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Anselm's Ontological Argument for God's Existence

Updated on August 15, 2014
Anselm of Canterbury
Anselm of Canterbury
Anselm of Canterbury
Anselm of Canterbury

Anselm's Ontological Argument and Classic Critiques

Here is the target text of the famous ontological argument, from the Proslogion by Anselm of Canterbury:

"Therefore, Lord, who grant understanding to faith, grant me that, in so far as you know it beneficial, I understand that you are as we believe and you are that which we believe. Now we believe that you are something than which nothing greater can be imagined.

Then is there no such nature, since the fool has said in his heart: God is not? But certainly this same fool, when he hears this very thing that I am saying - something than which nothing greater can be imagined - understands what he hears; and what he understands is in his understanding, even if he does not understand that it is. For it is one thing for a thing to be in the understanding and another to understand that a thing is.

For when a painter imagines beforehand what he is going to make, he has in his undertanding what he has not yet made but he does not yet understand that it is. But when he has already painted it, he both has in his understanding what he has already painted and understands that it is.
Therefore even the fool is bound to agree that there is at least in the understanding something than which nothing greater can be imagined, because when he hears this he understands it, and whatever is understood is in the understanding.

And certainly that than which a greater cannot be imagined cannot be in the understanding alone. For if it is at least in the understanding alone, it can be imagined to be in reality too, which is greater. Therefore if that than which a greater cannot be imagined is in the understanding alone, that very thing than which a greater cannot be imagined is something than which a greater can be imagined. But certainly this cannot be. There exists, therefore, beyond doubt something than which a greater cannot be imagined, both in the understanding and in reality."

Gaunilo of Marmoutiers launched a famous, and still studied, critical assault on Anselm's ontological argument while the ink was still wet, so to speak:

“...they say that there is the ocean somewhere an island which, because of the difficulty (or rather the impossibility) of finding that which does not exist, some have called the ‘Lost Island.’ And the story goes that it is blessed with all manner of priceless riches and delights in abundance . . . and . . . is superior everywhere in abundance to all those other lands that men inhabit. Now, if anyone tell me that it is like this, I shall easily understand what is said, since nothing is difficult about it. But if he should then go on to say, as though it were a logical consequence of this: You cannot any more doubt that this island that is more excellent than all other lands truly exists somewhere in reality than you can doubt that it is in your mind; and since it is more excellent to exist not only in the mind alone but also in reality, therefore it must needs be that it exists. For if it did not exist, any other land existing in reality would be more excellent than it, and so this island, already conceived by you to be more excellent than others, will not be more excellent. If, I say, someone wishes thus to persuade me that this island really exists beyond all doubt, I should either think that he was joking, or I should find it hard to decide which of us I ought to judge the bigger fool – I, if I agreed with him, or he, is he thought that he had proved the existence of this island ."


David Hume's critique is encapsulated in this passage from his Dialogues Concerning Natural Religion : "...there is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by any arguments a priori. Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable..."

Alvin Plantinga
Alvin Plantinga | Source
Charles Hartshorne
Charles Hartshorne | Source
David Hume - early critic of the Ontological Argument.
David Hume - early critic of the Ontological Argument. | Source

Ontological Arguments

The term "ontological argument" applies to any proof that purports to show actual existence of an entity on the basis singly of premises that are analytic (this term will be explained.) An alternative definition has this type of argument purporting to prove an empirical conclusion (which is, as such, knowable a posteriori) from premises all of which are knowable a priori. The two definitions are not interchangeable because, we know today, analytic and a priori do not mean the same thing.

