According to Brian Leftow, Anselm of Canterbury’s best argument for the existence of God rests upon the assertion that a perfect being must exist necessarily if it exists at all. Further, and still, according to Leftow, Anselm offers us a metaphysical theory of the truth-makers of modal claims, which supports this argument by providing the modal-logical axioms required to turn the argument into a valid proof. In the book Anselm’s Argument: Divine Necessity, Leftow puts forward a reconstruction of Anselm’s argument and his modal metaphysics and argues at length that the principle at work, “Perfect Being Necessity,” is sound and unassailable. The book offers a thought-provoking and potentially widely useful discussion of modal metaphysics, which should appeal to any philosopher of religion with an interest in perfect being theology, modalities, and modern analytic metaphysics. The book is also a very challenging read.
The first few chapters of the book focus on Anselm. In chapters 1 and 2, his account of modality is reconstructed in modern terms based on textual evidence. We find in Anselm a metaphysics of modal notions based on powers inherent in real objects. Leftow argues that this modal metaphysics constitutes an account of absolute necessity and possibility specifically. Further, Anselm’s metaphysics can be modelled using standard modal logic with possible worlds semantics. In this logic, axiom 4 is required, and so a logic as strong as S4 is required. It is important to note in this context that while we may here talk about possible worlds, they should be considered merely a façon de parler since Anselm (like Leftow himself) is a realist but not necessarily a realist about worlds: the truth-makers of modal talk are real powers, not worlds. In chapter 3, Leftow raises and responds to some objections to the reconstructed account.
In Leftow’s view, Anselm’s modal metaphysics grounds and elucidates Anselm’s “ontological” argument, and in particular the Perfect Being Necessity principle which the argument includes as a premise. The argument in question Leftow draws not from the Proslogion but from Anselm’s Reply to Gaunilo. In chapter 4, Leftow presents a formal reconstruction of it and then of a different, strengthened or refined argument. In chapter 5, the last chapter to focus on Anselm apart from perhaps the very last chapter, chapter 17, Leftow argues that Anselm is committed to an S5 logic, since an S4 logic was already in place, and we need the Brouwer axiom to make a premise of the reconstructed argument (necessarily) true. As Leftow notes, if we have S5, we need only the Perfect Being Necessity principle and a premise saying that a perfect being is possible in order to have a valid argument to the conclusion that the perfect being exists (p. 80). From “if a perfect being exists, then it is necessary that it exists” and “it is possible that a necessary being exists” we may directly conclude that “a perfect being exists.”
The rest of the book focuses on Perfect Being Necessity. Chapters 6 to 17 provide a rich and wide-ranging discussion aimed at bolstering this principle: Chapters 6–12 defend it against objections, and 13–17 provide arguments for the principle. Among the objections against this principle that are discussed are Hume’s argument against necessary existence in general, Kant’s arguments against an absolutely necessary being, and Swinburne’s arguments against the notion of divine necessary existence as a case of absolute necessity. In chapters 13–17, Leftow develops several lines of argumentation in favor of Perfect Being Necessity, and in doing so engages with thinkers such as David Lewis, Timothy Williamson, Arthur Prior, David Armstrong, Alvin Plantinga, and others. For the most part he makes his case by arguing that the assumption of a contingently existing perfect being leads to problems. Throughout he uses possible worlds-talk. Metaphysical theories that ground modal talk in powers come up very rarely and only in passing. Similarly, Anselm does reappear in the text, but excepting a short discussion about divine aseity (pp. 216–217), he is absent until the final chapter, chapter 17. Chapters 14 and 15 seem, somewhat surprisingly, to be about Zeus and not about Anselm's God. In chapter 17, finally, Leftow argues in an Anselmian spirit that necessary existence is a perfection or (perhaps equivalently) that having it entails not having a number of “less-makers.”
The book is not an easy read. The organization of the material can be confusing, and although I appreciate the unexpected twists and turns in the arguments, I am not seldom thrown off track by not finding what I expect and what I sometimes sorely need in order to fully follow the discussion. More often than not in these cases, I find a footnote referring me either to Leftow’s earlier work God and Necessity (2012) or to a forthcoming book titled Anselm’s Proof. The former I have often found myself nearly “necessitated” to consult. The latter seems to me would have been exceedingly helpful had it existed.
