LEARNING FROM ANSELM´S ARGUMENT

Abstract

Anselm’s original argument for the existence of God seems to pull in opposite directions. On the one hand, it is not easy to see what, if anything, is wrong with it; on the other, it seems incredible that the existence of a being like God could be proved entirely a priori. This paper presents a diagnosis of what seems to be wrong with Anselm’s original reasoning. The diagnosis is general enough to be of use elsewhere, and it is this: conceptual possibilities are inferential dead-ends, not free inference tickets to prove any substantial claim. It remains to be seen if other versions of Anselm’s original insight, both contemporary and not, fall into the same conceptual possibility trap.

Keywords:
Anselm; Ontological Argument; Conceptual Possibility; God

Anselm’s original argument for the existence of God is still fascinating. An indication of that is the sheer number of different interpretations of his original text, as well as different classical arguments inspired by his surprising move.¹ I will not discuss the current literature on Anselm’s reasoning, be it more exegetical or more logical in nature, nor will I discuss other classical ontological arguments. I will rather present a close reading of Anselm’s core text, along with a proposed diagnosis of his supposedly logical mistake.

More than one modern take on Anselm’s reasoning seems to fall into the trap diagnosed here. One of Plantinga’s two versions of Anselm’s argument uses the premise “There is a possible world in which maximal greatness is instantiated” (Plantinga 1974a: 111), and fails to disclose that at stake here is merely a conceptually possible world; if the diagnosis proposed in this paper is right, Plantinga falls prey of the same logical mistake of trying to infer a non-trivial conclusion from a merely conceptual possibility. In Plantinga’s other version, he uses the premise “Maximal greatness is possibly exemplified” (Plantinga 1974b: 213), and fails again to recognise that the possibility in question here is merely conceptual.

Perhaps some renderings of Anselm’s original reasoning, despite appealing to conceptual possibilities, are immune to the criticism presented here; that may be the case with Adams 1971. Other renderings do not even appeal to conceptual possibilities (e.g. Klima 2000), and are thus beyond the scope of the present discussion. In any case, this is not a review paper. Further inquiry will be needed to see which renderings of Anselm’s original reasoning, and which later classical ontological arguments, fall prey to the supposedly logical mistake diagnosed here.

Anselm’s reasoning is still fascinating, presumably because it has two features that pull in opposite directions. On the one hand, it seems incredible that the existence of a being like God could be proved using reason alone. It is not surprising of course to prove by reasoning alone that there exists a prime number smaller than 5 and bigger than 2. Other than mathematical, abstract entities, however, it seems incredible that one can really prove the existence of something out of thin air, so to speak. The so-called cosmological and design arguments for the existence of God rely on empirical data; perhaps they are not cogent, but they are not trying to pull a rabbit out of a hat. If they do not fail, it is not that surprising. Anselm’s argument is quite different; if it does not fail it is very surprising. It means that God is a very special sort of entity, not unlike mathematical and other abstract entities in that its very existence is provable by reasoning alone. And that is quite surprising. On the other hand, though, it is not easy at all to see where is the mistake in Anselm’s reasoning, if it is indeed mistaken. Thus, as Plantinga writes, “although the argument certainly looks at first sight as if it ought to be unsound, it is profoundly difficult to say what, exactly, is wrong with it” (1974a: 85-6).

In this paper, a tentative diagnosis of what is wrong with it is presented. If this diagnosis is right, the mistake has enough generality to be useful as a lesson in how not to reason philosophically from conceptual possibilities. The shocking diagnosis, if not wrong, is that the sort of reasoning Anselm seems to have had in mind is a closet version of a trivial fallacy: the appeal to ignorance. It remains to be seen if other versions of the so-called ontological argument make the same mistake, not to mention other philosophical arguments that appeal to conceptual possibilities.

