As noted by other users, this argument seems to be a version of Alvin Plantinga’s so-called Victorious Modal Ontological Argument for the Existence of God (presented at the end of Plantinga’s book The Nature of Necessity). There is a lot I could say about it, but, space being limited, I want to focus on what I view as the argument’s fundamental problem. Also, not knowing how much knowledge of modal logic’s symbolism I can assume, I’ll use “N” and “P” for “necessarily” and “possibly.” I’ll also abbreviate “a maximally great being exists” (or, in Plantinga’s way of speaking, “maximal greatness is exemplified”) as “MGE.” “P(MGE)” and “N(MGE)” will then mean “possibly, a maximally great being exists,” or equivalently “at some possible world, a maximally great being exists,” and “necessarily, a maximally great being exists,” or equivalently, “at all possible worlds, a maximally great being exists.” (Rather nonstandardly, I’ll use “A(MGE)” for “a maximally great being actually exists.”) To begin with, at least, I won’t worry here about what I view as the problematic definition of the property of maximal greatness as the necessary exemplification of maximal excellence but will simply assume its legitimacy, taking its use as a way of expressing the noncontingency of God’s existence used by Charles Hartshorne (the Hartshorne and Plantinga versions of the argument look very different but are essentially the same at their cores—as is Norman Malcolm’s and as are the rather lengthy Robert Maydole arguments I’ve seen). A maximally great being’s either necessarily existing or necessarily failing to exist—its existing at all possible worlds or at none—is then an expression of its noncontingency, which I will here treat as a premise (as the argument given does). First, I'll show that something must be wrong with the argument; second, I'll say what I think is the fundamental problem with it.
As the argument stands, we may symbolize it as
- P(MGE) (possibility premise)
- If P(MGE), then N(MGE) (noncontingency premise)
2a. N(MGE) (2, 1, modus ponens)
- If N(MGE), then A(MGE) (modal premise—that which is necessarily true
is actually true—modal axiom T)
- Therefore, A(MGE) (3, 2a, modus ponens)
But changing “P(MGE)” to “P(not-MGE)” (which, I note, is not the same as not-P(MGE)) yields the atheistic argument
1*. P(not-MGE) (possibility-not premise)
1**. not-N(MGE) (1*, duality)
2*. If P(MGE), then N(MGE) (noncontingency premise)
2a*. not-P(MGE) (2*, 1**, modus tollens)
2a**. N(not-MGE) (2a*, duality)
3*. If N(not-MGE), then A(not-MGE) (modal premise—that which is necessarily
true is actually true—modal axiom T)
4*. Therefore, A(not-MGE) (3*, 2a**, modus ponens)
(“Duality” refers to what in any classical modal logic amounts to logical equivalence. “Possibly, p” and “not necessarily not-p” is one duality; “necessarily, p” and “not possibly not-p” is another. As long as we are applying a classical modal logic—like Plantinga’s S5—the use of duality should be thought of as not changing anything but the form of a modal sentence—the way it “looks.”) Lines 1** and 2a** are merely rewritings of one statement into an equivalent form for the sake of applying modus tollens or modus ponens later; they are not new premises. Lines 3 and 3* are simply substitution instances of the modal axiom T, “if Np, then p.” All that has changed from the first argument to the second is that “P(MGE)” has changed to “P(not-MGE).” Instead of the first premise’s being that a maximally great being’s existence is possible, we have the new premise that a maximally great being’s nonexistence is possible. (Foreshadowing: But, if neither MGE nor not-MGE seems self-contradictory, why shouldn’t both be possible?)
One can also create parallel arguments for any other proposition thought to be noncontingent. Philosophers usually consider mathematical truths to be necessary truths. We do not know whether Goldbach’s conjecture is true or is instead false, but it must be one or the other, and if mathematical truths really are necessary truths, then either Goldbach’s conjecture is necessarily true or it is necessarily false. Using “GB” for “Goldbach’s conjecture is true,” we may then write the argument
- P(GB) (possibility premise)
- If P(GB), then N(GB) (noncontingency premise)
2a. N(GB) (2, 1, modus ponens)
- If N(GB), then A(GB) (modal premise—that which is necessarily true
is actually true—modal axiom T)
- Therefore, A(GB) (3, 2a, modus ponens)
Voilà! We have proven the truth of the Goldbach conjecture without doing any mathematics! But….
1*. P(not-GB) (possibility-not premise)
1**. not-N(GB) (1*, duality)
2*. If P(GB), then N(GB) (noncontingency premise)
2a*. not-P(GB) (2*, 1**, modus tollens)
2a**. N(not-GB) (2a*, duality)
3*. If N(not-GB), then A(not-GB) (modal premise—that which is necessarily
true is actually true—modal axiom T)
4*. Therefore, A(not-GB) (3*, 2a**, modus ponens)
And behold! We have now proven not only the truth of the Goldbach conjecture but also its falsity! And without doing any mathematics!
