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Two Varieties of ‘Possible’ and the Ontological Argument


The ontological argument for the existence of God has a long and well-discussed history. First clearly articulated by St. Anselm in 1078, it almost immediately generated lively debate, debate that continues to the present day. Attacks on the argument have been launched by Gaunilo, St. Thomas Aquinas, David Hume, Immanuel Kant, and others, and those attacks have forced supporters of the argument (including, but not limited to, Alvin Plantinga, William Alston, and David Bentley Hart) to present different formulations of it. This has sharpened the lines of demarcation between the two sides and made the issues involved clearer. I will try to address an aspect of the debate that I think has been largely ignored to this point.

Briefly, the argument runs along the following lines. God is that than which no greater can be conceived. It is more perfect to exist in the mind and in reality than to exist merely in the mind. God exists in the mind. Therefore, God exists in reality. Throughout this discussion, I will simply grant that the argument is deductively valid.

Peter Kreeft and Robert Tacelli have summarized several forms of the argument, and I will be relying on their presentation here. (Interestingly, while addressing the ontological argument, they say, almost apologetically: “We include it, with a minimum of discussion, not because we think it conclusive or irrefutable, but for the sake of completeness.”[1]) Here is their summary of Anselm’s original version:

  1. It is greater for a thing to exist in the mind and in reality than in the mind alone.
  2. “God” means “that than which a greater cannot be thought.”
  3. Suppose that God exists in the mind but not in reality.
  4. Then a greater than God could be thought (namely, a being that has all the qualities our thought of God has plus real existence).
  5. But this is impossible, for God is “that than which a greater cannot be thought.”
  6. Therefore God exists in the mind and in reality.[2]

I want to focus on line 3, “Suppose that God exists in the mind but not in reality.” In particular, I want to examine what it is, in Anselm’s sense, to “exist in the mind.” Without going too far out on a limb, I can say that four-sided triangles do not exist. I don’t merely mean that four-sided triangles do not exist in reality (though of course I think that). The important point is that the idea of a four-sided triangle is incoherent. That being the case, it can’t exist in my mind, either. The fact that I can form a sensible sentence about a four-sided triangle doesn’t mean I have a clear picture of one in my head for the simple reason that there is no such thing as a clear mental image of a four-sided triangle. It is a contradiction in terms; a four-sided triangle is impossible. So when Anselm says “Suppose that God exists in the mind,” he is at the very least saying “Suppose that God, as defined, picks out a coherent concept.” The use of “suppose” is important: Anselm offers as a premise the coherence of the God-concept.

Other authors have been more explicit in this regard. In Charles Hartshorne and Norman Malcolm’s version of the argument, the very first premise is “The expression ‘that being than which a greater cannot be thought’ … expresses a consistent concept.”[3] Their candor, I suppose, is laudable. Right out front, without argument, they baldly state that, unlike a four-sided triangle, the definition of God is logically well-formed. Kreeft paraphrases their entire argument like this:

  1. The expression “that being than which a greater cannot be thought” (GCB, for short) expresses a consistent concept.
  2. GCB cannot be thought of as (a) necessarily nonexistent, or as (b) contingently existing, but only as (c) necessarily existing.
  3. So GCB can only be thought of as the kind of being that cannot not exist, that must exist.
  4. But what must be so is so.
  5. Therefore, GCB (i.e., God) exists.[4]

Alvin Plantinga, too, follows this pattern. After setting out some preliminary definitions, he gives as his first premise: “There is a possible world (W) in which [God exists].”[5] Though Plantinga puts this in the language of possible worlds, rather than in that of coherent concepts, he is still simply asserting that the existence of God, as defined, is not impossible. His version of the argument, again as paraphrased by Kreeft, runs as follows:


Maximal excellence: To have omnipotence, omniscience, and moral perfection in some world.
Maximal greatness: To have maximal excellence in every possible world.