Let us start with Anselm’s own words:

If that than which a greater cannot be thought exists only in the understanding, then the very thing than which a greater cannot be thought is something than which a greater can be thought. But that is clearly impossible. Therefore, there is no doubt that something than which a greater cannot be thought exists both in the understanding and in reality. (Anselm, Proslogion, Chapter 2)

Anselm’s first move is to define God as that being greater than which a greater cannot be thought. ‘Greatness’ is here to be understood as excellence. Notice also that the thought is not directly that God is greater than all other beings; that would send the argument back to the empirical realm, just like the cosmological or design arguments. His idea is rather that the concept of God is such that there is no concept of a greater being. Whatever great being one can conceive, so great that no greater can even be conceived, that is what Anselm means by ‘God.’ It is this feature alone that supposedly proves that that being exists, not just as something that one thinks, but rather as something that exists apart from existing in thought. And that is so because the very hypothesis that the being thus defined does not exist lands one supposedly in contradiction: because, in that case, one can easily think of another being, just like the first, except that it exists. However, this supposedly contradicts the first thought, because we were explicitly assuming that God is that being greater than which no other can be thought - and we just thought about a being greater than God.

One initial and serious worry is that the very concept Anselm has in mind is perhaps impossible. After all, necessarily, there is no odd number greater than which no greater odd number can be thought: for any odd number one can think of, there is always a greater one. This is a serious worry, but it fails to present a diagnosis of what, if anything, is wrong in Anselm’s reasoning. Maybe Anselm’s concept of God is impossible - and maybe not. Even if it is impossible, though, there is presumably something wrong with his subsequent reasoning; and assuming his God is impossible does not explain what is it that is wrong with Anselm’s reasoning.²

Likewise, it is easy to come up with obviously flawed arguments that follow Anselm’s pattern. One thinks of that World Government greater than which none can be thought; that is easy enough. Alas, it does not exist, and it is doubtful whether it will ever exist. Following the same pattern of Anselm’s reasoning, though, we will reach the conclusion that that government exists. Again, this sort of reaction fails to explain what is wrong with Anselm’s reasoning. It tells us only that something is wrong.

Nothing serious depends on the issue of what exactly Anselm means by ‘cannot be thought,’ or what he should have meant. Still, a few clarifications are in order because that will be the heart of our scrutiny.

The first and obvious thing to say here is that it would be an uncharitable interpretation to claim that Anselm has in mind some narrow psychological understanding of thinking. Some people are, say, logically dim; they cannot think of a trivial logical proof of a given result. This does not mean that that proof cannot be thought by others. So ‘can be thought’ is to be understood as ‘some properly knowledgeable and competent agent is capable of thinking it, even if other agents are incapable of doing so.’

Still, this is problematic. Goldbach’s Conjecture may well be true, as far as we know, and in that case perhaps there is some way of proving it true. However, the most knowledgeable persons in this regard were not able, so far, to think of a way to prove Goldbach’s Conjecture. Does that mean that we are unable to think of a proof of Goldbach’s Conjecture? Not exactly. We should draw a distinction between thinking in the most general terms - which means basically that one can put forward some descriptive words to that effect - and thinking with some degree of detail. Presumably, no one is able to think with some degree of detail of a million-sided polygon; but most people can think about it in the most general way, just because they are able to articulate the concept and even say several true things about million-sided polygons. Perhaps Anselm has, or should have had, something like this in mind: some agents at least are able to think in the most general terms of a being greater than which none other can be thought, even if they would be hard-pressed to say anything detailed about it.

I will use ‘conceptual possibility’ to cover at least largely what Anselm seems to have in mind here. The thought seems to be that it is conceptually possible that there is a being such that no greater being is conceptually possible. A key difficulty here was already noted: it is not obvious that it is conceptually possible that there is a superlative being like that; perhaps for every very great being that is conceptually possible, there is another conceptually possible being that is even greater. Anselm may very well be asking us to think of a conceptually impossible being. That is a worry. But let us grant for the sake of argument that that being is not conceptually impossible. This will allow us to try and diagnose Anselm’s mistake.

