Holy Cyclops

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.