Many people define God as a perfect being. Some philosophers argue that it follows from this definition that God, if he (or she or it) exists at all, exists necessarily: he must exist. It might intuitively seem great or impressive if a being cannot possibly fail to exist, so it might seem that possibly not existing is a kind of vulnerability or flaw, and so a perfect being would have to exist, no matter what.
So far, atheists, who deny that God exists, might agree. Sure, if God (so defined) exists, then he can’t possibly not exist. But that doesn’t show that he actually does exist, does it? Well, some argue it does. That’s the conclusion of what are called modal ontological arguments for the existence of God.
1. Modal arguments and ontological arguments
Modal arguments are generally arguments that depend on claims about possibility, necessity, and impossibility, different “modes” of truth or existence. To say that “1+1=2” is necessarily true, or to say that a square circle can’t exist, is to make a modal claim. Similarly, the claim that God exists necessarily is a modal claim.
Modality is often understood in terms of possible worlds, basically, ways the world could have been. One possible world is the actual world, the world we live in. If a proposition is actually true, then it’s true in the actual world. If it’s true in all possible worlds, then it’s necessarily true: true no matter how the world is or could have been. If it’s not true in any possible worlds, then it’s impossible: there’s no way the world could have been that would have made it true.
Now, ontological arguments for God’s existence hold that something about God’s alleged perfection or greatness entails that God exists.
To combine these concepts, modal ontological arguments are ontological arguments that invoke possibility or necessity. These arguments define God in a way that entails that he exists necessarily if he exists at all, and the conclusion is that he does in fact exist necessarily.
The best-known modal ontological arguments are basically of this form:
(1) It is at least possible for God to exist.
(2) If God’s existence is possible, then necessarily, God does exist.
(3) Therefore, necessarily, God exists.
As noted, even most atheists will initially agree to (1). And it seems pretty clear that if (1) and (2) were true, then (3) would be true.
What might be really puzzling at this point is (2). Why would the mere possible existence of God entail his necessary existence?
2. The S5 Theory of Modal Logic
According to one popular theory of modal logic—a theory we can call the “S5 theory”—the following claim is true:
If a proposition could have been necessarily true, then it is necessarily true.
Thus, if we’re justified in accepting the S5 theory, then (2) is justified. For if God is defined as a necessarily existing being, then (2) is just an instance of that S5-based claim. It says that if possibly, God exists necessarily, then God does exist necessarily.
So is the S5 theory justified? It’s justified if we’re justified in believing that no matter which possible world turns out to be actual, any other possible world still could have been actual instead. Some will find it intuitive to think that what’s possible doesn’t depend on what’s actual. But that’s subject to some dispute.
3. Could God Have Existed?
Someone might also question whether God is possible.
If God is defined to be a necessarily existing being, then in some sense, it’s more “difficult” for him to exist than it is for beings like ourselves who only need to exist in one possible world. For a necessary God to exist, he must exist in all possible worlds.
Plus, God’s alleged attributes might be incompatible with each other, which would render him impossible. If God is defined as necessarily omnipotent and morally perfect, for example, then perhaps those attributes are in conflict with each other, the way “square” and “circular” are in conflict with each other. Then God, like a square circle, would be impossible.
In turn, some critics of modal ontological arguments have questioned whether we know, after all, that God is possible. At the very least, they observe that for all they know,
(1’) It is at least possible for God to not exist.
is just as plausible as (1).
But given God’s definition as a necessary being, then
(2’) If God could have failed to exist, then God does not exist.
Recall: If a necessary being exists, then it can’t possibly not exist. So if it’s possible for God to not exist, then he simply does not exist.
Thus, if (1) and (1’) are equally plausible, then (2) and (2’) may be equally plausible. But then the modal ontological argument has given us no reason to prefer theism over atheism.
For this reason, most of the recent literature about modal ontological arguments discusses whether God is possible after all.
As with other ontological arguments, one can parody this argument. Consider:
(1’’) There could have been a necessarily existing Atlantis.
(3’’) Therefore, Atlantis exists.
Given the S5 theory and (1’’), (3’’) follows, for the same reasons that (3) follows from (1) and the S5 theory.
Defenders of modal ontological arguments respond to parodies by arguing that something about God allows him to be a necessarily existent being, while (for example) Atlantis couldn’t have existed necessarily. For example, perhaps it seems intuitively impossible for a continent (or any other physical object) to exist necessarily, but not impossible for a being such as God to exist necessarily.
Modal ontological arguments arguably have a major advantage over traditional ontological arguments in that they seem to only require affirming that God could have existed. Despite this, they are generally not considered among the best arguments for theism. Still, they are interesting exercises in employing modal logic to attempt to demonstrate an important conclusion.
 Anselm 1965, chs. II-III. See Almeida (2008) for more on the conception of God as a perfect being. See also The Ontological Argument for the Existence of God by Andrew Chapman. Note that this definition is not intended to have a built-in commitment to the existence of God. See the next sentence: God, if he exists at all, exists necessarily.
 Plantinga (1974, 214).