One last preliminary is in order. Conceptual possibilities are here to be understood as sharply distinct from metaphysical possibilities. Suppose that the world is such that it is impossible for a thrown dice to change into an orange winged horse once it hits the table. That would be what I call a metaphysical impossibility. The world is such, under that reasonable assumption, that that event is simply impossible.

This contrasts sharply with conceptual possibility. That event is conceptually possible in exactly this sense: our knowledge of the truth-conditions of the relevant statement is not enough to know that that event will never happen. Thus, p is conceptually possible iff we are unable to know that ¬p merely on the basis of its truth-conditions. It is not conceptually possible for a dice that does not have exactly six sides to have exactly six sides - precisely because on the basis of its truth-conditions alone we know that the relevant statement is not true. Perhaps there are other sorts of conceptual knowledge, not linked in this way to our knowledge of truth-conditions. Still, it seems safe to bet that at the very least there is such a thing as linguistic knowledge, and that this is deeply dependent upon this sort of knowledge of truth-conditions. And that is all that is required to have a robust enough concept of conceptual knowledge, and therefore of conceptual possibility.

Are all conceptual possibilities metaphysically possible? Perhaps, yes - although there is scarcely any non-fallacious proof of that. Or perhaps some conceptual possibilities, like a thrown dice that changes into an orange winged horse once it hits the table, are metaphysically impossible. But even if all conceptual possibilities are metaphysically possible, the inference from ‘It is conceptually possible that p’ to ‘p is therefore metaphysically possible’ is invalid, at least if one understands ‘conceptual possibility’ as characterised above. So I am not assuming that some conceptual possibilities are not metaphysically possible; the point is that I am also not assuming naively that all conceptual possibilities are metaphysically possible. If one assumes this latter unproved and perhaps unprovable hypothesis, then it seems that there is no reason to believe Anselm’s reasoning, as parsed here, is invalid.

Let us now return to Anselm’s reasoning. He seems to believe that something like statement 1 is inconsistent with 2:

• 1. $◇\exists x\forall y[\neg (y=x)\to Fxy]$: it is possible that there is something greater than anything else.

• 2. $\neg \exists x\forall y[\neg (y=x)\to Fxy]$: there is no entity greater than anything else.

There is no inconsistency here, as one proves easily using a simple (alethic) modal logic built as an extension of classical logic. Reinterpreting Anselm’s premise, however, seems to give us the desired contradiction. The reinterpretation is the following:

• 1a. $◇\exists x\forall y[\neg (y=x)\to \square Fxy]$: it is possible that there is something that is necessarily greater than anything else.

This interpretation adds the necessity operator, to do justice to Anselm’s thought that necessarily there is no being greater than God; he does not mean to say only that there is no being greater than God, but rather that that is necessary. The very concept of God is that of a being such that no other being could possibly be greater.

Under this interpretation, which is still quite close to the text, we have what at first looks like the desired reductio ad absurdum: it seems that 2 is inconsistent with 1a, in which case we would have a neat logical proof that God (as defined by Anselm) exists. Here is a tentative proof:

$$\begin{gathered} \begin{array}{ccc}& ◇\exists x\forall y[\neg (y=x)\to \square Fxy]⊢\exists x\forall y[\neg (y=x)\to Fxy]& \\ & ◇\exists x\forall y[\neg (y=x)\to \square Fxy]& \\ & \neg \exists x\forall y[\neg (y=x)\to Fxy]& \\ & \forall x\exists y\neg [\neg (y=x)\to Fxy]& \\ & \alpha -\beta & \\ & {}_{\beta }\exists x\forall y[\neg (y=x)\to \square Fxy]& \\ & {}_{\beta }\forall y[\neg (y=a)\to \square Fay]& \\ & \exists y\neg [\neg (y=a)\to Fay]& \\ & \neg [\neg (b=a)\to Fab]& \\ & {}_{\beta }\neg (b=a)\to \square Fab& \\ & \neg (b=a)& \\ & \neg Fab& \\ & ╱╲& \end{array} \\ \begin{array}{ccc}{}_{\beta }(b=a)& & {}_{\beta }\square Fab\\ & & \text{Sym.}\\ & & \mathrm{\beta }-\mathrm{\alpha }\\ & & \underline{Fab}\end{array} \end{gathered}$$