  1. There is a possible world (W) in which there is a being (X) with maximal greatness.
  2. But X is maximally great only if X has maximal excellence in every possible world.
  3. Therefore, X is maximally great only if X has omnipotence, omniscience, and moral perfection in every possible world.
  4. In W, the proposition “There is no omnipotent, omniscient, morally perfect being” would be impossible—that is, necessarily false.
  5. But what is impossible does not vary from world to world.
  6. Therefore, the proposition “There is no omnipotent, omniscient, morally perfect being” is necessarily false in this actual world, too.
  7. Therefore, there actually exists in this world, and must exist in every possible world, an omnipotent, omniscient, morally perfect being.[6]

The point of all of this is that for the argument to work, the concept of God (that than which no greater can be conceived) must be coherent. A second point that one may see in passing is that no one seems willing to actually substantiate that this is the case. There is a third point to raise, but I will wait until the time is right.

Now, it is a happy fact for poetry and a sad one for logic that our language has many ambiguous terms, and ‘possible’ is one of them. Given that ‘possible’ has more than one normal meaning, allow me to address two of them.

Consider the question as to whether there exists an even number greater than 2 that cannot be written as the sum of two (not necessarily distinct) prime numbers. We might say, “Well, four equals two plus two, six equals three plus three, and eight equals three plus five. Hmm, I don’t know. It’s possible that all even numbers are like that.” On the other hand, we might just as easily say, “I don’t know; it’s possible that there is an even number greater than 2 that isn’t the sum of two primes.” (The proposition that every even number greater than 2 can be written as a sum of two primes is called the strong Goldbach conjecture.)

Philosophers call this sense of possibility “epistemic possibility.” Epistemic possibility is possibility with respect to one’s knowledge. In the situation just described, for all I know there may be an even number greater than 2 that cannot be expressed as a sum of two primes—or there may not be. Nothing I know rules out either alternative. Thus, both of “There is an even number greater than 2 that is not the sum of two primes” and “There is no even number greater than 2 that is not the sum of two primes” are simultaneously epistemically possible, and pronouncement of the (epistemic) possibility of either proposition is warranted.

There is another reading of the term, however. Consider the fact (assuming the Peano postulates, which are the standard rules for arithmetic) that either there is an even number greater than 2 that cannot be expressed as the sum of two primes, or there is not. Both propositions cannot be true, and one must be; we just don’t know which is which. There’s more: since the existence of our putative non-two-prime-summable even number greater than 2 is a matter of mathematics, whether there is such a number is invariant across all possible worlds. All this being established, one of the “possibilities” is not, in fact, possible. This second sense of ‘possibility’ is “logical possibility.”

To be clear, the existence of an entity or the occurrence of an event in at least one possible world makes that entity or event logically possible. The existence of an entity/occurrence of an event in no possible worlds makes that entity/event logically impossible, and its existence/occurrence in every possible world makes it logically necessary. This means that the existence of the Goldbachian number is not a question of logical possibility versus logical impossibility, but rather logical necessity versus logical impossibility. We should note that, unlike claims of epistemic possibility, statements of the form “X is logically possible” require at least some justification.

What has this to do with the ontological argument? Well, it is a fact that either a God as defined in the ontological argument exists, or that such a God does not exist. As with the putative not-a-sum-of-two-primes even number that does or does not exist (but we just don’t know which), we don’t know whether God exists. Still, precisely one horn of the dilemma is correct: either God exists or God does not exist. Further, like Goldbach’s number, the existence of God is a matter of logical necessity versus logical impossibility.

This last point is important, and it can be seen in each form of the ontological argument given above. Anselm derives a contradiction from the assumption that God does not exist in the world (“the” world, because possible world semantics had not yet been invented in Anselm’s day). Hartshorne and Malcolm deduce that God “cannot not exist”[7] and can only be thought of as “necessarily existing”[8] (that is, existing in all possible worlds). And Plantinga infers that “the proposition ‘There is no omnipotent, omniscient, morally perfect being’ would be impossible—that is, necessarily false”[9]—i.e., that God’s existence is logically necessary. In fact, the essence of the ontological argument is “[t]he claim that the very logical possibility of God’s existence entails His actuality.”[10]

I granted earlier that I would assume the deductive validity of the ontological argument, but for the purpose of deciding whether God or not exists, it’s not important if the argument is merely deductively valid—it must be sound. That is, it must be valid, and its premises must be true. The premise I want to focus on is the one involving the possibility of the existence of God. It doesn’t matter whose formulation we take, either the first half of Anselm’s line 3 (“Suppose that God exists in the mind”), Hartshorne and Malcolm’s line 1, or Plantinga’s line 1. They all boil down to “the concept of God, as defined, is logically possible.” Indeed, they cannot mean “God is epistemically possible,” for this means “For all I know, God could exist,” and that’s plainly not good enough. Imagine that someone told you, “For all I know, five-sided trapezoids exist.” Even if the statement is true (the speaker might be ignorant of the definition of a trapezoid or confused as to certain facts about the number 5), that in no way guarantees the actual existence (or even the logical possibility) of five-sided trapezoids.