Here is what is confounding about an ontological argument - pay attention to this and try to get the basic flavor even if you feel frustrated, at this point, at the jargon that was thrown at you in the preceding paragraph: To prove actual existence, one would naturally have to resort to methods of factual verification - something like checking whether so-and-so can be verified to actually exist. We are not talking here about possible misattributions of names to so-and-so or other confusions that could arise. The point is that, in principle, truths about existence of empirical items cannot be obtained by relying only on definitions and logical truths. Of course, to prove that a line segment with certain properties exists, we DO rely ultimately only on definitions and logical truths (counting as such also the axioms of the system.) But remember what we said: the challange is about empirical, factually verifiable objects. It's fine to be able to prove specific abstract items existing (as in geometry or in math in general); these objects are not empirical (you might not have known this). Moreover, the whole of mathematical endeavor is carried out a priori: a meaning is called "a priori" when it is known (for being true or false) REGARDLESS of experience or factual methods of verification. For instance, "2 + 2 = 4" is known a priori. Were you to actually experience a situation in which a fifth appears, or only three appear, when two and another two things of the same type are put together, you would protest that something is wrong: you would not let experience "correct" your view that "2 + 2 = 4" is true. Even if this were to happen again and again, you would consider the outcome odd: but it is odd because it goes against an a priori known truth. In Math, premises and conclusion are known a priori. The conclusion is in a sense included in the premises but it is not known as such until it is pulled out, so to speak: even so, the conclusion, like the premises are knowable independently of experience or a priori. Now realize how strange it is that we would have an a priori proof of the existence of a non-abstract entity: this would mean that this is also knowable a priori but how can a truth whose means of checking are empirical and factual be knowable without resort to experience? You could well be confined to a solitary existence in a prison cell, trying to see if you can prove, without any factual information, that something concretely definable exists out there in the world beyond your prison. There is no reasonable expectation of success in this endeavor, right? The exponents of the ontological type of argument agree with this in general but with an exception: when you define an entity G (name for "God") in a certain way, then you do have access, whether you know it or not, to a proof you can carry out without access to any factual information.

From a logical point of view, a meaning is analytic when it can be assessed for being true or false only on the basis of the meanings that are its components. For instance, as a competent user of language, you know the meanings of the words "three," "angles" and "triangle." These are the logical components of the meaning of the sentence "a triangle has three angles." So, you know whether this meaning is true or false - well, it is true of course. For the same reasons, you know that the meaning of the sentence "a triangle has two angles" is false. Indeed, these meanings, in our example, are logically necessarily true or false respectively. Suppose that you have a proof that starts with only analytic statements as premises; then you apply valid inferential rules and, deductively, you conclude that p. The conclusion, p, has to also be analytic; it cannot be empirical. Empirically verifiable statements can be true and can be false (not in the same context, of course.) "There is a triangle-shaped toy in the nexst room" is a non-analytic statement: such statements are called synthetic and are contingently, not necessarily, true or false. Such statements are possibly true and possibly false. We see, then, that a proof with analytic premises cannot possibly yield a contingent or synthetic statement as conclusion. And yet, Ontological Arguments terminate in a conclusion of the form "g exists" where "g" is a name (for "God" defined in a certain way.) So, the statement "g (for God) exists" should be analytic and logically necessarily true - since it is obtainable deductively from solely analytic premises. But how can statement about actual, empirically verifiable existence be analytic? Take "b" for "Bush": "b exists" is logically contingent - it can be true (as it is in the actual world) and it can also be false (in the sense that there is a logically conceivable world in which the statement is false there - the statement can be denied or negated without logical absurdity, and this is the case for any statement about the existence of a concrete object.) So, something must be wrong with the Ontological Argument - but it turns out to be fiendishly wrong to figure out what, if anything, is wrong.

The best known Ontological Argument is due to Anselm, a Medieval Bishop of Canterbury. It was immediately attacked by another monk known as Gaunillo. It has been assaulted and defended passionately over many centuries now. Its study requires attention to technical matters in philosophy and in logic - so, engaging it has broader interest.

Anselm might not have been the first thinker to come up with an ontological argument. There is at least one proof of the existence of what is defined as "soul" in Plato's dialogues, in which only the definition and logical truths are used. A definition is, of course, an analytic statement. We can also say that it is known a priori - because we must assume an idealized competent user of the language as agent who knows the meanings of the words used in the language independently of experience. Versions of an ontological argument for God's existence include an elaborate one by the famous logician Kurt Godel. A debate surrounds the issue whether we need a modal version of the argument - though Anselm himself was not clear on this detail. A modal statement is one whose evaluation as to true/false is sensitive to variable context. "Either it is raining or it is not raining" is not modal but "it is logically necessary that it is either raining or it is not raining" is modal. The prefixing modal "logically necessary" shows that grammatically but, from a logical point of view, what we have is that the second statement, unlike the first, comes out true if and only if the statement that is modalized ("either it is raining or it is not raining") is accepted as true in all logically possible contexts. The Ontological Argument (OA) has terms like "necessary" and "possible" in it. Moreover, treating the OA as a modal argument is beneficial to the view that takes this to be a good argument because we can find a deductively valid version of the OA. Nonetheless, as usual, the fight is over the soundness of the argument. Validity of deductive arguments is a matter of what logical form they have. A deductive argument that is invalid is finished! But even if valid, a deductive argument can be unsound (which means that at least one of its premises is not necessarily acceptable as true.)