A system of alethic logic trees is being used here, built as an extension of classical logic. The usual possibility (◇) and necessity (□) operators are used, along with the usual logical symbols. The arrow is to be read as the material conditional. Greek letters (α, β) are used as names of possible worlds, instead of the usual w ₁, w ₂; ‘α-β’ means that β is accessible from α, and ‘Sym.’ means that we are introducing symmetry in the accessibility relation between possible worlds. The prefix Greek letter indices means that a statement of that logical form is true at that possible world. Thus, if a statement of the form ◇p is true at α, then a statement of the form p is true at some world $\mathrm{\beta }\left({}_{\mathrm{\beta }}p\right)$ accessible from $\mathrm{\alpha }(\mathrm{\alpha }-\mathrm{\beta })$. I write ‘◇p’ instead of ‘_α ◇p’ because one always starts from α anyway, and it sure is a bit pedantic to always say ‘At the actual world, it is raining’ instead of just ‘It is raining.’

Now, the left branch is open just because an overly cautious version of what is sometimes called Leibniz’s Law (the substitution of identicals salva veritate) is being used here. Using that overly cautious version, though, one manages to prove quite easily the necessity of identity: if $a=b$, then $\square (a=b)$. Accepting this, in turn, allows one to close the left branch: one uses the identity ‘${}_{\mathrm{\beta }}(b=a)$’ to turn ‘$\neg (b=a)$’ into ‘$\neg (a=a)$.’ Thus, assume for the sake of argument that the left branch closes. What I want to discuss is the crucial fallacy in the right branch.

Symmetry was used to get from α-β to β-α. That is, since the possible world β is accessible from α, assuming symmetry between possible worlds we get that α is also accessible from β. In this logic trees system, being forced to use symmetry to close a branch means that one is upgrading from the simple K system of modal logic to KB (or B, if one also uses reflexivity).

So, the question is: is it fallacious to assume symmetry here? We accept Anselm’s thought that it is conceptually possible that God is necessarily the greatest, but this means that even if the second modality is metaphysical, the first is merely conceptual. That was not made explicit on purpose; had it been made explicit, the first logical form would read like this: $◇{}_c\exists x\forall y[\neg (y=x)\to \square Fxy]$. The ‘c’ index marks conceptual possibility; the necessity operator does not have an index because even if the box is read as conceptual necessity, it is quite reasonable to accept that conceptual necessity entails metaphysical necessity ($\square {}_{\mathrm{c}}p⊢\square p$), even if conceptual possibility does not entail metaphysical possibility.

Let us now see why there is good reason to reject symmetry in dealing with conceptual possibility. Consider any statement of the form $◇\square p\to p$. In axiomatic modal systems this is the logical form one adds to K to get KB. Assuming symmetry, any sentence with this logical form is a logical truth. Here is one way of proving that:

$$\begin{gathered} ⊢◇\square p\to p \\ \begin{array}{ccc}& & \neg (◇\square p\to p)\\ & & ◇\square p\\ & & \neg p\\ & & \mathrm{\alpha }-\mathrm{\beta }\\ & & {}_{\mathrm{\beta }}\square p\\ & & \mathrm{Sym}.\\ & & \mathrm{\beta }-\mathrm{\alpha }\\ & & \underline{p}\end{array} \end{gathered}$$

Consider Goldbach’s conjecture again. We do not know whether it is true or false, but it is rather reasonable to accept that perhaps it is necessarily true. Thus, it seems reasonable to accept that it is possible that Goldbach’s conjecture is necessarily true: ◇□p. From this, however, as long as one accepts symmetry, one proves that the conjecture is true. Since it is obvious that something is wrong with that supposed ‘proof,’ for otherwise every unproved mathematical conjecture would be easily proved, our task is to try and see what is wrong with it.

