Abstract
In this paper, we discuss formal reconstructions of Anselm’s ontological argument. We first present a number of requirements that any successful reconstruction should meet. We then offer a detailed preparatory study of the basic concepts involved in Anselm’s argument. Next, we present our own reconstructions—one in modal logic and one in classical logic—and compare them with each other and with existing reconstructions from the reviewed literature. Finally, we try to show why and how one can gain a better understanding of Anselm’s argument by using modern formal logic. In particular, we try to explain why formal reconstructions of the argument, despite its apparent simplicity, tend to become quite involved.
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Notes
An extensive overview can be found in Uckelman (2012).
Matthews and Baker (2010) hold that ‘much of [the] literature ignores or misrepresents the elegant simplicity of the original argument’. But the argument that is offered is, first, still an informal argument; therefore, it does not offer a possibility to understand why formal reconstructions tend to be complex. Second, it depends on a distinction between ‘mediated’ and ‘unmediated causal powers’, a distinction that we do not see in Anselm’s argument. The presented argument may be simple and elegant, but it is, in our terminology, an attempted emendation, not a reconstruction. (Oppy 2011 started a debate between Oppy and Matthews/Baker.)
Sometimes, proofs of the existence of God are accused of being question-begging, but this critique is untenable. It is odd to ask for a deductive argument whose conclusion is not contained in the premisses. Logic cannot pull a rabbit out of the hat.
Of course, it is notoriously hard to make precise what is to be counted as an analytic truth and it is even harder to reconstruct what a particular author would count as such. As concerns Anselm’s argument, we shall, in subsequent chapters, try to make more precise what we take to be Anselm’s understanding of key concepts in his argument, such as ‘existence’, ‘being greater’ etc., in order to make our assumptions plausible consequences of relations between these concepts.
The translation mainly follows Anselm of Canterbury (2008, p. 87f), but some minor changes are made to achieve unequivocal terminology. As we shall see in Sect. 2.4, there is good reason to translate ‘maius’ as ‘bigger’ rather than ‘greater’, but we will stick to the traditional translation in what follows.
Logan (2009, p. 176), for instance, attributes such a view to Graham Oppy.
Cf. Anselm of Canterbury (2008, p. 89).
Although it is not obvious that modern logic has the right tools for analysing Anselm’s argument, throughout this article, we will stay within the limits of established logical systems as we saw no need to introduce new formalism. (Henry 1972, for instance, does.)
Cf. Anselm of Canterbury (2008, p. 88).
Instead of distinguishing kinds of existence, in another fragment, Anselm distinguishes four ways of using ‘something’, concluding that ‘when that which is indicated by the name and which is thought of in the mind does in fact exist’, then this is the only way of using ‘something’ properly (Anselm of Canterbury 2008, pp. 477–479). However, we shall not pursue this line of reasoning. Alston (1965) elaborately discusses the adequacy of different conceptions of existence for Anselm’s proof.
Note that as our \(E!\) is supposed to stand for a substantial property, it is not to be confused with Russell’s existence predicate \(E!\), which is defined contextually and does not express a genuine property at all.
We are grateful to an anonymous referee for drawing our attention to this point.
If one insists on expressing being in the understanding by a predicate, it would be used in such a way that it could be ‘cancelled out’ in proofs (\(\exists x (Ux \wedge ...)\) and then \(\forall x (Ux\rightarrow ...)\)). For further discussion on this issue, see Oppenheimer and Zalta (1991).
Compare Logan (2009, pp. 18, 91, and 114). Logan (2009, pp. 15–17, pp. 125–127) also discusses at length what Anselm means by ‘unum argumentum’ and, in particular, argues that its form is, to a high degree, a product of Aristotelian dialectics. Therefore, it has the form of a syllogism (Logan 2009, p. 17):
God is X; X exists \(\therefore \) God exists
where X, the middle term, is Anselm’s aliquid quo. We will exclusively deal with the proof that aliquid quo exists.
