In this post I’ll explain both the motivation behind Alvin Plantinga’s Modal Ontological Argument, as well as its various steps. For my purposes, I’ll work with a simplified version of that argument which Plantinga presents in his book, The Nature of Necessity (Clarendon Press, 1974), pp. 216-17.

I begin with couple of definitions employed in the argument:

(D1) The property *unsurpassable greatness* is equivalent to the property *maximal excellence in every possible world* (ibid., p. 216).

(D2) The property *maximal excellence* entails the properties *omniscience*, *omnipotence*, and *moral perfection* (ibid., p. 215).

Commentary on the definitions. Both (D1) and (D2) are based on the intuition of “perfect being theology” which follows St. Anselm in thinking of God as “the being than which none greater can be conceived.” Thomas V. Morris explains the basic idea behind this Anselmian conception of God:

“The Anselmian conception of God is that of a greatest possible, or maximally perfect, being. On this conception, God is thought of as exemplifying necessarily a maximally perfect set of compossible great-making properties. To put it simply, a great-making property is understood to be a property it is intrinsically better to have than to lack. If, for instance, the exemplification of a state of knowledge is of greater intrinsic value than a lack of its exemplification, it will follow that one of the divine attributes is that of being in a state of knowledge. Likewise, if it is better to be omniscient than to be deficient in knowledge, God will be thought of as omniscient, and so forth. Traditionally, the Anselmian description has been understood to entail that God is, among other things, omnipotent, immutable, eternal, and impeccable as well as omniscient.” (Anselmian Explorations: Essays in Philosophical Theology (Notre Dame, 1987, p.12)

Traditionally understood, God is thought to be (among other things) omniscient, omnipotent, and morally perfect. Hence (D2) and the property *maximal excellence*.

But now consider two beings, A and B, where A is maximally excellent but contingently so, while B is necessarily maximally excellent (i.e., maximally excellent in all possible worlds). All other things being equal, which being is greater? It seems clear that B would be greater than A; thus, *unsurpassable greatness* is a great-making property and, according to the Anselmian conception of God, he has that property, too. A being that deserves the honorific title ‘God’ isn’t one that merely happens to be (i.e., contingently) omnipotent, omniscient, and morally perfect – it couldn’t have been otherwise. Hence (D1) and the property *unsurpassable greatness*.

So much for the definitions employed in the argument. Now for the argument’s logical structure (hold on to your seats!).

I introduce some (not too much) abbreviations and formal apparatus.

Where p is some proposition:

Necessarily, p = true, if and only if, p is true in all possible worlds (where a possible world is, roughly, a complete and consistent scenario).

Possibly, p = true, if and only if, p is true in some (i.e., at least one) possible world.

Where x is some object or thing:

UGx = x is unsurpassably great, i.e., x exemplifies *unsurpassable greatness*.

MEx = x is maximally excellent.

Now the argument itself (cf. _The Nature of Necessity_, p. 216; I renumber Plantinga’s premises; [technical note]: read the formulae below as asserting universal closures, or existence in a world, where appropriate):

(1) Possibly, UGx. [Premise]

(“There is a possible world in which unsurpassable greatness is exemplified.”)

(2) Necessarily, UGx if and only if necessarily MEx. [Premise: From (D1)]

(“The proposition *a thing has unsurpassable greatness if and only if it has maximal excellence in every possible world* is necessarily true.”)

(3) Necessarily, UGx. [From (1) and (2)]

(“*Possesses unsurpassable greatness* is instantiated in every world.”)

(4) UGx [From (3)]

(“But if [(3) is] so, it [the property UG] is instantiated in this world; hence there actually exists a being who is omnipotent, omniscient, and morally perfect and who exists and has these properties in every world.”)

Commentary on the parts of the arguments:

[A] Premise (2): It’s justified by (D1), whose rationale was given earlier in our discussion of Anselmian perfect being theology.

[B] The move from premise (3) to (4): It’s justified by the following modal axiom, which is very weak and included in all normal systems of modal logic:

(T) If necessarily p, then p.

In short, (T) says that whatever is necessarily true is true – which is hardly controversial. The move from (3) to (4) is good as gold.

[C] The move from (1) and (2) to (3): I introduce a few more abbreviations to facilitate my analysis. Following standard usage, let (the box) “[]p” = “necessarily, p” and (the diamond) “<>p” = “possibly, p.” Let “p <-> q” = “p if and only if q.” Then the move from (1) and (2) to (3) can be expressed slightly more formally as follows:

(1*) <>UGx

(2*) [](UGx <-> []MEx)

(3*) []MEx [from (1*) and (2*)]

This is the most technical step of the entire argument – and one that is most inaccessible to the reader without basic knowledge of modal logic. (In what follows, I shall do my best to explain the steps in layman’s terms!) The inference in question is valid in a system of modal logic called S5, which is the strongest of the normal systems of modal logic. Its characteristic axiom is the following:

(S5) If <>[]p, then []p (English: if it is possibly necessary that p, then it is necessary that p).

S5 is thought to characterize the logic of “broadly logical” or “metaphysical” necessity – the strongest kind of necessity there is. The basic idea behind the S5 axiom is this:

(INVARIANCE) “[T]hat what is necessary or impossible does not vary from world to world” (Ibid., p. 216).

To illustrate: consider the mathematical, and hence necessary, truth that 2+2=4. Since it is necessary, it is true in the actual world as well (cf. axiom T above). But because it is necessary, it can’t have a different truth value (i.e., falsehood) in some other, non-actual possible world. To think that it could have turned out to be false is to think of it as being contingently true, and not as a necessary truth. So, given that 2+2=4 is true in the actual world, it is also true in all possible worlds. This point, however, is not really about the *actual* world only. By the above line of reasoning (utilizing INVARIANCE), if we know that p is necessary and true in some possible world W (which may or may not be actual), then we know that it is true in all possible worlds – including the actual world. Hence, the S5 axiom: if there is a possible world in which p is necessarily true (i.e., <>[]p), then – given INVARIANCE – p is necessarily true (i.e., true in all possible worlds = []p).

[Really technical note (for the very observant reader; the average reader can skip this bracketed portion): How come (3) says []UGx while (3*) – which is supposed to be equivalent to (3) – says []MEx? Doesn’t (D1) assert that UGx = []MEx? (Answer: Yes.) So, isn’t []UGx = [][]MEx? (Again, yes.) Where did the extra “box” come from? Answer: from an axiom of a weaker modal system, S4, which is contained in the stronger system S5:

(S4) If []p, then [][]p (English: if it is necessary that p, then it is necessarily necessary that p, cf. INVARIANCE again.)

(S4) combined together with the aforementioned (T), if []p then p, gives us: []p <-> [][]p – which entails []UGx <-> []MEx.

Now back to our regularly scheduled program…]

[D] That leaves us with premise (1): <>UGx, that there is a possible world in which *unsurpassable greatness* is exemplified. Why believe that premise?

And there’s the rub – and, by my lights, the only really controversial point in the argument. I’ll leave you to dwell on that premise. 😉

Suggested reading: http://plato.stanford.edu/entries/ontological-arguments/