If the OA is valid and sound, we have unqualified success! This would be odd -- not accepting that God, defined as in the OA, would be irrational; faith drops out and, given a traditional understanding of Christianity, the roles are reversed: it is the intellectual, and not the

Proof by Reductio Ad Absurdum

This method is also known as Indirect Proof and Proof by Contradiction. It is rejected by the school of Mathematics known as Intuitionism.
This method is also known as Indirect Proof and Proof by Contradiction. It is rejected by the school of Mathematics known as Intuitionism.
The assumed premise (for reductio) in the OA is "it is not true that God exists." Other premises (given premises) are the definition of God and the principle that actual existence is a property that makes something "greater" .
The assumed premise (for reductio) in the OA is "it is not true that God exists." Other premises (given premises) are the definition of God and the principle that actual existence is a property that makes something "greater" .

Anselm's Proof

The definition of what is to be proven as existing is as follows: God is that than which nothing greater can be conceived to exist. Success in the proof does not show that what the Bible attributes to God is true by any means. The Biblical attributes of God are such that the above definition - or those in the Quran for that matter - are such that the above definition can be counted as acceptable to the monotheist. Unlike other purported proofs of God's existence, the OA, if successful, establishes uniqueness of the divine entity.

It is important that the definition itself be free of logical absurdity. We are to consider the definition as "closed" under its logical consequences: this means that whatever can be shown to follow logically from this definition (and also to follow in conjunction with other accepted statements) comes along for the ride regardless as to whether the author of the argument was aware of this or not. If logical absurdity is deducible from the definition, then the definition must be rejected and the proof never takes off as it were. The German philosopher Leibniz, an advocate of the ontological argument, was concerned that "greatness" could imply maximal ability in activities whose pursuit would cancel each other out at pain of logical contradiction: for instance, God may be the greatest protector of the innocent but also the greatest ravager of the innocent (even though he may choose, as a matter of benevolence, not to act on this latter ability.) Leibniz offered a rather convoluted way out of this difficulty. We could, however, postulate that the greatnesses attributed to God in the definition are about maximal abilities that are not mutually at odds.

Context is also important: if the entity to be proven to exist were of the category of natural numbers, for instance, then we would be able to obtain logical absurdity: as we know from the definition of natural number, the "greatest" natural number is not definable; and yet, we would be offered a definition of a natural number (God) who is said to be the greatest. There is no equivocation on "great" here because numbers are great in the way in which we use the word when we say that there is no natural number that is the greatest. Yet, God is not presumed as a number-like entity and the contradiction does not arise.

Anselm's proof is a priori: This means that he will try to prove that God exists by consulting only thinking and without any recourse to experience - like a proof in geometry, which you can run in your mind.

 Step 1: Anselm can start with a definition of God. This is his prerogative. He can define God any way he likes but then he will have to prove that this God EXISTS.

Anselm's definition of God is this: God is the greatest entity that can be conceived by the mind. "Greatest" means greatest in every conceivable way; most perfect. As indicated above, it is crucial that such concepts as "great" and so on are free of logical absurdity.

Notice that you don't have to conceive, or be able to conceive, all those perfections. All you know is that God is, by definition, the most perfect thing - the one entity with all the perfections to the maximum.

The perfections in teh definition are not matters that are up to subjective assessment. Think not of who the "greatest" basketball player ever is but of who the "greatest" basketball player is when it is specified what criteria for greatness are adopted objectively and how those criteria are ranked (for instance, "scored the greatest number of points in most successive seasons" and other criteria like this specified accordingly.)