Note that in our construal, the existence presupposition does no harm (even if id quo had to be construed as a definite description), for what is presupposed is only the ontologically neutral existence in the understanding, expressed by the quantifier.
‘In fact, everything else there is, except You alone, can be thought of as non existing. You, alone then, ...’ (Anselm of Canterbury 2008, p. 88).
In linguistics, devices with a similar function are called E-types (sometimes D-types). We thank an anonymous referee for this hint.
Concerning the question of why existence apparently makes ‘greater’, see Millican (2004), Nagasawa (2007) and Millican (2007). Whether existence makes everything greater is a question that cannot be discussed here at length, but Logan convincingly argues that both options can be maintained (Logan 2009, p. 94f).
Seneca is the only one of Anselm’s possible sources who also uses ‘maius’; all the others use the judgemental ‘melius’ (Logan 2009, p. 93). Whereas Descartes, Leibniz and others use ‘perfections’ or similar expressions and, hence, morally charged terms, Anselm uses a more neutral term. One might be tempted to translate ‘maius’ as ‘bigger’ instead of ‘greater’. This would be unusual and might sound a bit uncouth, but it would accommodate the neutrality of ‘maius’. (Of course, we are aware that ‘maius’ can also be understood as judgemental, but—unlike ‘melius’—it does not have to be understood this way.)
St. Augustine had already introduced something like a ‘scale of being’ and we know that Anselm was well aware of Augustine (see Matthews 2004, p. 64). In Proslogion, however, the logical structure of the argument does not involve different degrees of properties, but differences in the kinds of properties.
Otherwise, God would have to have both positive properties, \(P\) and \(Q\), in order to be id quo. But, on the other hand, if \(P\) and \(Q\) contradict each other, he cannot because, by assumption, nothing is both \(P\) and \(Q\).
A further property that seems to be mandatory is that ‘\(>\)’ be asymmetric. Thus, ‘\(>\)’ is likely to define some partial order. Whether any two objects can be compared as to their greatness as well—i.e. whether the ordering is total—cannot be decided on the basis of Anselm’s writings.
See, e.g., Schrimpf (1994, pp. 29–31).
Anselm of Canterbury (2008, p. 88f).
Henry (1972, p. 108f), for instance, argues that Anselm thinks of ‘being inconceivable not to be’ as something stronger than ‘being necessary’—a view he attributes to Anselm’s Boethian background. An interesting definition has been suggested in Morscher (1991), in which a conceivability-operator \(D\) is defined by means of the composition of an epistemic component \(D'\) (‘it is conceived that’) and an alethic component \(\lozenge \) (‘it is possible that’); thus: \(D p := \lozenge D' p\). The epistemic operator \(D'\) itself can be defined by \(D'p := \exists y D''_y p\), where \(D''_y p\) is supposed to stand for ‘\(y\) thinks that \(p\)’. Therefore, unpacking the definitions, Morscher’s notion of conceivability could be stated in the following way: It is conceivable that \(p\) if and only if it is possible that someone thinks that \(p\).
A similar distinction is discussed by Dale Jacquette in his Jacquette (1997). Jacquette favours—against Priest—a version that uses conceivability as an operator on propositions or propositional functions.
Maydole (2009) tries to provide a formal reconstruction of Anselm’s reasoning in (II.4)–(II.5). In the following, however, we shall exclusively deal with (versions of) Anselm’s argument as it is stated in (II.8)–(II.13), since it is here that we think the very heart of Anselm’s ontological argument is lying.
See Morscher (1991, p. 65).
