TheoLogica An International Journal for Philosophy of Religion and Philosophical Theology Published Online First: August 09, 2019 DOI: https://doi.org/10.14428/thl.v3i2.14803 1 Platonism about Goodness-Anselm's Proof in the Monologion JEFFREY E. BROWER Purdue University brower@purdue.edu Abstract: In the opening chapter of the Monologion, Anselm offers an intriguing proof for the existence of a Platonic form of goodness. This proof is extremely interesting, both in itself and for its place in the broader argument for God's existence that Anselm develops in the Monologion as a whole. Even so, it has yet to receive the scholarly attention that it deserves. My aim in this article is to begin correcting this state of affairs by examining Anslem's proof in some detail. In particular, I aim to clarify the proof's structure, motivate and explain its central premises, and begin the larger project of evaluating its overall success as an argument for Platonism about goodness. Keywords: Anselm, Platonism, goodness, universals, God In the opening chapter of the Monologion, Anselm offers an intriguing proof for the existence of a Platonic form of goodness.1 This proof is intended to provide the starting point for Anselm's larger project in the work as a whole, which is to show just how much of traditional theistic belief can be justified on the basis of reason alone.2 Anselm takes this proof to provide the ideal starting point for his project 1 By a 'Platonic form', I have in mind a familiar type of property or universal-namely, one that is "separate from" (rather than "immanent to") its possessors. I will have more to say about how I'm understanding such forms in §3.2 below. But I should make clear up front that, although I follow a standard convention in referring to certain properties or universals as 'Platonic' (as opposed to 'Aristotelian), I don't mean thereby to be taking a stand on the views of the historical Plato (or Aristotle). For a classic discussion of the latter, see Fine (1994). 2 Compare the opening lines of the Monologion: If someone does not know, as a result of either not hearing or not believing, that there is one nature, supreme among all existing things, who alone is self-sufficient in his eternal happiness, who through his omnipotent goodness grants and brings it about that all other things exist or have any sort of well-being, and a great many other things that we must JEFFREY E. BROWER 2 precisely because of the close connection that he takes to exist between such a Platonic form (which he identifies with the ultimate source of goodness) and God himself (whom he identifies with the supreme good). 3 Indeed, we can think of Anselm's proof as setting the stage for a distinctively Platonic argument for God's existence that gets developed over course of the Monologion as a whole: The Platonic Argument for God's Existence (1) There is a Platonic form of goodness. (2) If there is a Platonic form of goodness, then it is the supreme good. (3) If the supreme good exists, then God exists. ∴ (4) God exists. Anselm devotes chapter 1 of the Monologion to establishing the first two premises of this argument, and he devotes much of the rest of this work (especially chapters 2– 28) to establishing the third premise. In this article, I want to examine in some detail Anselm's proof of premise (1)- that is, his proof of what I shall refer to as Platonism about goodness. In particular, I want to do three things: clarify the structure of Anselm's proof, motivate and explain its central premises, and begin the larger project of evaluating its overall success as an argument for Platonism about goodness. In future work, I hope to say something about the place of this proof in the broader context of the Monologion, and hence about the role it plays in Anselm's defense of the other premises of his theistic argument-namely, (2) and (3).4 But for reasons that will become clear, I will have my hands full here just with the examination of Anselm's proof of (1). believe about God or his creation, I think such a person could convince himself of most of these things by reason alone, if he is even moderately intelligent. (Mono 1; I.13.5–11) All translations of Anselm are taken, with slight modification, from Williams (1997). References to the Monologion are given by chapter, followed by volume, page, and line numbers of the edition in Schmitt (1968). 3 Thus, as Anselm continues in the text immediately following that quoted in n. 1: There are many ways in which one could do this [sc. convince oneself of various things about God by reason alone], but I shall set forth the one that I think would be easiest for him. After all, everyone desires to enjoy only those things that he thinks good. It is therefore easy for [even a moderately intelligent person] to turn the eye of his mind sometimes toward investigating the source of the goodness of those things that he desires only because he judges that they are good. Then, with reason leading and him following, he will rationally advance toward those things of which he is irrationally ignorant." (Mono 1; I.13.12–15) 4 One obvious question here has to do with how Anselm can think that God (who is a particular) can be identified with a Platonic form (which is a universal). Doesn't such an identification involve a category mistake? Although I do not specifically address this puzzle here, the conception of PLATONISM ABOUT GOODNESS 3 Anselm's proof of Platonism about goodness strikes me as extremely interesting, both in itself and for the light it sheds on other aspects of his philosophy-in particular, his theory of universals and his metaphysics of goodness. Although this proof has not been entirely neglected in the secondary literature, it has not yet received the attention it deserves. What is more, even the attention that it has received fails to do justice to the subtlety, sophistication, and argumentative force of the proof.5 In what follows, I hope to begin correcting this state of affairs. 