 Step 2: Now Anselm must prove that God, as defined, exists. Anselm realizes that thinking of this God and defining this God does not prove that he exists. Anselm does not make the mistake so many students make in their exams - the mistake of thinking that if I have some idea in my mind, that means that the represented object of that idea exists. Well, it exists as an object of an idea in my mind, but this does not mean that it exists in the real world outside. Anselm knows all this, although he might not be able to avoid this very error in some way. At any rate, Anselm goes for a proof; he does not just assert that, because he can think of God it follows that God exists.

If the proof goes through, this would show that God is necessay in the logical sense. We should rather take it an error to be attributing logical necessity to items; rather, logical necessity, like logical possibility, characterizes propositions or meanings: it is logically necessary that God, as defined, exists... At any rate, a common error is to think that psychological necessity plays a role here - it does not! "It is necessary that God exists, since life would not make sense if God did not exist" - this sentence uses "necessary" not in the logical sense!

So where is the proof?

 Step 3: Anselm realizes how difficult it is to prove that this God, as defined, exists. Anselm uses a very smart method we know from geometry. It is called reductio ad absurdum: Assume the opposite of what you want to prove; then, draw conclusions from that in very carefully logical sequence; see if you draw some conclusion that is impossible - it contradicts something you have accepted or in any way is illogical and you cannot accept. Great! Since you were careful not to make any mistakes of logic along the way, then where is the problem? It must be the original assumption you made when you started with the opposite [contradictory] of what you wanted to prove. So, the opposite of what you want to prove is WRONG. Therefore, what you want to prove is RIGHT.

Usually, logical absurdity is obtained in reductio proofs by producing any sentence p and its negation not-p. The proof can terminate at that point and you have proved what you set out to prove - remember that you posited as presumed assumption the negation of what you meant to prove.

So, Anselm starts by assuming the negation of what he wants to prove. He assumes that God does not exist.

 Step 4: Now Anselm must produce a logically untenable result. - indeed, a contradiction. He goes on to say that he can think of another great entity. If God is G1, Anselm invites us to think of another entity G2. We said that G1 is the greatest, most perfect thing that ever exists. But I can think of a G2 which also has all the perfections, like G1, but in addition it also exists. [This does not mean that I proved or asserted that G2 exists. I can just think of it as existing. Why not?]

 Step 5: Now compare G1 and G2. Which one is more perfect of these two? Isn't it G2? G2, remember, is as great as G1 but, in addition, G2 exists and G1, as we said, does not. It is certainly greater to exist than not to exist. G2 is greater than G1.

 Step 6: But wait a moment: Didn't we say that, by definition, G1 is the greatest thing? Now we are saying that there is something else, G2, that is greater? We are contradicting ourselves. This is illogical. There must be a mistake somewhere.

 Step 7: Since we tried to be logical all the way, the only possible mistake must be the assumption with which we started. We assumed that God - the G1- does not exist. This is the only sentence that can be wrong it seems. Then, it is wrong to say that Goes does not exist. Therefore, it is true that God exists.

We have proven that God exists.

When you concentrate on the proof in this analytical and step-by-step way, it is difficult to tell where the mistake is.


Anselm's Ontological Argument

Are you persuaded, upon giving the proof due thought, that the Ontological Argument succeeds?

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Criticisms of the Ontological Argument

As indicated already, there is a critique of the overall ontological-type argument: since we can deny any existential claim regarding empirical objects without contradiction, then it should be possible to deny the conclusion of the ontological argument without contradiction. But this cannot be done if the conclusion follows validly from all analytical (logically necessary) premises! We have reached an absurdity - "we can and we cannot deny the existential statement that is the conclusion of an ontological argument." Therefore, something is wrong with the type of argument that raises this specter - something is wrong with the ontological type of argument as such. This seems to have been David Hume's approach.

On the other hand, IF ontological arguments are available, then we do have an example of an existential-empirical claim that cannot be denied without contradiction. Is the Humean critic of the OA begging the question as to whether there are ANY empirical-existential statements that are logically necessary? (To beg the question is, here, a species of fallacy akin to circularity: one presumes what one should be giving grounds for proving.)

