’s non-contingency and also accepts the possibility that God exists, then one is forced to conclude that God necessarily exists (and, therefore, that God exists).  But if one accepts the premiss of God’s non-contingency and also accepts the possibility that God doesn’t  exist, then one is forced to conclude that it is impossible  that God exists (and, therefore, that God doesn’t exist).  The premisses (1) God’s existence would be non-contingent, (2) it’s possible that God exists, and (3) it’s possible that God doesn’t exist, are mutually inconsistent.  One may assume (1) and (2), or one may assume (1) and (3), but one may not assume all three.  Given that we are accepting (1) (i.e., that God’s existence would be non-contingent), then when a theist, in writing out a modal ontological “proof” of God’s existence, assumes (2) instead of (3), he is implicitly assuming God’s necessary existence (and, therefore, his existence); and when a nontheist, in writing out an analogous modal ontological “disproof” of God’s existence, assumes (3) instead of (2), he is implicitly assuming God’s necessary nonexistence (and, therefore, his nonexistence).  Accepting (1) and (2) instead of (1) and (3), or accepting (1) and (3) instead of (1) and (2), amounts to assuming the conclusion one wants to get in the first place.  For entities defined as non-contingent, “possibly” and “possibly not” collapse to “necessarily” and “necessarily not.”

That’s it for the basic argument.  Now, for those who would like to see a bit more technical detail….

I’ll use “&” for the conjunction ”and,” “v” for the disjunction “or,” “~” for the negation “not,” and “->” for the material conditional “if…then” (or “only if”).   Then we have (see Dan Quattrone’s post in “Doing Things with Words,” at http://dtww.blogspot.com/2005/03/logic-is-for-tricking-people.html ), using “g” to mean “God exists,” 

1.  N(g->Ng)-> (Pg->PNg)          (A theorem of the modal logic K)
2.  [N(g->Ng) & Pg]->PNg          (1, exportation)
3.  PNg->g                               (Modal axiom B [which, along with modal axiom M {a.k.a. T}, extends K to KB], written in its dual form)
4.  [N(g->Ng) & Pg]->g         (2, 3, hypothetical syllogism [propositional logic])
5.  N(g->Ng)                       (Premiss:  The non-contingency of God’s existence, if he exists)
6.  Pg                                   (Premiss:  It is possible that God exists)
7.  N(g->Ng) & Pg                 (5, 6, conjunction)
8.  g                                     (4, 7, modus ponens)

But one might equally well argue

6*.  P(~g)                     (Premiss:  It is possible that God does not exist)
7*.  ~Ng                        (6, duality)
8*.  N(g->Ng)->(g->Ng)        (substitution instance of the modal axiom M [a.k.a. T], which is part of KB [and therefore also of S5]:  Np->p)
9*.  g->Ng                     (8*, 5, modus ponens)
10*.  ~g                        (9*, 7*, modus tollens)

One is reduced to asking which he finds more likely:  That it is possible that God exists, or that it is possible that God does not exist; or that it is necessary that God exists, or that it is impossible that God exists.  But that decision must be made entirely independently of the modal ontological argument itself.  Thus, unsurprisingly, this sort of modal ontological argument, trying to define God into existence, won’t help decide whether or not God actually exists.  
 

St. Anselm of Canterbury (1033-1109) was a monk and theologian who, in his Proslogion, gave an argument for the existence of God on the basis of the definition of God—i.e., on the basis of what it would mean to be  God.  The argument is fatally flawed and should convince no one, but, somewhat surprisingly, it has supporters even to this day.  St. Anselm’s ontological argument for the existence of God runs about like this:  (1)  God is that greater than which none can be conceived; (2)  an existent God would be greater than a nonexistent God; (3) if God did not exist, then something greater could be conceived—namely, a God that did exist; (4)  therefore, God exists.

Although it is possible to analyze this argument in some detail, I’ll just give the main problem here.  The argument fails to distinguish between an object or entity  and the mental conception of an object or entity.  (A lion wandering the African savannah isn’t the same as my mental conception of it; the former is a solid object, while the latter is only an idea.)  It might be that an object God would be greater than its mental conception, or that it would be greater for both the object God to exist and the mental conception of God to be conceived than it would be for the mental conception of God to be conceived without the object God’s existing; but the conception of God may be the greatest conceivable either way.  The object God’s existence or nonexistence doesn’t affect its mental conception’s being the greatest possible mental conception.  (Had St. Anselm argued that the conception of God is the greatest conceivable mental conception, and that the conception of God as existent would be greater than the conception of God as nonexistent, so that the conception of God must be conceived as existent, I’d've had no problem with the argument—but it wouldn’t have established God’s existence.  I conceive of Pegasus as winged, as maned, as white, as solid, as physical—as existent—but I do not thereby bring Pegasus into existence; and I may conceive of Pegasus as giving a ride to my nephew instead of to Bellerophon—I may conceive of Pegasus as nonfictional rather than as fictional—but, again, I do not thereby bring Pegasus into existence.  Conceiving of God as existent is not the same as affirming God’s existence; neither being conceived of as having existence-implying properties nor being conceived of as nonfictional implies actual existence.)

Even the language of the argument is deceptive.  One conceives  mental conceptions, but conceives of  objects; as phrased, the argument has the object God conceived.  Paying proper attention to the difference between existing  and being conceived of  makes that problem go away, but it also completely nullifies the argument.  “God is that entity whose mental conception is the greatest possible,” together with, “An existent God is greater than its mental conception,” doesn’t lead to a contradiction if God is assumed not to exist.