1. Brief Overview of Anselm's Proof Despite the central place that Anselm's proof of Platonism about goodness occupies in the Monologion, he devotes very little space to developing and defending it. In the prologue, Anselm suggests that the central ideas of his Monologion can all be found, at least in seminal form, in Augustine's De Trinitate. And indeed we can find something like the seeds for Anselm's proof of Platonism about goodness, as well as for his larger Platonic argument for God's existence, in the following two paragraphs of De Trinitate 8: Certainly you love only the good, because the earth is good by the height of its mountains, the moderate elevation of its hills, and the evenness of its fields; and good is the farm that is pleasant and fertile; and good is the house that is arranged throughout in symmetrical proportions and is spacious and bright; and good are the animals, animate bodies; and good is the mild and salubrious air; and good is the food that is pleasant and conducive to health; and good is health without pains and weariness; and good is the countenance of man with regular features, a cheerful expression, and a glowing color; and good is the soul of a friend with the sweetness of concord and the fidelity of love; and good is the just man; and good are riches because they readily assist us; and good is the heaven with its own sun, moon, and stars; and good are the angels by their holy obedience; and good is the lecture that graciously instructs and suitably admonishes the listener; and good is the poem with its measured rhythm and the seriousness of its thoughts. universals that I attribute to Anselm (§1; especially n. 7) would seem to allow for its resolution, since it identifies universals in terms of a broadly functional role they play, and hence can allow for entities of different ontological types to qualify as universal. It is important to note that this puzzle must be distinguished from that which Anselm himself addresses in Mono 26–27-namely, the puzzle as to whether God can be identified with any Aristotelian form or universal (or indeed, with any type of being in the Aristotelian category of substance). This latter puzzle is briefly touched on in §3.5 below (see esp. n. 23). 5 This judgment applies, I'm sorry to say, even to my own previous work on the proof. See Brower (2004, esp. pp. 229–30). JEFFREY E. BROWER 4 Why should I add still more? This good and that good; take away this and that, and see good itself if you can; so you will see God who is good not by another good, but is the good of every good. For in all these good things, either those which I have enumerated, or any others which are seen or thought, we would be unable to call one better than the other, if we judge in accordance with the truth, if the idea of good itself had not been impressed upon us, according to which we approve of something as good, and also prefer one good to another. Thus God is to be loved, not as this or that good, but as good itself. (De Trin 8.3; trans. Matthews and McKenna 2002) In the first of these two paragraphs, Augustine provides an extensive list of familiar goods, including not only various types of substance (e.g., earth, sun, moon, stars, animals, human beings, angels), but also various types of accident (e.g., health, justice, proportionality) and various types of artifact (e.g., farms, houses, poems, and lectures). In the second paragraph, Augustine goes on to suggest that, starting with such familiar goods, we can infer to the existence of something like a Platonic form of goodness, which is identical to God himself. Anselm's proof of Platonism about goodness is apparently intended to clarify the basis of the Augustinian inference from good things to a form of goodness. Even so, Anselm's own proof is also extremely compressed, occupying just a single paragraph of Monologion 1. In order to get our bearings, it will be useful to start by quoting the relevant paragraph in full. For ease of reference, I will hereafter refer to this paragraph as 'Passage A' and number the lines of its text as follows: Passage A There are countless goods, whose great diversity we both experience through our bodily senses and discern through the rational faculty associated with our mind. But are we to 2 believe that there exists some one thing through which all goods whatsoever are good? Or are different goods good through different things? Indeed, to all who are willing to pay 4 attention, it is most certain that whenever things are said to be more or less or equally a certain way, as compared to one another, they are said to be so through something that is 6 understood not as different but rather as the same in the diverse things, whether it is detected equally or unequally in them. For whenever things are said to be just as compared 8 to one another (whether equally or more or less so), they cannot be understood to be just unless it is through a justice that is not different in the diverse things. Therefore, since it is 10 certain that all good things, if they are compared to one other, are either equally or unequally good, they must all be good through something that is understood to be the same in the 12 diverse goods, even if things sometimes seem to be called good through different things. (Mono 1; I.14.5–18). 