The OA cannot get off the ground if absurdity can be shown to affect the definition of God or the concept of "greatness" that is in the definition. It is not clear-cut that this can be established. The OA also fails if that other premise - about existence as a property that confers greatness - is to be rejected. A common objection to this principle is that "some things are actually greater when they don't exist" but this is not a good objection for the following reasons. Take item x which is purely imaginary. The principle we are examining has this item, x, as defined, assumed now to exist. The objection to this principle often is based on failure to observe the requirement that it is the same item, x, that is then taken to be in existence. So, we cannot count as an objection that things assumed great turn out to be less so when existing - in those cases it is not the same description or definition that applies both to the imaginary and to the existent item. Also, an objection is raised along the following lines: a bad thing, a monster for instance, is greater when it does not exist than when it does exist. This confuses what our own reactions to the existent monster are (it is better for US is the monster did not exist) for what existence does as a property of the thing that is conceived of as existing: an existing monster is surely "greater" than a non-existing one.

The decisive attack, most students of the OA think, was launched by the German philosopher Kant. The pivot for that attack is this: existence is treated as a property in the OA, this is quite clear. Anselm withholds this property from the entity he imagines and calls "God" and then adds this property to the other properties God has to generate another imaginary entity. But, Kant argues, existence is not a property. To be more precise, existence is not a logical property. When we talk about objects, we do actually treat existence as a property. But the point here is about constructed items. Suppose that you learn - which is true although most people don't realize it - that the ideal triangles of geometry do not exist. In a 3-d universe, we cannot have two-dimensional or flat triangles. It gets even worse: the geometry of modern physics is not Euclidean - so, after all, the triangles of ideal Euclidean geometry are not instantiated in the actual universe. Yet, this does not affect what we mean by a triangle. A triangle, as defined in Euclidean geometry, is still to be picked out on the basis of the definition regardless of whether we have any such items to pick out or not. Suppose that our physics changed and we came around to thinking that the geometry of the universe is Euclidean after all and, by some additional fluke of spatial arrangements, the triangles of this geometry ARE instantiated - such triangles can and do exist in actuality. Once again, however, we have no impact on the definition of the Euclidean triangle. Whether such triangles exist or not, the definition of the ideal triangle is not affected. Since we define things like triangles by predicating properties or attributes of them, existence is not a logical property: whether existence is conferred or not, the definition is not affected. So, Kant's point is, existence is not a logical property and the OA fails because it uses existence as such a property.

An attempt to elucidate Kant's critique might go something like this (mixing in some views held by a 20th-century philosopher named P. F. Strawson): suppose someone gives you a description of a person you have every good reason to want to meet; after all the properties or qualities of such a person have been laid down with relish, then you are told that -- by the way, this person does not exist! But this works out strange the other way too: suppose they tell you at the end that this person exists! Why does this sound out of place - in both cases? Because existence is assumed in language about things that are laid down, defined through their properties, in conversation. In the first case, the oddity is that existence is taken out while you were reasonable to assume all the while that the person defined exists. In the second case, the oddity results from the redundant reference to the person's actual existence - such existence had been assumed all along of course. Once again, existence as a predicate doesn't seem to be acting in any identifiable way; it can serve as a presupposition in the pragmatic aspects of language that regulate usage but, as a logical property, existence appears to be inert - as Kant had opined.

There was a stronger view in the past - according to which the technical tools of Predicate Logic also show clearly that existence is either redundant and trivial, when explicitly conferred, or absurd and leading to contradictions when denied. This, however, shows something rather about the system of Predicate Logic in use and not about existence-as-a-predicate. In predicate systems known under the categorial name Free Logics, this problem does not arise at all. There is a view some share that there is some hope in all this: after all, Kant's point itself might be unnecessarily narrow, constrained by a specific way of thinking about how predicate logic is to be set up rather than by any deeper reasons. This does not mean that the OA does not come to grief also when translated into the language of Free Predicate Logic. ("Free" means actually "existential-assumptions free." Unlike the standard Predicate Logic that is taught in introductory textbooks, Free Logics do not permit automatic instantiation of predicate sets: a set like "all color things" is not to be treated straightforwardly as having at least one member unless this is also stated explicitly. From the lay point of view, this sounds like a more natural approach actually.)

