A Presentation of the Ontological Argument

Over recent months I have given special attention to the ontological argument for the existence of God [1], drawing upon a framework first purposed by St. Anselm of Canterbury (1033-1109). In this post, I hope to deal with Anselm’s argument in an explanatory manner as well as deal with in some length on the nature of modal logic and its application to this argument. Robert Maydole in his essay regarding the ontological argument in The Blackwell Companion to Natural Theology [2] has been an interesting champion as well influence in terms of my consideration of this argument. Since, I tend to agree with Plantinga at some points when he suggests that the argument seems to “smack too much of magic” (Plantinga 1967, 26) [3].

However, my lack of respect (or reverence maybe) for the argument came from Aquinas’s rejection of the argument as well as Kant’s criticism. Gordon Clark provides an interesting summary of this point:

Thomas Aquinas and Bishop Berkeley both believed in God, but neither of them believed that this argument proved the existence of God. The seventeenth-century rationalists thought that the argument was sound, and a few Protestant theologians have treated it with respect. Kant analyzed it with extreme care and decided it was a fallacy. But Hegel, though not sharing Anselm’s faith in the Atonement and the other Christian doctrines, had a system that is nothing but a transformed and expanded ontological argument. (Clark 1957, 257)

Interestingly, Aquinas rejected the argument on the grounds that we can not have direct access to the essence of a thing. Jan A. Aertsen [4] writes in explaining Aquinas’ view as that though “the senses are not the total cause of all of our knowledge, [ … ] they do provide the indispensable material from which the intellect abstracts the intelligible content” (Aertsen 1997, 32). So then, it follows from this that “human beings cannot know the essence of a substance that is not perceptible by the senses” (Ibid., 32). Our knowledge of God is only based by His effects in accordance to creation.

However, Kant in particular will be examined in the following sections to come, with his own respectful analysis. Since my opinions on those critiques have been changed, I do owe my respect to particular works of literature such as Gordon Clark’s Three Types of Religious Philosophy, Alvin Plantinga’s God and Other Minds, and [of course] Robert Maydole’s paper, The Ontological Argument, as discussed above.

To skip any more introduction on the matter, let’s first examine the central figure associated with the Ontological Argument: St. Anselm of Canterbury.

St. Anselm of Canterbury

Anselm was born in 1033 near Aosta, in those days a Burgundian town on the frontier with Lombardy. Though little is known about his early life, he later became a Benedictine monk who taught theology and philosophy just starting in his 20’s. In 1093, he finally became Archbishop of Canterbury. In his Monologion (1077), Anselm offered a number of arguments such as moral and cosmological arguments for theism, along with many others. Most famously known for his Ontological Argument in chapter 2 of his Proslogion (1077-78), St. Anselm has also made numerous of other contributions to the philosophy of theology (and philosophy in general as well).

His major works include some of my favorite: Cur Deus Homo (1095–98), De conceptu virginali (1099), De processione Spiritus Sancti (1102) and many more. His main contributions to the philosophy of theology include his work on the divine attributes, sin, freedom, and theistic proofs.

However, as for our focus in particular, William Lane Craig writes in regards to Anselm’s formulation of particular arguments for the existence of God and his dissatisfaction with them. According to his essay on the Ontological Argument, “Anselm remained dissatisfied with the complexity of his demonstration and yearned to find a single argument which would on its own prove that God exists in all his greatness” [5]. This would later set up what Anselm would call [according to his conception of God] as “the greatest conceivable being” (aliquid quo nihil maius cogitari possit). Anselm writes:

Hence, even the fool is convinced that something exists in the understanding, at least, than which nothing greater can be conceived. For, when he hears of this, he understands it. And whatever is understood, exists in the understanding. And assuredly that, than which nothing greater can be conceived, cannot exist in 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.