In particular, even on this rough reconstruction, Anselm’s argument is immune against Gaunilo’s famous ‘island-objection’, claiming that, were Anselm’s argument sound, we could, by analogy, also prove the existence of a ‘most perfect island’. Now, to see that this is not the case, let Gaunilo’s island \(g\) be defined by \(Ig \wedge \lnot \exists y (Iy \wedge y > g)\), expressing that \(g\) is an island such that no conceivable island is greater than \(g\). It can be seen quite easily that we cannot prove a contradiction from Greater 1 and ExUnd from the assumption that \(g\) does not exist unless we strengthen Greater 1—rather implausibly—to \(\forall x (\lnot E!x \rightarrow \exists y (Iy \wedge y > x))\), expressing that whenever some being does not exist in reality, there is some island which is greater than this being. Similar remarks can be made concerning all the reconstructions that will follow.
Sobel (2004, pp. 60–65), for instance, presents the argument in this form, though his aim is not to present a faithful reconstruction.
Throughout the following sections, we shall assume, as usual, that the second-order comprehension axioms are satisfied—i.e. for every formula \(\phi (x)\), it holds that (CA) \(\exists F \forall x (Fx \leftrightarrow \phi (x))\). In other words, every simple or complex formula defines a property. However, most of the time, we shall be concerned with quantification over the properties in the restricted class \(\mathcal P\) only. If there are only finitely many properties in \(\mathcal P\), then the quantifiers can be replaced by finite conjunctions.
Maydole (2009), who also uses a second-order predicate \(P\), is simply speaking of ‘great-making properties’.
There are earlier reconstructions, which have followed a similar line of understanding Anselm’s argument. Alvin Plantinga, for instance, suggests that Anselm’s reasoning might be based on the following premiss:
(2a) If A has every property (except for nonexistence and any property entailing it) that B has and A exists but B does not, then A is greater than B (Plantinga 1967, p. 67)
Plantinga later (cf. Plantinga 1974, p. 200) comes to question his earlier attempt to reconstruct Anselm’s argument by means of (2a). In any case, our problem here is how such an informal premiss might be spelt out in formal terms.
Given an explication of the properties in \(\mathcal P\), we may regard Greater 3 even as a definition of the notion of ‘being greater than’. As we shall see though, the right-to-left direction suffices for the purpose of proving that aliquid quo exists. Also, with Greater 3 at hand, \(>\) defines a partial order, a fact that seems to be welcome. As mentioned earlier, Anselm’s greater relation should clearly determine an order of some kind.
Here, the restriction to primitive properties is essential. Suppose we were to use an unrestricted universal quantifier \(\forall F\) in the definition of quasi-identity. Then, one instance of the comprehension scheme for second-order logic will be \(\exists F \forall x ( Fx \leftrightarrow \lnot E!x)\) and, hence, the property of non-existence \(\lnot E!\) will be among the values of the variable \(F\) in the definition of quasi-identity. Suppose, then, \(x \equiv _{E!} y\). Since \(E! \ne \lnot E!\), we would then have \(\lnot E!x \leftrightarrow \lnot E!y\)—i.e. \(E!x \leftrightarrow E!y\). Therefore, we could no longer assume, as needed in Anselm’s argument, that one of the ‘two’ compared things exists and the other does not. Notice also that a full account of quasi-identity would have to add some characterisation of which properties are essential for an object and which properties belong to an object only accidentally. Of course, this problem has already been discussed by ancient philosophers and cannot be discussed here in detail.
See also Anselm’s second reply to Gaunilo (Anselm of Canterbury 2008, p. 113).
In what follows, we use \(X=Y\) as shorthand for \(\forall x (Xx \leftrightarrow Yx)\).
From \((**)\), it follows that \(\forall _{\mathcal P} F (F \ne E! \rightarrow (Fa\rightarrow Fg))\). From the first conjunct of \((*)\), it follows by weakening that \(\forall _{\mathcal P} F (F \ne E! \rightarrow (Fg \rightarrow Fa))\) and, hence, \(\forall _{\mathcal P} F (F \ne E! \rightarrow (Fg \leftrightarrow Fa))\)—i.e. \(a \equiv _{E!} g\).