One way to block that attempted proof is to reject that the relation of accessibility between possible worlds is symmetric. But this is a bit too much. Perhaps metaphysical possibility is symmetric; one does not need to go that far. Rejecting that conceptual possibility is symmetric is enough. It is reasonable to accept that if Goldbach’s conjecture is possibly necessary, then it is true; the point is that we do not know whether it is possibly necessary, precisely because we do not know whether it is true or not. If it is false, then it is not possibly necessary; it will be impossible. The air of truth when one utters the statement ‘Goldbach’s conjecture is possibly necessary’ results from a merely conceptual reading of ‘possibly.’ Goldbach’s conjecture is possible in the sense that we do not know that it is false; we have no proof of that. We also have no proof that it is true. So, we do not know.

Reading the possibility operator here as merely conceptual possibility makes sense and allows one to block the attempted proof without going as far as rejecting symmetry across the board. It is just that conceptual possibility is not symmetric, even if metaphysical possibility is symmetric. Thus, we should write ‘◇_c □p,’ to say explicitly that we do not know whether p is necessarily true, but we accept that that is conceptually possible.

The general morals here could be summed up as ‘Mind your diamonds! Have you checked to see whether they are merely conceptual possibilities?’ The point is that if a given possibility is merely conceptual, reasoning on that basis is likely to be fallacious, unless one realises that that operator’s logical job is to block most inferences, not to allow them. Bearing in mind that ‘◇_c p’ is to be understood as ‘$\neg K_c\neg p$’ makes this obvious. (Using ‘K _c p’ to mean that an unspecified agent knows conceptually - i.e., a priori, i.e., merely linguistically - that p.) It is invalid to conclude that Goldbach’s conjecture is this or that just because we do not know conceptually that it is not so.

We have thus a general moral here. It is not just that conceptual possibility is not symmetric. Not being symmetric is just a special case of a general feature of conceptual possibility: its utter inferential weakness. This weakness is much more obvious if one translates this fancy philosophical term of art - ‘conceptual possibility’ - into its real content, putting in place the conceptual knowledge operator.

It can easily be seen now that to attempt to prove Goldbach’s conjecture starting from the conceptual possibility that it is necessarily true is falling flat in a form of the appeal to ignorance fallacy: I do not know conceptually that Goldbach’s Conjecture is not necessarily true, therefore it is necessarily true, and therefore it is true. The same mistake is at work in this other reasoning pattern:

$$\begin{gathered} ◇{}_{c}p \\ \square p\bigvee \square \neg p \\ \therefore p \end{gathered}$$

It is conceptually possible that the Goldbach’s Conjecture is true. However, as is the case with other mathematical statements, Goldbach’s Conjecture is either necessarily true, or necessarily not true. Therefore, it is true! Here is the attempted, fallacious ‘proof’:

$$\begin{gathered} ◇{}_{c}p \\ \square p\bigvee \square \neg p \\ \begin{array}{ccc}& \neg p& \\ & ╱╲& \\ \square p& & \square \neg p\\ \mathrm{Refl}.& & \mathrm{\alpha }-\mathrm{\beta }\\ \mathrm{\alpha }-\mathrm{\alpha }& & \mathrm{\beta }p\\ \underline{p}& & \underline{\mathrm{\beta }\neg p}\end{array} \end{gathered}$$

Note that in this case we do not use symmetry. So that is not the mistake here. And there is nothing wrong in the left branch - one just assumes reflexivity, upgrading thus to the T modal system. The mistake is in the right branch. Assuming that p is conceptually possible does not allow one to infer that there is a possible world β at which p is true. Perhaps there is no possible world at which p is true, if p happens to be impossible, even if p is conceptually possible. As we see, mixing conceptual possibility with metaphysical modality is tricky. Allowing the move in the right branch makes it easy to seemingly ‘prove’ that conceptual possibility entails metaphysical possibility:

$$\begin{gathered} ◇{}_{c}p \\ \neg ◇p \\ \square \neg p \\ \mathrm{\alpha }-\mathrm{\beta } \\ {}_{\mathrm{\beta }}p \\ \underline{{}_{\mathrm{\beta }}\neg p} \end{gathered}$$