Therefore, if that, than which nothing greater can be conceived, exists in the understanding alone, the very being, than which nothing greater can be conceived, is one, than which a greater can be conceived. But obviously this is impossible. Hence, there is no doubt that there exists a being than which nothing greater can be conceived, and it exists both in the understanding and in reality. (Deane 1962, p. 8)

According to Anselm, the greatest conceivable being is one who exists in the understanding alone (assumption). Noticeably along with other Middle Age philosophers, properties regarding existence-in-reality, power, goodness, completeness and so forth were all apart of a great chain of being, where “God is an upper bound to the great chain of being” (Maydole 2012, 554). Anselm writes in his Monologion:

Furthermore, if one considers the nature of things, one cannot help realizing that they are not all of equal value, but differ by degrees. For the nature of a horse is better than that of a tree, and that of a human more excellent than that of a horse [ … ] It is undeniable that some natures can be better than others. None the less reason argues that there is some nature that so overtops the others that it is inferior to none (Charlesworth 1998, p. 14).

Thus, saying that a being existing in reality is greater than the greatest conceivable being existing in the understanding emerges as a contradiction. Since, the greatest conceivable being can have no “ontological superiors”; especially, according to Anselm, as existence in reality is greater than existence in the understanding alone. Therefore, this being must have existence in the understanding as well as in reality.

In the following sections to come, I hope to analyze exactly why I believe the argument is sound, demonstrating more so that I believe the argument is far more powerful once seen in its logical form. I do hope to also try my best in expositing some matters regarding possibilities, necessities, with another brief section regarding modal logic – since its application to the ontological argument I believe has increased its fire power.

Schematization of Anselm’s Argument

Anselm’s argument can be structurally stated as such [6]:

  • (1) God exists in the understanding but not in reality
  • (2) Existence in reality is greater than existence in the understanding alone.
  • (3) A being having all of God’s properties plus existence in reality can be conceived.
  • (4) A being having all of God’s properties plus existence in reality is greater than God – from (1) and (2)
  • (5) A being greater than God can be conceived (3), (4).
  • (6) It is false that a being greater than God can be conceived – by definition of “God”
  • (7) Hence it is false that God exists in the understanding but not in reality – (1)-(6), reductio ad absurdum.

A few respective comments are in order to mount an explanation. In regards to conceivable, I simply mean to say that “there is no logical impossibility in the supposition that it obtains” [7]. In other words, there is nothing logically hindering something from being the case – indeed, it is conceivable. Furthermore:

…to say specifically that a being having all of God’s properties plus existence in reality is conceivable, is simply to say that it is possible that there is a being having all of God’s properties plus existence in reality – that is, it possible that God exists [8].

However, Maydole gives special attention to this matter. In respect to the validity of Anselm’s ontological argument (1a.),

it is inconceivable that one and the same thing could have both existence-in-reality and existence-in-the-understanding. Things that have existence-in-reality are very different from kinds of things that have existence-in-the-understanding. A table, for example, is different from the concept or idea of a table. Likewise, it is impossible to think of God, even qua pure spirit, as having existence-in-the-understanding, even if God fails to have existence-in-reality or even if “the fool is convinced that something exists in the understanding, at least, than which nothing greater can be conceived.” Either Anselm has been mistranslated or he misspoke and should have said that even the fool has existence-in-the-understanding; and instead of saying “whatever is understood, has existence-in-the-understanding” and so on. [9]

Of course, Anselm’s (what possibly might be construed as) Neo-Platonism could have lead him to say that “things in existence-in-reality and things that have existence-in-the-understanding might share many or most of the same properties” [10]. However, another note regarding what Anselm means by existence in reality requires further understanding (no pun intended).

It is not to be inseparably understood that existence in reality is necessarily the same thing as existence in general. For Anselm, “mental things that have existence-in-the-understanding exist just as much as nonmental things that have existence-in-reality, but in a different way” [11].

Anselm’s argument is thence suggesting that (2) existence in reality is greater than existence in the understanding alone. This isn’t simply a matter of intuition that the adherent of the argument hopes his interlocutor agrees with, but connotes Anselm’s idea regarding the chain of being that we saw back in his statement in the Monologion. Thus, an acceptance of (2)  leads us to a consideration of premises (3)-(6), where, taken along with (1) and (2), collectively make the contradiction that a being existing in reality is greater than the greatest conceivable being, which exists in the understanding.

In the next section, I would like to examine some criticism’s of the argument that has come particularly from the pen of German philosopher, Immanuel Kant, in his Critique of Pure Reason.

