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The Ontological Argument

Grant that God is possible and he exists. Grant that he is possibly absent and he doesn’t. The argument is valid both ways.

Defendants
Anselm of Canterbury, Proslogion II–III (1078); with René Descartes, Meditation V (1641), and Alvin Plantinga, The Nature of Necessity (1974), as successors
Claim
A being than which nothing greater can be conceived must exist, since if it existed only in the mind a greater one could be conceived: the same being, existing.
Charge
Putting existence into a definition and reading it back out. In the modal version, resting on a premise that is equivalent to the conclusion.
Verdict
Guilty of begging the question. Every version worth the name is valid. None gives a doubter a reason to believe, and the best modern defender conceded that a rational person may reject his key premise.

The argument, in their words

Anselm wrote the Proslogion as a prayer, and addressed the argument to the fool of Psalm 14 who “has said in his heart, there is no God”. Even the fool understands the phrase “a being than which nothing greater can be conceived”, so that being exists at least in his understanding. Then:

And assuredly that, than which nothing greater can be conceived, cannot exist in the understanding alone. For, suppose it exists in the understanding alone: then it can be conceived to exist in reality; which is greater. — Anselm, Proslogion II (trans. Deane)

Chapter III goes further. God cannot even be conceived not to exist:

For, it is possible to conceive of a being which cannot be conceived not to exist; and this is greater than one which can be conceived not to exist. — Anselm, Proslogion III

Descartes made existence a perfection and God the being with every perfection:

… it is not less impossible to conceive a God, that is, a being supremely perfect, to whom existence is awanting, or who is devoid of a certain perfection, than to conceive a mountain without a valley. — Descartes, Meditation V, para. 8 (trans. Veitch)

Plantinga opens his chapter by admitting that the argument “looks, at first sight, like a verbal sleight of hand or a piece of word magic” (p. 196). His version replaces conceivability with possible worlds. Maximal excellence includes omniscience, omnipotence and moral perfection. A being has maximal greatness if it has maximal excellence in every possible world. Then:

  1. The property has maximal greatness entails the property has maximal excellence in every possible world.
  2. Maximal excellence entails omniscience, omnipotence, and moral perfection.
  3. Maximal greatness is possibly exemplified.

Therefore: there exists a being that has maximal excellence in every world. (Plantinga, pp. 213–216.)

Premises (34) and (35) are definitions. All the weight is on (36), and on the principle that what is possible or necessary does not vary from world to world. That principle is the modal logic S5, and it licenses the step from “possibly necessary” to “necessary”.

Rebuke 1: the same proof delivers an island

The first reply came from a monk of Marmoutier named Gaunilo, writing “in behalf of the fool”. Picture the most excellent island, “which, because of the difficulty, or rather the impossibility, of discovering what does not exist, is called the lost island.” Run Anselm’s steps on it. An island that existed would be more excellent than one that didn’t, so the most excellent island exists.

If a man should try to prove to me by such reasoning that this island truly exists, and that its existence should no longer be doubted, either I should believe that he was jesting, or I know not which I ought to regard as the greater fool: myself, supposing that I should allow this proof; or him, if he should suppose that he had established with any certainty the existence of this island. — Gaunilo, In Behalf of the Fool, s. 6

A proof that also proves the lost island proves too much. Something in it is wrong, even if we cannot yet say what.

Anselm’s response

Now I promise confidently that if any man shall devise anything existing either in reality or in concept alone (except that than which a greater [cannot] be conceived) to which he can adapt the sequence of my reasoning, I will discover that thing, and will give him his lost island, not to be lost again. — Anselm, Reply to Gaunilo, ch. III

That is a promise rather than a reply. The step that does the work, “it can be conceived to exist in reality; which is greater”, says nothing about God. If it fails for islands, the reason must be something about islands, and Anselm does not say what.

The usual later answer is that islands have no maximum. You can always add another palm tree, so “the greatest conceivable island” describes nothing. I agree. But that answer concedes that the argument only works for a concept that is coherent, and whether “the greatest conceivable being” is coherent is the question. It returns in Rebuke 3.

Russell’s summary of where most readers end up:

… it is easier to feel convinced that it must be fallacious than it is to find out precisely where the fallacy lies. — Russell, A History of Western Philosophy (1946), ch. on Leibniz

Rebuke 2: existence adds nothing to a concept

The classical diagnosis is Kant’s. Saying that a thing exists does not describe it further. It says the description is satisfied.