Therefore, instead of Realization, we would adopt \(\forall _{\mathcal P} \mathcal{F}\exists x \forall _{\mathcal P} F (\mathcal{F}(F) \rightarrow Fx)\).
Matthews (2004, p. 72).
Anselm, for instance, argues that id quo is in the understanding, and therefore it is conceivable that id quo exists in reality. If, by conceivability, Anselm here would mean conceivable by the fool, there would be no reason to accept this inference. After all, the restricted mental capacities of the fool might detain him from being able to conceive of the existence (in reality) of id quo, even though id quo is in his understanding.
Cf. footnote 30.
Hartshome (1962, pp. 49–51).
See Anselm of Canterbury (2008, p. 88).
A similar premiss is mentioned in Maydole (2009) as well as in Lewis (1970). Logan, on the other hand, thinks that such a principle is problematic, ‘since a chimera can be understood, although it cannot exist, since it is by definition a mythical beast. Nor does Anselm mean that understanding the term ‘a square circle’ involves the possibility of its actual existence’ (Logan 2009, p. 94). Here it is important to keep in mind the distinction between ‘being in the understanding’ insofar as the ‘word signifying’ is understood and ‘being in the understanding’ in the sense that the signified object itself is understood. (See our discussion in Sect. 2.5.) Logan’s counterexamples against the assumption that every understandable being conceivably exists make it reasonably clear that he takes the locution ‘can be understood’ in the first sense. However, as our earlier discussion should have made clear, this is not what PossEx is supposed to express (and it is, of course, not what Anselm is saying). So there is no need to disagree with Logan since the round square, though understandable in the first sense, cannot be understood in the second sense.
The informal discussion will draw on intuitions concerning possible world semantics; therefore, it should be read with a grain of salt. In particular, it does not commit us to attribute a view to Anselm, according to which—in Proslogion—he was arguing in terms of possible worlds.
The comparison principle from Sect. 3 Greater 2 \(\forall x \forall y (E!x \wedge \lnot E!y \rightarrow x > y)\) does not suffice to imply God!: Although Greater 2 is sufficient to produce a contradiction from the assumption of the non-existence of a God in the context of classical logic, it is not sufficient in modal logic in conjunction with Def M-God 1, ExUnd and PossEx. The reason for this is that ExUnd and PossEx assure us only of the possible existence of a God. Hence, it is consistent with Greater 2 that there is a God \(g\), which conceivably exists without existing actually. Greater 2 simply does not say anything about possibly existing objects. Moreover, there seems to be no reasonable modification of Greater 2 that does. Now, this version is not convincing for another reason that we have already discussed in earlier sections. Recall that, in (II.10), Anselm does not want to compare something with something else as to its greatness.
By ExUnd , we are given some \(g\) in such a way that (i) \(\lnot \exists z \lozenge (z > g)\). By PossEx , then, \(\lozenge E!g\). Suppose for reductio that \(\lnot E!g\). Then, by Greater 4, it follows from \(\lozenge E!g\) and \(\lnot E!g\) that \(\exists z \lozenge (z > x))\), thereby contradicting (i). Thus, we may conclude that \(E!g\); hence, \(\exists x (Gx \wedge E!x)\). A version which is similar to this reconstruction and which uses a premiss that is similar to Greater 4, can be found in Maydole (2009, p. 556).
The problem with Lewis (1970) is that his proposed reconstructions are framed in a non-modal language, where modal claims are reformulated by means of explicit quantification over possible worlds. However, it seems to us that Anselm’s argument should be rephrased in terms of a modal operator \(\lozenge \), corresponding to Anselm’s ‘it is conceivable that’, for it is only such an operator that shows up in Anselm’s actual argument. Lewis’ reconstructions, therefore, violate requirements (1) and (3).