Kant’s Critique of Anselm’s Argument

In his Critique of Pure Reason (A598/B626), Kant argues that stipulating existence as a predicate to its subject does not give the subject actual existence. For instance, supposing the statement “all triangles have three sides” is true (which it is), a contradiction emerges once I deny the predicate in respect to its subject. In other words, I cannot be consistent in suggesting that “triangles do not have three sides”, which would then no longer constitute a triangle.

However, a denial of the subject (triangle) in respect to its predicate is not a contradiction. It is in the same way with God – you cannot stipulate the predicate of existence to the subject “God” and then suppose that it exists in reality. This is a simple understanding in regards to the concept of things. Namely, that things may or may not exist or things must or must not exist. Respectively, contradictory things cannot exist in reality (or any other for that matter).

Gordon Clark in his analysis of the ontological argument writes that “the idea of God is a heuristic principle, a principle by which we conduct investigations, particularly investigations into morality. But this is far from establishing the objective reality of a corresponding entity” [12]. Heuristic simply meaning that some given ‘principle’ is able to be learned or discovered  by a person. Kant’s view however regarding the idea of God is that it being a heuristic or methodological principle “does not follow validly from the thesis that necessary concepts may correspond to no objective reality” (Clark, 36). Kant addresses the issue regarding the subject-predicate aspect of Anselm’s argument:

If in an identical judgement, I annihilate the predicate in thought, and retain the subject, a contradiction is the result; and hence I say, the former belongs necessarily to the latter. But if I suppress both subject and predicate in thought, no contradiction arises; for there is nothing at all, and therefore no means of forming a contradiction. To suppose the existence of a triangle and not that of its three angles, is self-contradictory; but to suppose the non-existence of both triangle and angles is perfectly admissible. And so is it with the conception of an absolutely necessary being. Annihilate its existence in thought, and you annihilate the thing itself with all its predicates; how then can there be any room for contradiction? [13]

Thus, Kant’s example regarding the triangle is that a denial of the subject annihilates the three angles while the main matter is pertaining to a denial of God the subject. If we are saying that God has existence just as a triangle has three angles, and we are denying subjects, does existence vanish along with God? Not exactly. Clearly stated, Kant is saying that an “annihilation” of God’s existence is an annihilation of God himself. However, is it possible to “annihilate to God in thought?” Clark interestingly writes:

What does it mean to annihilate something in thought? Does it mean merely not to think about it? Refusing to think surely is a prescription for avoiding self-contradiction, but what bearing such a refusal has on God, or even on angles, is hard to say. Maybe then annihilation means denial of existence. If I positively deny that there are any triangles, it is not absurd to deny triangularity. If now I positively deny that God exists, it is not absurd to deny his existence. But this doesn’t fit the triangle example. With respect to triangles Kant has insisted that the denial of the subject eliminates the predicate. But when Kant turns to the main matter and demands, Deny God exists; he asks us to deny, not the subject, as in the triangle example, but the predicate. Thus this argument does hang together. [14]

However, Clark goes on to say that an annihilation of God in thought is impossible because according to Augustine, or at least as he noticed, truth is not only possible, but “inevitable and inescapable.” Thus, if we think at all, we think the laws of logic [15]. Truth is inherent or innate in the mind. But, God himself is the ontological foundation for truth because He is the truth. Therefore, it is impossible to annihilate God in thought since he “exists so truly that he cannot even be conceived not to exist” (Clark, 39). Thus, Clark suggests that Kant has failed in three ways:

  • (1) Kant has not substantiated his view of what it means to “annihilate” something in thought.
  • (2) Kant did not disprove the existence of God (nor Anselm’s argument in particular).
  • (3) Kant has not exposed an adequate critique of Anselm’s argument.

In respect to (3), an interesting shed of light can be given to this failure of Kant’s. The simple response seen from philosophers like Clark, Plantinga, Lowe, and others, is namely that Kant’s criticism is essentially a red-herring to Anselm’s argument.  To quote E.J. Lowe:

Nothing in… the ontological argument implies that… existence… must be a divine attribute or property, in the way that omniscience or omnipotence are… [T]he Kantian objection… is just a red herring with no real bearing on the soundness of the ontological argument. (Lowe 2007, 337)

Plantinga as well even proceeds to rightly assert an observation that is crucial to analyzing the criticism of Kant:

Unfortunately, it seems to have no particular bearing on Anselm’s argument. For Anselm can certainly agree, so far as his argument is concerned, that existence is not a real predicate in the explained sense. Anselm maintains that the concept the being than which none greater can be conceived is necessarily exemplified; that this is so is in no way inconsistent with the suggestion that the whole concept of a thing diminished with respect to existence is equivalent to the undiminished whole concept of that thing. Anselm argues that the proposition God exists is necessarily true; but neither this claim nor his argument for it entails or presupposes that existence is a predicate in the sense just explained (Plantinga 1967, 36).