Being is evidently not a real predicate, that is, a conception of something which is added to the conception of some other thing. — Kant, Critique of Pure Reason (trans. Meiklejohn)

“A hundred real dollars contain no more than a hundred possible dollars.” The real ones are better to have. They are not a different hundred dollars.

Descartes saw the objection coming. In the very next paragraph he writes that “my thought imposes no necessity on things; and as I may imagine a winged horse, though there be none such, so I could perhaps attribute existence to God, though no God existed.” His answer is that the cases differ: “because I cannot conceive God unless as existing, it follows that existence is inseparable from him, and therefore that he really exists” (paras. 9–10).

Kant’s triangle disposes of that answer. A triangle must have three angles. You may not keep the triangle and drop the angles, but “to suppose the non-existence of both triangle and angles is perfectly admissible.” Likewise you may not have a God who lacks existence, but you may have no God. The mountain needs its valley. The landscape needn’t have either.

Hume put the general point in the mouth of Cleanthes:

Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. — Hume, Dialogues Concerning Natural Religion, Part IX (1779)

Schopenhauer called the proof “really a charming joke”, and described the trick. Someone assembles a concept from predicates, “among which however he takes care to include the predicate actuality or existence, either openly stated or wrapped up for decency’s sake in some other predicate, such as perfectio, immensitas, or something of the kind.” He then extracts it again by logic.

Kant’s own closing line:

… we may as well hope to increase our stock of knowledge by the aid of mere ideas, as the merchant to augment his wealth by the addition of noughts to his cash account. — Kant, Critique of Pure Reason, “Of the Impossibility of an Ontological Proof”

Rebuke 3: in the modal version the premise is the conclusion

Plantinga’s version sidesteps Kant. It never treats existence as a perfection. It needs only possibility, necessity and S5. So the question is no longer whether a step is valid. Every step is. The question is whether premise (36) is any easier to accept than the conclusion. It is not. Given the definitions, the two are equivalent.

Write G for “a maximally great being exists”, ◇ for “possibly” and □ for “necessarily”. Maximal greatness is maximal excellence in every world, so the definitions give □(G → □G): wherever such a being exists, it exists everywhere. Here is the argument, and its mirror.

Argument A (Plantinga)
  1. ◇G  premise (36)
  2. □(G → □G)  definition of maximal greatness
  3. ◇□G  from 1 and 2
  4. □G  from 3, by the S5 principle ◇□p → □p
  5. G  from 4
Argument A′ (the mirror)
  1. ◇¬G  premise: possibly, no maximally great being
  2. □(G → □G)  same definition
  3. ◇G → □G  steps 1–4 of A, with 1 discharged
  4. ◇¬G → □¬G  contrapositive of 3
  5. □¬G, so ¬G  from 1 and 4

In plain terms: in S5 every world can see every other, so the definition says the being exists in all worlds or in none. “Possibly it exists” picks out one world, which means all of them, which means this one. “Possibly it doesn’t” picks out one world without it, which means none has it, including this one.

Both arguments are valid. They share premise 2 and differ only in premise 1. And since G implies ◇G trivially, the first argument shows that, given the definition, ◇G and G are the same claim. Whatever credence you give the premise, you must give the conclusion. An argument whose only contested premise is equivalent to its conclusion begs the question in the strict sense.

Plantinga says as much about a parallel case. Of the proposition that there could be possible but unactual objects, he writes: “This proposition resembles (36) in that if it is possible, it is true and indeed necessarily true. The same goes for its denial” (p. 220).

The premise sounds modest. Surely it is at least possible that there is a God? That is the “possible” of “for all I know”. The ◇ in S5 means metaphysically possible, and for this being, metaphysically possible means actual. The agnostic who says “maybe” has granted the first kind and not the second. The entry in the Stanford Encyclopedia names the open question: “under what conditions can one reasonably believe something of the form, ‘it is possible that F’ without merely deriving that conclusion from ‘it is actual that F’?”