For the ontological argument to work, it must boil down to “If God is logically possible, then God exists.” That is to say, the validity that I granted at the beginning of our adventure assumes the use of logical possibility, not epistemic possibility. To demonstrate that the argument is sound (and to be entitled to conclude “God exists”), one must show “God is logically possible” is true; failing absolute proof, good reasons for thinking that it is true constitute good reasons for thinking that the argument is sound.

So why isn’t any effort given to showing that the definition of God given in the argument is coherent? We read things like:

The crucial Premise, therefore, is … that it is possible that a maximally great being exists. To refute this Premise, one would need to show that the very concept of an infinitely great being is somehow logically incoherent—like a “married bachelor.” Since no argument to that effect has been forthcoming, however, it follows necessarily and inescapably that “Therefore, a maximally great being exists.”[11]

I trust that I am not alone in thinking that something slippery has just occurred. The fact that no argument showing that the concept of God is incoherent is on offer at best allows us to say that for all we know, God’s existence is possible. This is a textbook case of epistemic possibility. The question, then, is what are we to make of this?

Such a statement is not subject to an infinite number of interpretations, and I see two charitable readings. The first is that an unintentional case of equivocation is in play: ‘possible’ (explicitly and by implication) is being used in both the epistemic and logical senses. While this hypothesis is hard to square with the author’s direct use of “logical possibility” mere paragraphs above the declaration of triumph just quoted, an argument for the epistemic possibility of God is given (i.e., no one has proved that God is impossible).

The second escape route that I see is that the author doesn’t feel that he or she bears the burden of proof to show that the God-concept is coherent. That is, s/he may be reasoning along the following lines. “The term ‘God’ has been used for thousands of years, and it has been used by many over that time in the way that I am using it: to denote an eternal, omnipotent, omnibenevolent, omniscient creator. Perhaps not everyone in my faith tradition has explicitly used Anselm’s (or Hartshorne and Malcolm’s, or Plantinga’s, …) definition, but the ones they use are closely related. There exists no real confusion when we use the term ‘God’, and use of the term contributes to productive conversation. This constitutes a prima facie case that ‘that than which no greater can be conceived’ is well-defined. That being the state of things, no defense of the coherence of the definition is needed unless or until there appears to be a problem.”

The first response to this line of defense is to note that just because a potential problem is not obvious doesn’t mean that there isn’t a problem. There is a profound problem with either “There is an even number greater than 2 that may not be written as a sum of two primes” or its denial. Not only is one of them false, there is no possible world in which it is true. One of them is, logically speaking, totally, irrevocably, and catastrophically compromised. Which one is it? We don’t know; it’s not obvious.

If that was all that one could say, then theists would still be in a strong position. Saying that there might be a problem, just not a clear one, goes nowhere in undermining the theistic position that there is no reason to doubt that their definition is coherent. What is needed to force theists to defend their “crucial premise” is not necessarily proof that “that than which no greater can be conceived” is a flawed definition, but merely solid reason to think that it might be.

To try to do this in a meaningful way, we have to know what “that than which no greater can be conceived” means. What do theists have in mind? If whatever it is is coherent, the game is over; we will know what God is and that God exists. There seems to be no better person to start with than Anselm himself. In chapter 5 of Proslogion, Anselm says:

What then are you, Lord God, that than which nothing greater can be thought? But what are you if not that which is the greatest of all things, who alone exists through himself, who made everything else from nothing? For whatever is not this, is less than what can be thought. But this cannot be thought about you. For what good is lacking to the supreme good, through which every good thing is? And so, you are just, truthful, happy, and whatever it is better to be than not to be.[12]