 The most important ones are Malcolm’s (1960), Anselm’s (1965, ch III), Gödel’s (1967), Plantinga’s (1974: ch. 10), Hartshorne’s (2011) (cf. Goodwin 1978), and Maydole’s (2003 and 2009). See Oppy (2021: §§ 7-8) for general discussion of modal ontological arguments.
 For more on modality in general, see Possibility and Necessity: An Introduction to Modality by Andre Leo Rusavuk.
 See n. 2 above. Specific instances tend to offer unique defenses of the possibility premise.
 This is a simplification to deal with the fact that using the word “possibly” and related terms can be misleading (see Engel [2020, 106-107] for some discussion), in that some people will interpret “possibly” in the epistemic, “for all we know” sense. Normally the first premise says something like, “Possibly, God exists,” or “It is possible for there to be a being with God’s properties,” or “There is a possible world in which maximal greatness is instantiated” (cf. Plantinga 1974, 214). Sophisticated formulations of modal ontological-arguments are careful in their statements of the possibility premise, but the more careful they are, the less understandable they are to lay readers. Therefore, I’ve elected to state the possibility premise in the way I have here. However, crucially, everything else I say in the entry works if we replace it with, e.g., “it could have been the case that a God-like being existed,” “there is a possible world in which maximal greatness is instantiated,” or any of the other standard replacements, mutatis mutandis. That is, the key objections I note (namely, that God may not be possible and that S5 may not be sound) apply to arguments with more-sophisticated analogues of (1).
 The discussion here is greatly simplified from what it could have been. “The S5 theory” is the theory that system S5 is sound, i.e, that all theorems of S5 are true. See Garson (2021, § 2) on modal logic in general and Lewis and Langford (1959) for the name “S5.”
 That’s stronger than it needs to be. I have described the situation in which no matter which world turns out to be actual, any other possible world could have existed instead. We describe that situation as “universality”: accessibility—i.e., which possible worlds are possible relative to which—is universal. But strictly speaking, accessibility doesn’t need to be universal; it only needs to be reflexive, symmetric, and transitive, or reflexive and Euclidean. Indeed, you can get something a lot like the modal ontological argument with mere symmetry: from “possibly, God exists necessarily,” you can infer that (actually) God exists. See Kane (1984). See also Garson (2021, §§ 6-7) for more on these topics.
 For example, arguably, if the past cannot be changed, then accessibility is not universal (cf. Metcalf 2017). Given that the Berlin Wall fell, it is impossible that: the Berlin Wall never fell. But it could have been the case that the Berlin Wall never fell: that world is possible in itself but not possible given the actual world. If that’s correct, then this shows that accessibility is not symmetric, and so a fortiori it’s not universal. Also, some philosophers have questioned transitivity; see e.g., Salmon (1986) and Jacquette (2006). Questioning S5 is not normally considered a major objection to the modal ontological argument, however; cf. Maydole (2009, §§ 3-5).
 For example, can God intend to do evil? Arguably, if “yes,” then he’s not necessarily morally perfect, and if “no,” then he’s not omnipotent. If “yes and no,” then that appears to be a contradiction, and arguably, contradictions are impossible. See for example Morriston (2001) and Metcalf (2004). See also The Concept of God: Divine Attributes by Bailie Peterson.
 See e.g., Findlay (1948). See also Oppy (2021, § 8).
 Malcolm 1960; Maydole 2003 and 2009; Metcalf 2005.
 See Devine (1975) and compare the discussion of existent lions in Descartes (2006, 51 ff). If we define an “elion” as an existent lion, then it appears as if one cannot deny that “elions” exist. Descartes replies that existence is essentially a part of God’s nature in the way that the concept of existence isn’t essentially part of a lion. Compare Plantinga’s (1977, 90-91) reply on behalf of Anselm as well, which is that greatest possible islands are incoherent in a way that a perfect being is not.
 See PhilPapers n.d. Very few philosophers in general, or philosophers of religion in particular, believe that ontological arguments are the strongest arguments for theism. Indeed, there is some discussion of whether they have any worth at all (Engel 2020).
Gödel, K. (1967). The completeness of the axioms of the functional calculus of logic. In J. van Heijenoort (Ed. and Tr.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (pp. 582-591). Harvard University Press.
Salmon, N. (1986). Modal paradox: Parts and counterparts, points and Counterpoints. In P. French et al. (Eds.), Midwest Studies in Philosophy XI: Studies in Essentialism (pp. 75-120). University of Minnesota Press.
The Concept of God: Divine Attributes by Bailie Peterson
The Ontological Argument for the Existence of God by Andrew Chapman
Possibility and Necessity: An Introduction to Modality by Andre Leo Rusavuk
Modal Epistemology: Knowledge of Possibility & Necessity by Bob Fischer
About the Author
Tom Metcalf is an associate professor at Spring Hill College in Mobile, AL. He received his PhD in philosophy from the University of Colorado, Boulder. He specializes in ethics, metaethics, epistemology, and the philosophy of religion. Tom has two cats whose names are Hesperus and Phosphorus. http://shc.academia.edu/ThomasMetcalf