A way to clearly see the fallacy in this supposed ‘proof’ is to unpack the concept of conceptual possibility, as suggested above. Using the operator K _c for knowledge on the basis of truth-conditions alone, the above is rewritten thus:

$$\begin{gathered} \neg K{}_c\neg p \\ \neg ◇p \\ \square \neg p \\ \mathrm{Refl}. \\ \mathrm{\alpha }-\mathrm{\alpha } \\ \neg p \end{gathered}$$

The logic tree does not close now, despite using reflexivity, because the negation of a knowledge operator does not allow any simplifying inference. No conclusion regarding p or ¬p follows from the fact that p is not known (either on the basis of truth-conditions alone or on some other basis too). Thus, any reasonable logic that includes an operator for conceptual possibility will regard it as an inferential dead end, not as a free inference ticket.

Conceptual possibility is mainly a disabling operator, not an enabling one: its main logical role is to disable inferences. This contrasts with conceptual necessity, that enables inferences. From the conceptual necessity of p one infers validly that p, but from the mere conceptual possibility that p one does not infer anything relevant at all. And this is so even if instead of p we have □p. From the conceptual possibility that p is necessary, it does not follow that p is necessary. Again, this is so because the claim that □p is conceptually possible is just the claim that ¬□p is not conceptually necessary, which in turn is just the epistemic claim that we do not know on a linguistic basis alone that it is not necessary that p.

We have now a clarifying explanation of what is wrong with Anselm’s original insight. It certainly looks as if the sheer possibility of a necessary existent entails its very existence, and this was perhaps Anselm’s main thought. However, this is an illusion. From conceptual possibilities no non-trivial conclusion follows because a conceptual possibility is just the absence of conceptual knowledge. And from the absence of knowledge in general, be it conceptual or not, no non-trivial conclusion follows. Far from being free inferential tickets to prove the existence of God or anything else, conceptual possibilities are inferential dead-ends. As was hinted at the beginning, further inquiry will be needed to determine whether other versions of Anselm’s reasoning - both contemporary and not - fall in the same conceptual trap.

References

• Adams, R. M. (1971) “The Logical Structure of Anselm’s Arguments.” The Philosophical Review 80.1: 28-54.
• Anselm, Proslogion. In Basic Writings. Ed. and trans. by T. Williams. Indianapolis, IN: Hackett, 2007.
• Klima, G. (2000) “Saint Anselm’s Proof: A Problem Of Reference, Intentional Identity And Mutual Understanding”, in Medieval Philosophy And Modern Times, ed. G. Holmström-Hintikka. Dordrecht: Kluwer, pp. 69-87.
• Oppy, G. (2021) “Ontological Arguments’, in The Stanford Encyclopedia of Philosophy (Fall 2021 Edition), ed. E. N. Zalta, https://plato.stanford.edu/archives/fall2021/entries/ontological-arguments/
  » https://plato.stanford.edu/archives/fall2021/entries/ontological-arguments/
• Plantinga, A. (1974a) God, Freedom, and Evil. New York: Harper and Row.
• Plantinga, A. (1974b) The Nature of Necessity. Oxford: Oxford University Press.

• 1
  For an overview of the current literature, see Oppy 2021.
• 2
  Note that to say that there is something wrong with the way Anselm tries to prove God’s existence is quite different from saying that there is something wrong with his starting point. I am in the business not of challenging his starting point, but rather a particular way of parsing his reasoning. The distinction at stake here is that between unsound valid reasoning and invalid reasoning. If Anselm’s definition of God is defective, his reasoning is unsound even if valid. On the other hand, if his reasoning is invalid, then even if his definition of God is not defective, he is unable to prove in that particular way that God exists.

• Article info

  CDD: 189

Publication Dates

• Publication in this collection
  13 Dec 2021
• Date of issue
  Oct-Dec 2021

History

• Received
  21 June 2021
• Reviewed
  29 Sept 2021
• Accepted
  07 Oct 2021