In reaction to Viger (2002), who claims that Anselm’s argument would fall prey to Russell’s paradox, Nowicki (2006) presents another modal reconstruction of Anselm’s argument. Yet, Nowicki’s formulation does not make transparent what Anselm seems to be arguing for—namely, that God himself could be conceived to be greater, if we would assume that he does not exist. The same is true of the reconstruction provided in Morscher (1991), p. 68. In addition, Morscher’s reconstruction heavily relies on his conviction that Anselm’s id quo has to be rendered as a definite description, a view which we have already discarded earlier. A version which is similar to Morscher’s can be found in Jacquette (1997). Jacquette (1997) also contains a discussion of general aspects concerning intensionality in Anselm’s argument.
Russell (1905, p. 489).
Equivalently, \(\forall x (x = \mathfrak s(y) \rightarrow \lozenge (\mathfrak s(y) > x))\).
This definition essentially corresponds to premiss 3D of Lewis (1970, p. 180).
Consider the following constant-domain counter model \(\mathfrak M\): Let \(\mathfrak M\) consist of three possible worlds \(a\) (the actual world), \(v\) and \(w\). The domain of the model consists of three objects, \(1, 2\) and \(3\). We stipulate that \(1\) does not exist in the actual world \(a\), whereas \(2\) and \(3\) do, and furthermore, that \(1,2\) and \(3\) exist in each of the other worlds. We further stipulate that \(\mathfrak g_a (1)=1\); \(\mathfrak g_a (2)=2\); \({\mathfrak {g}}_a (3)=3\); \(\mathfrak g_w (1)=2\); \(\mathfrak g_w (2)=\mathfrak g_w(3)=1\); \(\mathfrak g_v (1)=2\); \(\mathfrak g_v (2)= \mathfrak g_v(3)=3\) and that \(\succ _a = \lbrace \rbrace \); \(\succ _w = \lbrace (2,1) \rbrace \) and \(\succ _v = \lbrace (3,1) \rbrace \). Clearly, \(1\) is aliquid quo in the sense of Def M-God 2 (whereas \(2\) and \(3\) are not), because for each world \(u\), \(\mathfrak g_u(1)\) is not exceeded (in \(u\)) by the greatness (in \(u\)) of any other object. Therefore, ExUnd is satisfied. Furthermore, since each object exists in some world, PossEx is satisfied as well. With respect to Greater 5, note that the actual greatness of \(1\) (the only non-existing object in \(a\)), viz., 1 itself, is exceeded (in \(w\)) by the greatness of \(1\) in \(w\) (=\(2\)) and the actual greatness of \(1\) is in \(v\) exceeded by the greatness of \(2\) and \(3\) in \(v\)—namely, \(3\). Hence, Greater 5 is satisfied as well, for each conceivably existing object can be conceived to be greater than the only non-existing object \(1\). Thus, each of the premisses is satisfied; yet, by the definition of the model, the only being than which nothing greater can be conceived (i.e. \(1\)), does not exist in the actual world \(a\).
In his Lewis (1970), Lewis argues that Def M-God 2 was no plausible explication of aliquid quo. According to Lewis, Def M-God 3 (or, rather, the premiss corresponding to our Def M-God 3), should be seen as the correct translation of aliquid quo. Recall that Def M-God 3 says that a being \(x\) is a God if \(x\)’s actual greatness is not exceeded by the greatness of any being in any possible world. The problem Lewis has with this is that it would give undue preference to the actual world over other possible worlds. Consequently, he thinks that, although the argument based on Def M-God 3 is valid, it does not establish the existence of a being reasonably to be called ‘God’, on the ground that there is no reason to prefer a certain possible world to some other. However, Lewis cites no reasons in Anselm’s writings in support of this view.
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Eder, G., Ramharter, E. Formal reconstructions of St. Anselm’s ontological argument. Synthese 192, 2795–2825 (2015). https://doi.org/10.1007/s11229-015-0682-8
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DOI: https://doi.org/10.1007/s11229-015-0682-8