To give one last stressing of this point, Maydole also writes in respect to Kant:

He is right that existence is not a property in the usual sense of being includable in the concept of a thing. His explanation is that existence is not a property at all. A better explanation would be that we beg the question of the thing’s very existence if we include existence in its concept or essence. True, we do not add to the concept of a thing when we say that it exists. So existential propositions are indeed synthetic. But, contrary to Kant, I think that we do predicate something new of a thing when we say that it exists (Maydole 2012, 570 – emphasis mine).

Other philosophers have taken different approaches in respect to critiquing Anselm’s ontological proof. A.J. Ayer in his Language, Truth, and Logic writes that “existence is not an attribute. For, when we ascribe an attribute to a thing, we covertly assert that it exists: so that if existence were itself an attribute, it would follow that all positive existential propositions were tautologies, and all negative existential propositions self-contradictory; and this is not the case” (Ayer 1952, 43). This will have a brief examination in the following section.

Ayer’s Critique of the Ontological Proof 

To quote Ayer in full force:

A similar mistake has been made in connection with such propositions as “Unicorns are fictitious.” Here again the fact that there is a superficial grammatical resemblance between the English sentences “Dogs are faithful” and “Unicorns are fictitious,” and between the corresponding sentences in other languages, creates the assumption that they are of the same logical type. Dogs must exist in order to have the property of being faithful, and so it is held that unless unicorns in some way existed they could not have the property of being fictitious.

But, as it is plainly self-contradictory to say that fictitious objects exist, the device is adopted of saying that they are real in some non-empirical sense – that they have a mode of real being which is different from the mode of being of existent things. But since there is no way of testing whether an object is real in this sense, as there is for testing whether it is real in the ordinary sense, the assertion that fictitious objects have a special non-empirical mode of real being is devoid of all literal significance.

It comes to be made as a result of the assumption that being fictitious is an attribute. And this is a fallacy of the same order as the fallacy of supposing that existence is an attribute, and it can be exposed in the same way (Ayer 1952, 43).

First, it is important to note what I mean by a negative existential. In short, a negative existential claim is “one denying the existence of some thing: e.g., ‘Pegasus does not exist’ and ‘There are no round squares'” [16]. In other words, in respect to Ayer’s critique, provided we were to have some existential proposition such as “dogs are faithful””or “unicorns are fictitious” we would be positing an affirmative tautology and having their negations function as self-contradictory, which Ayer takes as “following grammar beyond the boundaries of sense” (43). Thus, statements such as “God exists” are thence tautologous, and negative existential claims are self-contradictory.

However, the problem with Ayer’s analysis (despite its irrelevancy like that of Kant’s) of these kind of existential propositions is that they ignore the consequence of the truth or falsehoods of them. For, according to Ayer, negative existential propositions if true express nothing particularly meaningful. It is why Ayer for example says that “unless unicorns in some way existed they could not have the property of being fictitious.”

Of course, considering if the proposition “unicorns do not exist” were true, then no subject-predicate propositions of any sort are about unicorns. As Plantinga similarly notices in regards to negative existentials of this kind according to Ayer’s critique, “this is plainly outrageous” (Plantinga 1967, 39). Why couldn’t one talk or think about beings that do not exist? Maydole even interestingly comments that “the burden of proof of a claim that a word or phrase is meaningless must always fall on the challenger, especially when the word appears to be used with understanding by a great many people (Maydole, 557).

C.D. Broad in his Religion, Philosophy and Physical Research states that “[t]he Ontological Argument presupposes that existence is a quality or power” (1953, see 182-183). Surely, though his critique is in line with that of A.J. Ayer and John Wisdom [17], the essential red-herring associated with this line of reasoning is to suggest that “existential propositions are of logically the same form of characterizing propositions” (Broad, 182). For more on this subject, see Plantinga 1967, 38- 43.