There is actually a valid logical form in a system of Modal Logic which is the form of a version of the OA. So, there is a version of the OA that has logical form which is formally valid in some deductive logical system. The modal system in which the form is logically valid ("correct" or guaranteeing true conclusion on the basis of true premises) is the system known as S5 which tracks logical modalities. We can take these to be analytic modalities (not everyone agrees...) Analytic, remembers, means that the meaning of the statement is immediately true/false (no turning to the empirical world to check!); this is because of the meanings of the words in the statement. It's like this: you don't wait, perhaps nervously, for the umpires of any given baseball game to determine whether they will play by the "three-strikes" rule! It is insanity, or perhaps funny, to turn to historical or day-to-day empirical confirmation in this case. Given the defined meanings of words like "baseball," "strike," "three," and so on, the statement about the three-strikes is analytic and logically necessarily true. The statement "two strikes and you are out in baseball" is also analytic and logically necessarily false (the meanings in the words, not empirical recourse, suffice for determination of the truth-value of the statement.)

It seems right that the correct modal system for the OA is S5 - the system that tracks analytic modalities: notice the role played in the OA by the definition of the concept "God." A certain meaning of a word is laid down. Logical tautologies are also analytic - in this case the words whose meanings settle things are words like "and," "not," "possible," "necessary," "all," etc.. You can glimpse a symbolic representation of this logical form below - in propositional modal logic S5. We need, however, to ascend to modal predicate logic and this complicates things. David Lewis, a late philosopher, came up with several disambiguating translations of the OA, which result in predicate modal, and rejected the one valid form he found. You cannot reject a valid form as invalid but a deductive argument that has a valid form can still be unsound (not having supported or necessarily all true premises.) Lewis' objection was that the one version that had a valid logical form made an unwarranted assumption: that there is something special about our actual state of affairs. To understand what is going in here consider the following: our technical definition of "possibly p" should rather be "there is at least one logically possible state of affairs in which p is true." Thinking of God, for instance, we trace this being which, by definition, maximally perfect, along several possible worlds (it ought to be the maximally perfect being in each one of these worlds, you'd think.) These are controversial issues, because the whole picture seems laden with old-fashioned metaphysical claims about possibilia. Still, our logical machinery has us running over definable alternate states of affairs (which we also in everyday speech when we talk about how things could have been or might have been, and so on.) Although it is clear that in some respects the actual world is privileged, it is not clear that the actual world is also privileged in the sense that counts for studying logical matters like validity of logical forms. Lewis' critique is based on his discovery that the valid logical form for a version of the OA would be according special status to the actual world - which, Lewis thought, condemns this valid version to unsoundness.

This is a valid logical form in the modal system S5 which tracks logical/analytic modalities. This form permits a valid variant of the OA. This is a propositional modal form. Things get more intricate when we ascend to modal predicate.
This is a valid logical form in the modal system S5 which tracks logical/analytic modalities. This form permits a valid variant of the OA. This is a propositional modal form. Things get more intricate when we ascend to modal predicate.
In the valid form we plug in a well-formed predicate formula. In modal predicate logic, we now have several alternatives depending on the relative positions (and scopes) of the quantifier ("some") and the modal operators "necessarily" and "possibly."
In the valid form we plug in a well-formed predicate formula. In modal predicate logic, we now have several alternatives depending on the relative positions (and scopes) of the quantifier ("some") and the modal operators "necessarily" and "possibly."

A Valid Modal Version of the Ontological Argument

How are we to read the logical form?

"If (given that p then necessarily p), then (given that possibly p then p.)"

Breaking it down:

  1. if p, then necessarily p
  2. it is the case that possibly p
  3. therefore, p

1: by definition of "God", if (p) "God exists" then (necessarily-p) "God cannot not exist or exists as a matter of logical (analytic) necessity" (it is a matter of how the word "God" is defined.)

2: possibly-p: "it is possible that God exists": this means that the concept is consistent or it not possible to derive absurdities from it; it is logically possible that God exists, as "God" has been defined.

3. p: "God exists" is the conclusion.

© 2014 Odysseus Makridis


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