The titular Anselmian parts of the book offer discussions that are full of thought-provoking and original points, but they also seem to me curiously preliminary or even somewhat incomplete. In his discussion of Anselm’s views on modalities, Leftow starts off with Anselm’s remarks on necessity as compulsion and possibility as power, but it is not clear how he then uses these to establish that Anselm is committed to a complete analysis of modal language that lines up with modern modal logic. Precisely on the critical point of the details and coherence of Anselm’s semantical theory, Leftow’s discussion turns tentative and hypothetical. Here more of Anselm’s own text, and also more of the existing scholarly discussion, would have been helpful. According to what I think we may call the standard reading, Anselm never develops a full systematic treatment of alethic modalities. The reason he does not is that he thinks statements of the form “it is necessary that […]” and “it is possible that [...]” are inherently vague. If Anselm is right, this means providing unequivocal truth-conditions for such statements is a non-starter. The best we can do – and this is what we find Anselm suggesting – is to try and sort out for each such statement the various (reasonable) interpretations under which the statement is true and false. Now as Leftow notes in footnotes (p. 23, n. 27; p. 28, n. 46), Anselm did apparently mean to treat modalities more systematically, and in his Lambeth Fragments we find what looks like the beginnings of such a project. On this topic, Eileen Serene’s pioneering study of the fragments is still relevant (1981), and Simo Knuuttila’s overview of Anselmian modalities in the Cambridge Companion to Anselm could also have been helpful (2004). For instance, a discussion of Knuuttila’s account of Anselmian partial powers could have helped clarify whether Leftow’s introduction of the notion of “primed” powers should be taken as a piece of Anselm interpretation or as an improvement on Anselm.
Leftow’s formalizations of Anselm’s Reply to Gaunilo argument raise similar concerns. Little is offered in terms of explanation as to how we get from Anselm’s text to the 4-premise, 8-line formalization offered. Nor is it entirely clear what motivates Leftow's move from this first formalism to a second, revised, 5-premise one. The latter is presented as an improvement on Anselm’s, and this is the one Leftow says he defends as his own. The two formalizations are intriguing. Notably, both employ quantified modal logic, but it is very unclear why this should be necessary. The first argument is as valid with the modal symbols removed, and so it is unclear why we should consider this a modal argument, strictly speaking (pp. 66–67). A more detailed account of how we get from Anselm’s text to the formalization would have helped to clarify the matter, I believe. The second, improved argument seems indeed to be a modal argument, but here it is unclear what job if any is done by the quantifiers. Only the two last premises seem necessary, and these get us a valid deduction if we have the Brouwer axiom, at least if we ignore the quantifiers. What we then get is the familiar argument in ordinary sentential modal logic from g → □g and ◊g to the conclusion g (where g = “God exists”). As mentioned above, Leftow admits as much at the end of the succeeding chapter 5, where he says that “Perfect Being Necessity, a possibility premise and Brouwer are enough” (p. 80). Here I think again some engagement with the literature would have been welcome. For instance, I believe that a discussion of Jaako Hintikka’s “Kant on Existence, Predication, and the Ontological Argument” would have been very helpful in sorting out the various issues involved in applying quantified modal logic to Anselm's argument (1981).
In conclusion, Anselm’s Argument: Divine Necessity presents a formidable but stimulating and rewarding read. At its best, it amicably and tirelessly challenges its readers to engage in debate, inviting them to scrutinize each argument, objection, and counterexample in wonderful detail. (And sometimes it provokes a frenzied search for more nuanced and charitable readings of the presented opponents’ arguments!) At its worst, it provokes a mixture of curiosity and frustration by pointing the reader to forthcoming publications. I think, however, that perhaps in the future the book will come into its own as the second installment of a trilogy, starting with God and Necessity and ending with Anselm’s Proof, hopefully and absolutely possibly making my criticisms obsolete.