Plantinga’s response

He anticipated the mirror. He calls it Argument D, built on “no-maximality”, the property of being such that there is no maximally great being. His answer:

Logic tells us that A and D cannot both be sound; but it also tells us they cannot both be unsound; one is sound and the other is not. — Plantinga (1974), p. 219

True, and it tells us which way to bet only if we already know whether God exists. Plantinga then concedes the point himself, which is to his great credit:

It must be conceded, however, that Argument A is not a successful piece of natural theology. — Plantinga (1974), p. 219
… a sane and rational man who thought it through and understood it might none the less reject it, remaining agnostic or even accepting instead the possibility of no-maximality. — Plantinga (1974), p. 220, of premise (36)
They cannot, perhaps, be said to prove or establish their conclusion. But since it is rational to accept their central premiss, they do show that it is rational to accept that conclusion. And perhaps that is all that can be expected of any such argument. — Plantinga (1974), p. 221

Given the equivalence, it is rational to accept the premise exactly when it is rational to accept the conclusion. The argument has carried rational acceptability from one to the other, a distance of zero. The same carriage is available to the atheist, who can accept ◇¬G on exactly the same terms. The current Stanford Encyclopedia entry, by Graham Oppy, Joshua Rasmussen and Joseph Schmid, puts it this way:

Absent such a symmetry breaker, however, the MOA is dialectically toothless—quite clearly, if you don’t already accept the claim that God exists, you won’t agree that (i) is more acceptable than (i*) absent some symmetry breaker. — Oppy, Rasmussen and Schmid, SEP (2024), s. 8

The same entry makes the point with arithmetic:

… it is not plausible that ‘God exists and 2+2=4 therefore God exists’ is a successful argument, even though all monotheists accept that it is (classically) sound. — SEP (2024), s. 13

Rebuke 4: Gödel moved the premise into the axioms

Gödel’s version, from a 1970 manuscript, replaces “possibly God” with axioms about “positive” properties, and derives the possibility from those. The dispute moves with it. A property that is necessarily uninstantiated cannot be positive under those axioms, so anyone who thinks omnipotence, say, is impossible must deny that it is positive. The SEP entry again:

Unless we have an independent, non-question-begging way to discern which properties are positive, this looks like game over for Gödelian ontological arguments. — SEP (2024), s. 9

The argument has since been checked by machine. Christoph Benzmüller and Bruno Woltzenlogel Paleo ran it through higher-order theorem provers. Dana Scott’s 1972 variant of the axioms is consistent and the conclusion follows. Gödel’s own manuscript axioms turned out to be inconsistent, so they prove that God exists and also that he doesn’t. Scott confirmed to the authors that he had not noticed.

The provers also confirmed the modal collapse that J. H. Sobel found in 1987: the axioms imply that every truth is a necessary truth. Nothing could have been otherwise.

Scope of the rebukes

None of this shows that God does not exist. The mirror argument is exactly as bad as the original, for the same reason. The rebukes are to a method: getting a conclusion about what exists from a definition plus a premise that is the conclusion restated.

Kant’s objection is contested, and it does not obviously touch the modal version, which is why Rebuke 3 carries most of the weight here. Whether S5 is the right logic for metaphysical possibility is also contested. I have granted it, since the argument needs it and the mirror uses it too.

Plantinga’s narrow claim, that it can be rational to accept premise (36), is not refuted here. It is a claim about permission, and the permission is symmetric. The SEP authors allow that “it may well be that theists have—or can have—good reasons for thinking that there are sound ontological arguments.” What no version supplies is a reason for anyone else. Arguments from design, cause or experience are separate and are not assessed here.

In their favour

The arguments are valid. That is more than most of the defendants on this site can claim, and it is why the ontological argument has occupied logicians for nine centuries. Russell, who rejected it, wrote: “Clearly an argument with such a distinguished history is to be treated with respect, whether valid or not.” Gödel thought it worth formalising.

Anselm answered his critic point by point. Descartes stated the obvious objection to his own proof himself, in the next paragraph. And Plantinga said plainly what his version does and does not show, in the same chapter as the argument. Few philosophers concede in print that their best argument for a conclusion they care about is not a proof.

The argument also did philosophy a service. Explaining why it fails forced a clear account of what “exists” means, from Kant to Frege and Russell, and the modal version showed exactly what S5 commits you to.

Disclosure: this page was researched and drafted by a language model, which can conceive of a great many things that do not exist. Check the quotations.

References