(This might be a good place to note that the relevant kind of possibility is not subjective but is objective. Were it subjective, the contingency premise would be judged false by anyone who was not already convinced of the truth of N(MGE). Its being objective renders the plea “But surely, it’s at least a millionth likely that MGE, or a billionth likely, or a trillionth likely—and therefore P(MGE) must be true” pointless—just as “But surely, it’s at least a millionth likely that GB, or a billionth likely, or a trillionth likely—and therefore P(GB) must be true” pointless. Not to mention that “Surely, it’s at least a millionth likely that, etc.” could just as easily be said of P(not-MGE) and of P(not-GB), so that it doesn’t decide between P(MGE) and P(not-MGE) or between P(GB) and P(not-GB).)
I hope that these parallel arguments are enough to convince you that something must be wrong with the ontological argument originally given. Next, I'll say what I think fundamentally goes wrong with it.
If we can see that something must go wrong with the ontological argument given—essentially, Plantinga’s so-called Victorious modal argument—what goes wrong with the argument given?
If changing only the premise P(MGE) to the premise P(not-MGE) allows the conclusion of A(not-MGE) instead of the conclusion A(MGE)--if P(MGE) and P(not-MGE), when combined with the noncontingency premise, result in contradictory conclusions—then we must suspect that something about the combination of the two premises must be to blame. We could, of course, simply jettison the noncontingency premise—in fact, I find it dubious—but if we keep it, what goes wrong?
The noncontingency premise makes the following true: either N(MGE) or N(not-MGE). If N(MGE), then P(MGE) is also true, but P(not-MGE) is not; if N(not-MGE), then P(not-MGE) is true, but P(MGE) is not. This is by design: Hartshorne argued that it was not possible for God to merely happen to exist, or to merely happen not to exist. Plantinga’s maximal greatness was designed to bring about that same noncontingency. But notice that then P(MGE) is equivalent to N(MGE), and notice that then P(not-MGE) is equivalent to N(not-MGE). If MGE is possible, then not-MGE is not; if not-MGE is possible, then MGE is not. The noncontingency premise collapses possibility and necessity for a maximally great being.
But then the premise P(MGE) is equivalent to N(MGE), and any reason to think that P(MGE) is true is also reason to think that P(not-MGE) is false. That’s not usually how the justification of a claim of possibility works. Here, though, sufficiently strong reason to think that P(MGE) is also sufficiently strong reason to think that N(MGE) (and to think that not-P(not-MGE)). But if one had sufficiently strong reason to think that P(MGE) as to accept P(MGE) (and therefore to go ahead and use the argument), he would also have to have sufficiently strong reason to think that N(MGE); and if he had that, he wouldn’t need the ontological argument. And if one did not have sufficiently strong reason to think that N(MGE), he wouldn’t have sufficiently strong reason to think that P(MGE); and lacking that, he couldn’t use the argument. The argument is therefore either superfluous (if one already has sufficiently strong reason as to accept N(MGE)) or toothless (if one does not already have sufficiently strong reason as to accept N(MGE)). And as a practical matter, that’s what’s fundamentally wrong with it.
There are, of course, other things one might notice about it. One might notice that if neither “possibly, a maximally great being exists” nor “possibly, a maximally great being does not exist” (neither P(MGE) nor P(not-MGE)) seems self-contradictory, and yet if one of them must be not only false but necessarily false, then it seems that some statements can be necessarily false without being self-contradictory. One may then choose among options: (a) one of the two really is self-contradictory despite not seeming to be, or (b) the premise of God’s noncontingency is false, or (c) for some statements, being necessarily false does not imply being self-contradictory. (Anyone wanting to keep (c) will then have to choose between (a) and (b).) As the same issue arises in the arguments involving Goldbach’s conjecture, and would do so in any argument having a similar noncontingency premise, option (b) has to be a more general option—perhaps that no non-tautologies are necessary truths and no non-contradictions are necessary falsehoods, thereby avoiding that sort of noncontingency premise altogether.
Or, one might think of the term “possible worlds” not as denoting worlds but as denoting world-descriptions, and he might think of its being impossible for an entity described one way as exemplifying a property defined in terms of how it is described in other ways. So, for instance, one might think that a tiger could be described as striped, because stripedness is a within-world property, but he might think of it as illegitimate to describe a tiger as striped at every possible world, i.e., world-description, even if he thought that tigers were indeed striped at every possible world, because although he might think of “Necessarily, tigers are striped” as true—he might think of “At every possible world (i.e., world-description), tigers are striped” as true—he might think of “Tigers exemplify necessary stripedness” as badly phrased, saying something nonsensical. He might suppose the necessity of tigers’ being striped to be a feature of the collection of world-descriptions rather than a feature ascribed to tigers in any individual world-description. One might think of Plantinga’s move from maximal excellence—maximal power, maximal knowledge, maximal goodness, or whatever one wants to include in the term—to necessary maximal excellence (which constitutes maximal greatness), thought of as a property of an object at a possible world rather than as characterizing the entire collection of world-descriptions, as simply illegitimate.
But the fundamental problem—that the argument is either superfluous or toothless—remains, even if one keeps the noncontingency premise in place and doesn’t worry about anything else that could be noticed about the argument. Because of the collapsing of possibility and necessity and the consequent equivalence between “P(MGE)” and “N(MGE),” so that sufficient justification for accepting the former must also be sufficient justification for accepting the latter, you can’t use the argument unless you don’t need it, and if you need it, you can’t use it.
(I don't mind the copying of my posts elsewhere, but please attribute them to "Keith Brian Johnson (MindWalk)")