14 There is a lot going on in this passage, much of which we'll only become clear as we proceed. But for now, let us focus on the overall structure of the passage, which PLATONISM ABOUT GOODNESS 5 is fairly straightforward. Anselm begins, in lines 1–2, by calling our attention to a specific class of things with which we are all familiar-namely, good things-and by remarking on the sheer number and diversity of its members. Unlike Augustine, Anselm doesn't bother to list any specific examples of such goods. On the contrary, he simply assumes that we will all know the sorts of things he has in mind (perhaps because he assumes that we will already be familiar with Augustine's own list). Next, in lines 2–4, Anselm asks us to reflect on the source of the goodness of this class's members. What accounts for the fact that they are all good, in spite of their "countless" number and "great diversity"? Finally, in lines 4–13, Anselm defends what he takes to be the only plausible answer to this question-namely, that there is "some one thing" that accounts for their goodness. As he recognizes, this is not the only possible answer that could be given. On the contrary, he notes that it could be suggested instead that "different goods [are] good through different things". Even so, Anselm thinks this suggestion can be ruled out once we appreciate a certain fact about the members of the class of good things-namely, that they can be "compared to one another" in respect of their goodness, and in particular as "more or less or equally" good. To a first approximation, we can represent Anselm's proof of Platonism about goodness as follows: Proof of (1)-Initial statement (1a) There is a class of good things, which contains members of many different types. (1b) The members of the class of good things can be compared to one another. (1c) If the members of the class of good things can be compared to one another, there must be some one thing through which they are good- namely, a Platonic form of goodness. ∴ (1) There is a Platonic form of goodness. As this initial statement is intended to make clear, Anselm's proof is similar in structure to certain standard realist arguments for universals.6 Like such arguments, Anselm's proof begins with the existence of a certain class of things (in this case, good things) and attempts to infer the existence of something shared in common by the members of this class (in this case, a Platonic form or universal of goodness). Like such arguments, moreover, Anselm's proof attempts to justify the inference by 6 For an introduction to standard arguments for realism about universals, see Armstrong (1989) and Loux (2006). JEFFREY E. BROWER 6 identifying some distinguishing feature of the class in question that requires a realist explanation (in this case, that the members can all be compared in a certain respect). In short, Anselm's proof appears to be an argument from good things to a Platonic form or universal of goodness. Although Anselm doesn't explicitly identify the Platonic form of his proof-that is, the "one thing" through which all good things are good-with a universal, it seems clear that this is the sort of entity he has in mind. For a universal, as standardly understood both now and in Anselm's time, just is an entity shared in common by the members of some class.7 Thus, if there is a universal humanity, it is an entity shared in common by the class of human beings; if there is a universal whiteness or goodness, it is an entity shared in common by the class of white things or good things. More generally and a little more precisely: In addition to clarifying the overall structure of Anselm's proof, our initial statement enables us to identify the main interpretive challenge that it poses-that of making sense of its central premises, (1b) and (1c). As we shall see, moreover, meeting this challenge will require getting clear about the answers to two different questions: 7 As it happens, this understanding corresponds closely to a standard medieval definition of universals that Anselm himself would have been familiar with-namely, Boethius's definition, according to which a universal is some one thing that is common to many, at the same time, and in such a way as to the constitute the nature or substance of its possessor. The definition occurs in Boethius's Second commentary on Porphyry's Isagoge, I.10 (trans. in Spade 1994, p. 22, ¶18). For discussion of the historical sources of this definition, which does not appear to be original to Boethius, see Spade 1996, pp. 3–6. Note that insofar as this definition characterizes universals as an entity that is somehow "one over many", it appears to be a broadly functional conception of universals-that is, one that could in principle be satisfied by entities of different specific ontological types, provided they can play the relevant nature-constituting role. Universals • A universal F-ness =def An entity that can be shared in common by members of some class of F things insofar as they are F. PLATONISM ABOUT GOODNESS 7 I refer to Q1 as the comparison question, since it asks about Anselm's understanding of what the comparison of good things consists in. