Disclaimer: I almost completely renounce this post and the methodology I use. Since writing it I've come to think most of the philosophical framework used in here is dubious, especially as it appears the van Inwagen's "Fregean" argument. I've decided to keep the post up anyway though as it does express plausible reservations if you don't share my newfound metaphilosophical gripes.

The Failure of the Ontological Argument

Introduction

Anselm’s famous ontological argument (often called the ontological argument) and its modernizations are a popular subject of debate between philosophers, apologists, and online communities. Compared to other common arguments for the existence of God, Anselm’s argument has a reputation for being conceptually difficult. So, the first task of this essay is to translate Anselm’s argument from his Proslogion into modern terms so that it can be thoroughly understood. Anselm’s argument is also notorious for being “sketchy,”  in that for many, it is uncompelling, but it is not immediately apparent where it goes wrong. Thus the second task of this essay is to explain some of its faults. I propose two problems that use the tools of modern logic and metaphysics to show that the argument is unsuccessful against the modern non-theist.

1. What is Anselm’s Argument

Ontological arguments are an attempt to deductively prove the existence of God from the conceptual contents of God alone. This means the arguments are supposed to show that when we fully extract every property and idea folded into the concept of God, we have no choice but to conclude that God exists. Anselm’s argument is credited as being the first of this kind, and its most popular formulation is the one that comprises chapter 2 of his Proslogion. He writes

Well then, Lord, You who give understanding to faith, grant me that I may understand, as much as You see fit, that You exist as we believe You to exist, and that You are what we believe You to be. Now we believe that You are something than which nothing greater can be thought. Or can it be that a thing of such a nature does not exist, since “the Fool has said in his heart, there is no God?” [Psalms 14, l. 1, and 53, l. 1.] But surely, when this same Fool hears what I am talking about, namely, “something-than-which-nothing-greater-can-be-thought”, he understands what he hears, and what he understands is in his mind [intellect, understanding], even if he does not understand that it actually exists. For it is one thing for an object to exist in the mind, and another thing to understand that an object actually exists. Thus, when a painter plans beforehand what he is going to execute, he has [it] in his mind, but does not yet think that it actually exists because he has not yet executed it. However, when he has actually painted it, then he both has it in his mind and understands that it exists because he has now made it. Even the Fool, then, is forced to agree that something-than-which-nothing-greater-can-be-thought exists in the mind, since he understands this when he hears it, and whatever is understood is in the mind. And surely that-than-which-a-greater-cannot-be-thought cannot exist in the mind alone. For if it exists solely in the mind even, it can be thought to exist in reality also, which is greater. If then that-than-which-a-greater-cannot-be-thought exists in the mind alone, this same that-than-which-a-greater-cannot-be-thought is that-than-which-a-greater-can-be thought. But this is obviously impossible. Therefore there is absolutely no doubt that something-than-which-a-greater-cannot be-thought exists both in the mind and in reality. (Anselm 1965, chapter 2, translation by M. J. Charlesworth.)

A potential problem that both Oppy and Sobel notice is that there is an ambiguity between existential and universal generalizations in the argument, such that a fair reading of it may leave it without existential import. Drawing primarily from Sobel 2003, I will interpret the argument charitably to guarantee that it validly concludes that a being-than-which-a-greater-cannot-be-thought (hereafter an MGB) really exists. To do this I will separate the argument into three sub-arguments, the first showing that a MGB exists in the mind, the second showing that a MGB cannot exist in the mind alone, and the third concluding that a MGB exists in reality.

1.1 First sub-argument. 

I take it that the function of the fool in Anselm’s argument is as a rhetorical tool to argue that even for those who deny the existence of a MGB this proof should be compelling. Thus the argument’s first premise makes use of the fool alone to show that a MGB exists in the understanding. The following excerpt is the first sub-argument in the passage:

 Even the Fool, then, is forced to agree that something-than-which-nothing-greater-can-be-thought exists in the mind, since he understands this when he hears it, and whatever is understood is in the mind.

I have regimented this segment as such:

(1) A MGB is understood.

(2) Whatever is understood exists in the mind. 

(3) A MGB exists in the mind. (Modus ponens, from 1 & 2)

(1) introduces “is understood” as a predicate of predicates. I have not used the modal language that is often used in interpretations of Anselm’s argument because I believe Oppy 2006 is right to say this is plausibly stronger than what Anselm had in mind. It is ambiguous what exactly it means for something to be understood, but I will return to this in section 3.

Anselm several times in the passage separates existence between existence in the mind and in reality, which I interpret as an implicit espousal of what is now called a thick ontology or metaontological pluralism–the view that there are multiple modes of being. This will be the focus of section 2. If (2) is right, then we can infer that a MGB does exist in the mind.

1.2 Second sub-argument.

    The next sub-argument from the excerpt is a reductio:

And surely that-than-which-a-greater-cannot-be-thought cannot exist in the mind alone. For if it exists solely in the mind even, it can be thought to exist in reality also, which is greater. If then that-than-which-a-greater-cannot-be-thought exists in the mind alone, this same that-than-which-a-greater-cannot-be-thought is that-than-which-a-greater-can-be thought. But this is obviously impossible.

    The excerpt suggests an argument like this:

(4) (Assume for reductio) a is a MGB that exists in the mind but not in reality.

(5) Anything which exists in the mind alone would be greater were it to exist in the mind and in reality.

(6) a would be greater were it to exist in the mind and in reality. (Modus ponens, 4 & 5)

(7) a can be thought to be greater. (6)

(8) a cannot be thought to be greater. (4)

I will grant (5) in this post because the objections I intend to bring to light are not challenges to Anselm’s axiological suppositions. (7) is not derived from (6) through any rule of inference, but I think Anselm is correct to suppose that if we are writing out a way in which something could have been better, it can be thought to be greater. (8) follows from (4) because a MGB is one than-which-a-greater-cannot-be-thought and a is a MGB. Because (7) and (8) are inconsistent conclusions, and (6) is a derivation, the only candidate for a false premise is (4) (of course, excluding (5)). To show that contradictions flow downstream from (4) is to show that it cannot be that a MGB exists in the mind but not in reality. 

1.3 Third sub-argument 

This gets us to the universal generalization that

(9) If a MGB exists in the mind, it also exists in reality.

But of course, from the first sub-argument, it is known that an MGB does exist in the mind. Therefore Anselm derives that (10) A MGB exists in the mind and reality. (Modus ponens, from 3 & 9) (11) A MGB exists in reality. (10) 

2. Through Thick and Thin

The above formulation of Anselm’s argument relies on a significant metaontological supposition: metaontological pluralism. This manifests itself in Anselm’s argument by the separation of existence into existence in the mind and existence in reality. Some other examples of metaontological pluralism are some Meinongian views that distinguish between being and existence, Aristotelian views that distinguish between potential and actual existence, and Heidegger’s ontology which distinguishes between a lot of German stuff that I don’t understand. 

The thick ontology of Anselm’s argument is quite different from Meinongianism, however, in that the thick ontology required for Anselm’s argument does not need to divorce being from existence. The argument is perfectly compatible with Peter van Inwagen’s Quinean thesis that “Being is the same as existence” (van Inwagen 2009, p. 480), whereas this thesis is intended to rule out Meinongian views. I will suppose for this section that Anselm’s metaontology includes the claim that there is no distinction between being and existence. The thesis that the proponent of Anselm’s argument must contest, however, is that being is univocal. Hence, in the remainder of this section I will explore two possible problems with the equivocal metaontology of Anselm’s argument and how these problems interact with the argument’s logical structure.

2.1 Merrick’s Dilemma

The first problem is inspired by Trenton Merricks’ arguments against metaontological pluralism, the view that there are multiple modes of being. It presents itself when translating (3) into symbolic logic. As a refresher, the premise is

      (3) A MGB exists in the mind.

A formulation of the premise where “exists in the mind” is modeled as a first-order predicate looks like this

      (3’)  (∃x) MGBx & Mx

My concern here is that (3’) implies a somewhat idiosyncratic type of existential pluralism, one where the mode of being “existence in the mind” is a modeled by a predicate (as is “existence in reality”), but there is a different ontological concept expressed by the existential quantifier. The existential quantifier is entirely unrestricted, implying a kind of generic existence or being that all the xs that exist in the mind and all the ys that exist in reality share.

I will avoid stepping into the debate of whether existence is a property, and instead talk about how Merricks approaches this view.Merricks argues that a pluralism that maintains multiple modes of being in addition to generic existence like that modeled by unrestricted quantification undermines the historical support for pluralism (Merricks 2019). This is because many of the motivations for pluralism begin with the observation that some things that are, are not in the same ways in which other things are. If, despite this, there is still a generic type of being in which everything participates, then everything that is, is in the same way everything else is. Merricks shows that, “This implication is clearly in tension with the sorts of views that virtually all pluralists have tried to articulate and defend” (Merricks 2019, p. 19).

Additionally, Merricks builds on van Inwagen 2014 to argue that the introduction of multiple modes of being is needlessly profligate if it is understood that all things share in a generic type of being. If both tables and relations generically exist in the same way, it is unnecessary to introduce multiple modes of being to distinguish between, say, abstract and concrete existence because the difference between abstracta and concreta can more comfortably be couched in the differences between their properties rather than in their modes of being. Van Inwagen complains that Russell ought to “Stop trying to do something more when there’s nothing more to be done: stop trying to express the vastness of the difference between relations and tables by saying that they have different kinds of being” (van Inwagen 2014, p. 23). I expand this argument further in section 2.2.

If we find these worries compelling, then the argument is a non-starter as we would not think anything satisfies either of the existential predicates used in this formulation of Anselm’s argument.

But we can instead model Anselm’s modes of being in terms of quantification. I would then translate (3) as:

      (3’’)  (∃Mx) MGBx

where ∃m quantifies over only those things that exist in the mind. So far so good. What about deriving (3’’) though? How should we formalize

      (2)  Whatever is understood exists in the mind

My first intuition is to read (2) like this

      (2’)  (∀φ) (Uφ ⇒ (∃Mx) φx)

    (2’) uses the standard universal quantifier. This is a problem because it implies that the argument, at some step, still requires the use of unrestricted quantification over a domain of which everything, whether existent in the mind or in reality, is a part. Either this domain is populated by those things that generically exist or the standard universal quantifier is stipulated to mean something other than quantification over those things that generically exist. I will return to the second possibility shortly. If the former interpretation of the standard quantifier in (2’) is the proper interpretation, then I can redirect us back to the concerns I had with (3’)’s acceptance of generic existence. Perhaps there is instead a way to translate (2) that restricts the universal quantifier like

      (2’’)   (∀Mφ) (Uφ ⇒ (∃Mx) φx)

If ∀M quantifies over the the same entities that ∃M does, then ∀M quantifies only over those things that exist in the mind. In English, (2’’) says “for all properties which exist in the mind, if they are understood, then there is something that exists in the mind which instantiates them"[1]. I’m not sure at all if restricted universal quantification is how Anselm would want the word “whatever” to be understood. Nonetheless, let us suppose that (2’’) avoids the metaontological problems of pluralism plus generic existence. Merricks would argue that pluralists who think that everything either exists in the sense of ∃M and/or in the sense of ∃R cannot actually state this position without the use of unrestricted quantification (and therefore without being burdened with the problems of (3’)).

To summarize how Merricks’ argument applies to the metaontology of Anselm’s argument, ask yourself how Anselm could formally state that “everything exists in the mind and/or in reality.” The first approach is (12) ∀x (∃My(y=x) or ∃Ry(y=x)) & ∃Mx(x=x) & ∃Rx(x=x)

(12) makes use of the standard universal quantifier. The Anselmian can accept that this is unrestricted quantification over things that generically exist and deal with the consequences, or they can stipulate that the formula ∀x(Fx) is shorthand for ∀Mx(Fx) & ∀Rx(Fx). But how can it be stated that ∀ quantifies only over the domains quantified over by ∀M and ∀R? With this definition of ∀, (12) is shorthand for (13) ∀Mx (∃My(y=x) ∨ ∃Ry(y=x)) & ∀Rx (∃My(y=x) ∨ ∃Ry(y=x)) & ∃Mx(x=x) & ∃Rx(x=x)

This avoids the problems of (3’). But it does not actually state that “everything exists in the mind and/or in reality.” (13) is perfectly consistent with a view where there is a third mode of being, ∃3, where it is true that ∃3x(x=x), because the first two conjuncts of (13) are trivially true: ∀Mx is quantifying over only the xs such that ∃My(y=x) and ∀Rx quantifies only over the xs such that ∃Ry(y=x). Therefore, this interpretation of the standard universal quantifier in (12) does not state Anselm’s view, because it is true under any view where there are at least the modes of being ∃M and ∃R.

David Builes argues that the first horn of the dilemma–conceding generic existence–may be more palatable than it first appears, by fleshing out a view where unrestricted quantification corresponds to a non-joint-carving generic existence that is not fundamental, while the specific modes of existence such as existence in the mind and existence in reality are fundamental. At the cost of parsimony, this would somewhat preserve the historical supports for pluralism because it allows the pluralist to say that ideas and dogs (or tables and prime numbers) do exist in fundamentally different ways, but that they also share in a generic, non-fundamental being. 

Builes shows though that this relief is short-lived based on two theses thoroughly defended in Sider 2011. Purity is the first thesis: fundamental facts contain only fundamental notions. The second thesis is completeness: each non-fundamental fact is true in virtue of some fundamental fact. Builes asks of the two-ways-of-being-pluralist “is (12) a fundamental fact?” If (12) is a fundamental fact then by purity it must only contain fundamental notions. This requires that the generic being which corresponds to unrestricted quantification must be fundamental. This redirects the pluralists back to the original issues with pluralism plus generic existence. If (12) is not a fundamental fact, then by completeness it must be true in virtue of some fundamental fact, which by Builes and Siders’ lights requires that there is a fundamental fact φ such that (12) is true in L if and only if φ. And by purity this fundamental fact cannot contain the unrestricted quantification that the two-ways-of-being pluralist takes to be non-fundamental, which would redirect the two-ways-of-being pluralist to the second horn of Merricks' original dilemma as they would have to find a way to state (12) without the use of unrestricted quantification.

Merricks shows how his dilemma presents itself in every alternate way of stating two-ways-of-being pluralism, such that a two-ways-of-being pluralism where there is no generic being cannot be stated without invoking unrestricted quantification. Unlike these pluralisms, however, “[...] we can state monism” (Merricks 2019, p. 24). Merricks decisively defends this claim, painting a vastly more compelling picture of monism than pluralism.

2.2 Van Inwagen’s Fregean Argument

Van Inwagen draws from Frege to support the idea that there is some kind of being which everything, whether mental or “real,” shares. He starts his defense by noting that numbers aren’t equivocal. The number ‘six’ means the same thing when I say there’s six dogs as it does when I say there’s six ideas in my mind. Van Inwagen then argues that any exists-statement can be translated into a number of-statement that makes use of univocal ‘number of’ phrases:

To say that unicorns do not exist is to say something very much like this: the number of unicorns is 0; to say that horses exist is to say essentially this: the number of horses is 1 or more. And to say that angels or ideas or prime numbers exist is to say—more or less—that the number of angels, or of ideas, or of prime numbers, is greater than 0. (Van Inwagen 2009, p. 482.)

If all exists-statements are translatable into univocal number of-statements, it suggests that there is some generic being that everything shares that is expressed by number of-statements. 

When either ideas or dogs are inputted into the number of-sentence “the number of Fs is greater than 0,” the mode of being that each has is properly communicated. For the proponent of generic being, this is straightforward: everything shares in exactly the same kind of being, and “the number of Fs is greater than 0” is simply another way of stating that “there are Fs.” The two phrases are picking out exactly the same thing. For those who deny this generic kind of being, it’s not clear why exists-statements about different things can be translated into these univocal number of-statements: why is it that just the sentence “the number of Fs is greater than 0” can imply at least two vastly different modes of being depending on what F is? What is a number of-statement actually picking out?

I will provide an analogy to supplement this point. Suppose ‘sprints’ were a univocal word with only one meaning and that ‘runs’ had exactly two vastly different meanings–as the the two-ways-of-being pluralist supposes is true of, say, ‘exists.’ The first meaning of ‘runs’ is the way that an athlete runs and the second meaning of ‘runs’ is the way that a computer runs. The only definition of ‘sprints’ is the way that an athlete sprints. I cannot translate every single matter-of-fact runs-statement into a matter-of-fact sprints-statement without some of those sentences being false. For example, I can change the runs-statement “the trackstar runs” to “the trackstar sprints,” but not the runs-statement “the computer runs” to “the computer sprints.” Despite this, imagine a “runs-pluralist” who thinks that any runs-statement can be translated into a sprints-statement and that that sprints-statement means exactly the same thing as the runs-statement. I take this to be analogous to the metaontological pluralist who thinks that every exists-statement can be turned into a number of-statement and that the number of-statement means exactly the same thing as the exists-statement, because like ‘sprints’ in this analogy, the numbers in number of-statements are univocal.

The pluralist might respond that number of-statment “the number of ideas is greater than 0” is actually an incomplete translation of the exists-statment “ideas exist in the mind,” and that a complete translation would say that “the number of ideas in the mind is greater than 0.” I think there are two paths for the proponent of this objection. The first is that there is a generic existence that is properly expressed by the incomplete number of-statement, but that there is also a second mode of being that is specified with the complete translation. I think this avoids van Inwagen’s argument. But it does so at the cost of opening itself up to the problems of pluralism plus generic existence put forward by Builes and Merricks. The second horn is that the incomplete number of-statement only expresses the same meaning as the exists-statement because for any F, the particular mode of being which it may have is intrinsic to F. For example, an idea may only exist in the mind, thus “the number of ideas is greater than 0” tells us that some ideas have some underspecified positive ontological status that is inferred to be “existence in the mind” because this is the way the ideas must exist.

This prompts the question: in virtue of what? Why is it that ideas must exist in the mind? The answer must be found in the properties of ideas. It is because of all of the properties that we take ideas to have that it would only make sense for them to exist in the mind. But if the difference between ideas and dogs is entirely secured by the properties of ideas which determine their mode of being, what reason is there to suppose that ideas and dogs exist differently in the first place? That is–ideas seem intangible, conceptual, and perhaps abstract, while dogs are tangible, physical, and concrete, and this is what motivates our distinguishing them from dogs. However, in listing the properties that separate ideas from dogs we have accounted for all of the conceptual distance between them. Allowing dogs and ideas different modes of becomes superfluous. This is the heart of van Inwagen’s rant against Russell.

Additionally, this second horn of the objection against van Inwagen’s argument does not avoid Merricks’ charge that its metaontology cannot be stated.

Merricks’ dilemma against pluralism when directed towards Anselm’s argument leaves its proponent with three horns: accept a pluralism that contradicts pluralism's primary motivation, accept a pluralism that cannot be stated, or reject Anselm’s pluralism. When supplemented with Builes’ second dilemma, the first two horns look almost unacceptable by themselves. Van Inwagen’s argument doubles down to demonstrate that the third horn is the only viable option. But choosing the third horn is tantamount to conceding that Anselm’s argument is unsound because the premises of existential import in Anselm’s argument are reliant upon metaontological pluralism.

3. Conceivable! You Keep Using that Word

My next objection to Anselm’s argument is that anyone who denies the real existence of a MGB cannot accept all of its premises. Sobel puts it nicely:

I take Gaunilon to say, can the person who hears the words, but is not sure that they identify anything in reality, be sure that he has in mind an object, an unsurpassable being, or a blessed isle that must by its nature exist not only in his mind, but also in reality. He cannot deny that he has in mind the words, or that he has in mind what they mean, for he understands them. But he can wonder whether he has in mind a thing described by them, for he can understand that he does have a thing in mind, given how these words describe things, if and only if such things exist in reality.” (Sobel 2003, p. 65.)

The fool’s position is characterized by Anselm with the premise that (14) There does not exist a MGB in reality

This premise, when conjoined with

      (9)  If a MGB exists in the mind, it also exists in reality.

entails the conclusion that

(15) There does not exist a MGB in the mind. (Modus tollens, 18 & 9)

If we grant (9), then immediately upon encountering Anselm’s argument, the fool has reason to reject

      (3)  A MGB exists in the mind. 

(3), of course, is a derivation from (1) and (2), meaning rejecting (3) is tantamount to rejecting either[2] (1) A MGB is understood. or (2) Whatever is understood exists in the mind.

Sobel thinks (2) is more problematic out of the two. The words “being-than-which-a-greater-cannot-be-thought” and the meaning they express may be properly understood, but it is dubious at best to say that therefore something answers to that meaning–particularly in this case, because it would require that such a being exists in reality too. And, of course, this supposition is reliant on Anselm’s problematic metaontology, meaning even the theist may have good reason to doubt the soundness of the argument[3].

Rejecting (1) is instead resisting the idea that the concept of a MGB is understood. This is plausible because Anselm does not adequately specify the conditions for something to be understood. What exactly does the fool have to understand about a MGB for such a being to be “understood?” Different notions of understanding or conceivability may leave (1) in poor or good condition, but it is up to the proponent of the argument to choose the most favorable one. Furthermore, whether a target of Anselm’s argument will accept that a MGB is understood depends on their axiological views. If the description MGB makes indispensable use of axiological properties, then what it expresses, if anything at all, is going to rest upon the axiological suppositions through which it is interpreted, and what it expresses is going to determine whether that expression is understood. Some axiological views will render a "being-than-which-a-greater-cannot-be-thought" as unintelligible or expressing something different for every person or nothing at all.

These problems do not only limit the effectiveness of Anselm’s argument against the fool. An agnostic who is not decided between (11) and (14) has good reason to be undecided between (3) and (15). This would undermine Anselm’s conclusion that “there is absolutely no doubt that something-than-which-a-greater-cannot be-thought exists both in the mind and in reality.” In fact, because the acceptance of (3) is contingent in this way on whatever views the interlocutor antecedently holds towards (11) and (14), the argument does exactly nothing for the non-theist who will only be motivated towards the existence of a MGB by Anselm’s argument to the same degree that they were motivated towards the existence of a MGB before they heard the argument.

Conclusion

I have presented two problems with Anselm’s ontological argument. The first is that it rests on a pluralist metaontology. Merricks gives a dilemma against a pluralism like Anselm’s: either the main historical motivations for pluralism are undermined, or the view cannot be stated. I have also shown how van Inwagen’s Fregean argument against pluralism applies to Anselm’s metaontology. Finally, I showed that any non-theist has good reason not to accept at least one premise of Anselm’s argument, rendering it inert against its original targets--including the fool.

Footnotes

[1] At the time of writing, it is opaque to me what metaphysical commitments come from properties “existing in the mind,” but I suspect that projecting the conceptual framework of contemporary analytic philosophy onto thousand-year-old writing will, no matter what, produce strange results.

[2] Alternatively, the fool could reject (9), which is essentially resistance towards some of Anselm’s axiological assumptions.

[3] I have omitted another objection to (2) because I am not confident that it works. The idea is that (2) opens the ontological floodgates to the existence of anything understandable whose description contains real existence. For example, a really existent king of France, for whom a parody argument can be made. Possibly, Gaunilon’s maximally great island objection could be formulated as an argument against (2). I am not confident in either of these approaches, because I am not sure how symmetrical a really existent king of France or a maximally great island to a MGB given Anselm’s metaontology and the ambiguity of (2).

Works Cited

Builes, David (2019). “Pluralism and the problem of purity.” Analysis 79 (3):394-402.

Charlesworth, Maxwell John (1979). St. Anselm's Proslogion: With a Reply on Behalf of the Fool by Gaunilo and the Author's Reply to Gaunilo. Notre Dame Press.

Merricks, Trenton (2019). “The Only Way To Be.” Noûs 53 (3):593-612.

Oppy, Graham (2006). Arguing About Gods. Cambridge University Press.

Sider, Theodore (2011). Writing the Book of the World. Oxford: Oxford University Press.

Sobel, Jordan Howard (2003). Logic and Theism. Cambridge University Press.

van Inwagen, Peter (2009). “Being, Existence, and Ontological Commitment.” Metametaphysics, Oxford University Press, pp. 472-506.

van Inwagen, Peter (2014) “Modes of Being and Quantification,” Disputatio 6: 1-24.