Anselm has been taken to mean that God has each attribute that it is better to have than not. He adds further qualities later, describing God as (among other things) omnipotent, merciful, wise, and eternal.[13] René Descartes, who put forth his own version of the ontological argument, includes omnipotence, omniscience, immutability, eternality, and simplicity among the divine attributes.[14] We have seen that Plantinga lists omniscience, omnipotence, and moral perfection as descriptive of a maximally great being, and famed logician Kurt Gödel ascribed all “positive properties,” though he does not say specifically what they are.[15]

Not just any list of divine attributes will do. Naturally, they must be compatible, but there is more: “If any of the properties that are conceptually essential to the notion of God do not admit of an intrinsic maximum, then Anselm’s argument strategy will not work because … the relevant concept of God is incoherent,” and therefore “Anselm’s argument works, if at all, only for concepts that are entirely defined in terms of properties that admit of some sort of intrinsic maximum.”[16] It hardly needs saying, too, that the attributes ascribed to God must be worthy of the greatest thing imaginable. Omniscient, omnipotent, and omnibenevolent? Surely.

Let me tell you a secret: I doubt that there is an even number greater than 2 that cannot be written as a sum of two primes. Do I have proof of my position? No, though everlasting fame would be mine if I did. What I do have is this: all candidate numbers up to and including 400 trillion have been tried, and every single one has been summable as two primes. I understand that 400 trillion isn’t more than the smallest step toward infinity, but when your record is 200-trillion-minus-1 wins and no losses, you are justified in being optimistic about a perfect season (though, again, you are not guaranteed one).

Are proponents of the ontological argument in this position? Has it been a millennium of attempted debunking by skeptics, with each attempt having been easily swept aside? No. In fact, there is a sizeable body of literature that suggests that an omniscient, omnipotent, omnibenevolent agent cannot possibly exist because there are logical issues with these properties, both singly and in combination. I will make no effort to summarize all of the arguments, but I will supply a few examples.

Let us begin with omnipotence and the famous question, “Can God make a rock so heavy that he can’t lift it?” Here the questioner is attacking the definition of omnipotence as the ability to do anything. If God can create such a rock, the argument goes, God is not omnipotent (because he can’t lift it). If God cannot create an unliftable rock, God is again not omnipotent. It is an easy enough move for the theist to simply (and reasonably) stipulate that the definition of omnipotence is the ability to do anything that is logically possible. But then the questioner can ask, “Can God tell a lie?” The classic theistic reply is that God’s omnipotence is the ability to do anything logically possible that does not run counter to his essential nature. At this move, the theist’s problems only get worse, for now the defense relies on essentialism, a doctrine that has been controversial since its inception over 2000 years ago. Further, and more than a little ironically, Plantinga himself provides us with a retort: Consider Mr. McEar, a being with the essential property that he may only scratch his own ear. Using the latest definition, Mr. McEar is omnipotent.

It is important to remember that it is not necessary to prove that the definition of God is incoherent. We must only cast enough doubt to remove the theistic point of view from the position of “null hypothesis.”

Now consider omniscience and omnipotence taken together. If God is omniscient, God knows the future. For example, if God knows that it will rain tomorrow, then it must rain tomorrow. If it must rain tomorrow, then God is incapable of arranging for fair skies on the appointed day. If such is the case, then God is not omnipotent. Generally speaking, if God knows the future, then he cannot change it. Therefore, if God is omniscient, he cannot be omnipotent.

It gets worse still if we pursue this further. If God is omniscient, God knows the future. If God knows the future, he has no freedom at all, as each of his future acts is determined. Even if the theist wants to say that God is free if he is doing only things that he wants to do, one is left with another question: If I am locked in a room that I want to be in, am I free to leave?

While it may be possible to once more massage definitions to avoid these pitfalls, omniscience all by itself is problematic. As a first approximation, define the belief-set of an omniscient being to be the set consisting of all and only true propositions. Patrick Grim gives a Cantor-style argument with the conclusion that there can be no set of all and only true propositions.[17] Grim investigates several other definitions of omniscience, but concludes that there are none (of which he is aware) that are both faithful to common usage and that escape his paradox.[18]

Again, it may be possible to avert each of these paradoxical traps if the theist is allowed to modify definitions as we go, but each redefinition invariably restricts the powers of God and/or does violence to the generally accepted meanings of the words (e.g., Anselm says the ability to lie is a weakness[19]). Sometimes the damage is not too great, and other times God’s essence comes out unrecognizable. As a rule, more nuanced paradoxes require more limiting definitions and tortured meanings.