An Overview of Modal Logic

The vocabulary we are considering in respect to constructing a model of modality are the pair of interdefinable operators (□, ◊). Since, “from a syntactic point of view, both function exactly as the sentential negation operator (~) does” [18]. We can construct very well-formed formulas such as the ones in the following:

  • □∃x[Fx] ~ ◊∀x[Fx v Gx]
  • ∃x[□Fx]
  • ◊□∃x[Fx] – □∃x[Fx]]

With □ meaning “necessarily” and ◊ meaning “possibly”, these formulas (from top to bottom) can be understood as: (1) It is necessary that there is something that is F; it is not possible that everything is (F or G); (2) There is something that is necessarily F, (3) and if it is possible that it is necessary that there is something that is F, then it is necessary that there is something that is F. However, what do we mean by possible, and necessary?

In short, a logical necessity “is a truth that follows directly from logic, and whose denial leads to a logical inconsistency.” In other words, some given law like that of p ⊃ p is necessarily true where the truth-hood of what obtains (given the respective of whatever p is) is the case in every possible world. Possible worlds, to note, is not some literal planet or universe but rather a “conjunction which comprises every proposition or its contradictory, so that it yields a maximal description of reality – nothing is left out of such a description” [19]. Thus, by negative particular conjuncts in a maximal description we come to different possible worlds (“&” meaing “and”, and “~” meaning “it is not the case that”):

  • W1: p & q & r & s . . .
  • W2: p & ~q & r & ~s . . .
  • W3: ~p & ~q & r & s . . .
  • W4: p & q & ~r & s . . .

According to Craig’s analysis, “[o]nly one of these descriptions will be composed of conjuncts all of which are true and so will be the way reality actually is, that is to say, the actual world” (Craig 2004, 126). Thus, simply taking the proposition “God exists”, we are suggesting that the proposition is comprised “by some maximal description of reality” (Craig, 126).

However, now referring to the syntax for modal logic, the formal system with special operators (or symbols) identified above – ◊ and □. Within the formal language that “includes them, these symbols are ‘meaningless’. But then we give them meaning by constructing reference points. This part is the semantics” [20]. To quote Poythress in full:

The semantics consists in models for modal logic, that is, models with specific correspondences to the elements in the formal language of modal logic [ …] In a model for ordinary classical logic, we have to correlate between propositional symbols p and true or false values in the model. In addition, if we include quantification theory, the model has to have individuals belonging to some set U. We have to use propositional logic when using modal logic, because modal logic is intended to be an extension of propositional logic. [21]

In light of Poythress’s analysis of the relationship between modal and propositional logic, we can finally begin to scheme some particular models. However, as noted above, we have to use the formal symbols □ and ◊ in some given manner within the modal so that it correlates with something. In accordance to model theory, modal logic starts with the models used for propositional logic (see Poythress, 457). Let us see some examples.

Sticking to the thought of Poythress, W includes a set S = {T, F} for the truth values that will be assigned to propositions. “It also includes a function f that establishes a correlation between each propositional symbol in the formal language on the one hand, and one of the two elements T and F on the other hand” (Poythress, 508). So, for example, if some given formal language includes the symbols P(1), P(2), P(3), etc., you may have the function f assign a truth value to each of these symbols:

  • f(P(1)) = T,
  • f (P2)) = F, and
  • f(P3)) = F

These assignments thence allows us to calculate the truth value of compound propositions (such as those seen in Poythress, 508). Respectively, though I don’t wish to analyze the matters regarding necessity and quantification, variations and enriched logics, I think we can move onto Anselm’s argument schemed according to its modal interpretation. Which, to use as a lexicon, we can say,

Ux = x is understood
Sy = the concept of y exists-in-the-understanding
Ex = x exists-in-reality
Gxy = x is greater than y
Fxy = x refers to y
Dx = x is a definite description
d = the definite description
P(Y) = Y is a great making property
c = it is conceivable that… [22]

A Final Scheme of Anselm’s Argument

Maydole thus goes on to structure the argument as the following [23]:

  • (A1) The definite description “that than which it is not not conceivable for something to be greater” is understood.
  • (A2) “That than which it is not conceivable for something to be greater” refers to that than which it is not conceivable for something to be greater.
  • (A3) The concept of whatever a definite description that is understood refers to has existence-in-the-understanding.
  • (A4) It is conceivable that something is greater than anything that lacks a great-making property that it conceivably has.
  • (A5) Existence-in-reality is a great making property.
  • (A6) Anything the concept of which has existence-in-the-understanding conceivably has existence-in-reality.
  • (A7) It is not conceivable that something is greater than that than which it is not conceivable for something to be greater.


  • (A8) That than which it is not conceivable for something to be greater exists-in-reality.

In respect to its modal form, we can also translate the argument to its similar deductive form [23]:

  • (Dd & Ud)
  • Fdg
  • (x)(y)((Dx & Fxy & Ux) ⊃ Sy)
  • (x1) (Y)[(P(Y) & ~Yx1 & c Yx1) ⊃ c (∃x2) Gx2x1]
  • P(E)
  • (x) Sx ⊃ c Ex)
  • ~c(∃y) Gyg


  • Eg

Concluding Thoughts

In respect to Anselm’s argument, I think it is safe to say that criticisms lashed out against the argument have failed in what appear to be at points “preliminary ways”. To reflect, (1) Kant’s response to Anselm fails in respect to his inability to complete what he means by an “annihilation of thought” as well as its irrelevancy. The same thing goes with (2) Ayer and Broad’s critique that namely existence is not an attribute – which is itself, not a prospect found throughout Anselm’s argument. Lastly, recent schemes over the last half century in respect to quantified forms of modal logic and so forth applied to the argument have done very well in the succession of the OA.

A serious amount of consideration I believe is owed to this taken for granted piece of natural theology.



  • [1] You can see my paper The Ontological Argument and The Irrelevance of Kant (May 2013) here.
  • [2] The Blackwell Companion to Natural Theology, ed. W.L. Craig and J.P. Moreland (Wiley-Blackwell: 2012)
  • [3] Maydole makes this interesting comment in respect to Ontological Arguments: “Ontological Arguments are frequently the target of parodies, more so than any other argument in philosophy.” (Maydole 2012, 553)
  • [4] From Aquinas’ Philosophy in it’s Historical Setting, in The Cambridge Companion to Aquinas, ed.  Norman Kretzmann and Eleonore Stump (Cambridge: 1997) p. 32
  • [5] from The Ontological Argument in To Everyone An Answer, ed. W. L. Craig, F. Beckwith, and J.P. Moreland (IVP Press: 2004) p. 124
  • [6] Argument scheme has been taken from Alvin Plantinga, God and Other Minds (Cornell University Press: 1967) p. 29
  • [7] Plantinga, 29
  • [8] Plantinga, 29
  • [9] Maydole, 555
  • [10] Maydole, 555
  • [11] Maydole, 555
  • [12] Gordon Clark, Three Types of Religious Philosophy (Trinity Foundation: 1973) p. 36
  • [13] From The Impossibility of an Ontological Proof, in Critique of Pure Reason (p. B. 629)
  • [14] Clark, 39
  • [15] I wouldn’t like to lead my reader to think that I am going off my own argument in this thought. Clark earlier in his analysis of  the rationalists stated that Descartes’ “first truth is not really ‘I think,’ as he said. His first truth is really the laws of logic. ‘I think’ must be true because a person who denies it exemplifies it in his denial and so has contradicted himself. ‘I think’ depends on the laws of logic” (Clark, 30).
  • [16] Quine Notes, Winthrop University (see PDF).
  • [17] see John Wisdom, Interpretation and Analysis (London: 1931) p. 62
  • [18] John Divers in Alvin Plantinga, (Contemporary Philosophy in Focus: 2007) p. 72
  • [19] W.L. Craig, The Ontological Argument in To Everyone Answer, ed.  W. L. Craig, F. Beckwith, and J.P. Moreland  (IVP Press: 2004) p. 126
  • [20]Vern Poythress, Logic: A God-Centered Approach to the Foundation of Western Thought (Crossway: 2013) p. 507
  • [21] Poythress, pp. 507-8
  • [22] The conceivability operator according to Maydole, “need not be made explicit for this argument, since the deduction shows that the argument is valid in nonmodal first-order quantification theory” (Maydole, 556)
  • [23] See Maydole 2012, 556-7 for the deduction that validates this argument.

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