I suggest, then, that a definition of God based on a list of divine attributes is plagued with enough (at least potential) problems to not deserve to be presumed logically solid. Are there ways to coherently define “that than which no greater can be conceived” without resorting to such a list? This is surely possible. Is it possible to do this in a way that captures the notion of God to the satisfaction of both theists and atheists? We don’t know, because no one has tried. We do know that the theists’ preferred approach to the problem, what we may call the appeal to divine attributes, has, to put it mildly, not proceeded without difficulties. In spite of this, nowhere in accumulated theistic literature of nearly 1,000 years is there to be found a general defense of the definition of God, a pushback to particular criticisms, or an alternative to enumeration of divine traits.

This total lack of a defense of the theist’s premise that God is logically possible cannot be justified. It isn’t as though there hasn’t been time to mount a defense. Gaunilo, a contemporary of Anselm, gave a counterargument in real time, and objections to omnipotence date to no later than the 13th century. There has been time to formulate drop-dead responses; it’s just that no one has done it. The best case for theists is that their presumption of coherence has been to some degree undermined, and therefore it is reasonable to ask that some positive account of the coherence of God be given before the ontological argument can be given any weight at all.


[1] Peter Kreeft, “20 Arguments for God’s Existence” (n.d.). Peter Kreeft website. <https://www.peterkreeft.com/topics-more/20_arguments-gods-existence.htm>.

[2] Kreeft, “20 Arguments for God’s Existence.”

[3] Kreeft, “20 Arguments for God’s Existence.”

[4] Kreeft, “20 Arguments for God’s Existence.”

[5] Kreeft, “20 Arguments for God’s Existence.”

[6] Kreeft, “20 Arguments for God’s Existence.”

[7] Kreeft, “20 Arguments for God’s Existence.”

[8] Kreeft, “20 Arguments for God’s Existence.”

[9] Kreeft, “20 Arguments for God’s Existence.”

[10] All About God Ministries, “Ontological Argument” (n.d.). All About Philosophy website. <https://www.allaboutphilosophy.org/ontological-argument.htm>.

[11] All About God Ministries, “Ontological Argument.”

[12] Gregory Sadler, “Anselm of Canterbury” in The Internet Encyclopedia of Philosophy ed. James Fieser (Martin, TN: University of Tennessee at Martin, n.d.). <https://www.iep.utm.edu/anselm/>.

[13] Sadler, “Anselm of Canterbury.”

[14] Lawrence Nolan, “Descartes’ Ontological Argument” in The Stanford Encyclopedia of Philosophy (Spring 2020 edn.) ed. E. N. Zalta (Stanford, CA: Stanford University, 2020). <https://plato.stanford.edu/archives/spr2020/entries/descartes-ontological>.

[15] Graham Oppy, “Ontological Arguments” in The Stanford Encyclopedia of Philosophy (Spring 2019 edn.) ed. E. N. Zalta (Stanford, CA: Stanford University, 2019), §7
(“Gödel’s Ontological Argument“). <https://plato.stanford.edu/archives/spr2019/entries/ontological-arguments/>.

[16] Kenneth Einar Himma, “Anselm: Ontological Argument for God’s Existence” in The Internet Encyclopedia of Philosophy ed. James Fieser (Martin, TN: University of Tennessee at Martin, n.d.), §2
(“The Classic Version of the Ontological Argument“). <https://www.iep.utm.edu/ont-arg/>.

[17] Patrick Grim, “Truth, Omniscience, and the Knower.” Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition Vol. 54, No. 1 (July 1988): 9-41.

[18] Patrick Grim, “Problems with Omniscience” in Debating Christian Theism (pp. 169-180), ed. J. P. Moreland, Chad Meister, and Khaldoun A. Sweis (Oxford, UK: Oxford University Press, 2013).

[19] Thomas Williams, “Saint Anselm” in The Stanford Encyclopedia of Philosophy (Spring 2016 edn.) ed. E. N. Zalta (Stanford, CA: Stanford University, 2016), §3.2 (“The Consistency of the Divine Attributes“). <https://plato.stanford.edu/archives/spr2016/entries/anselm/>.

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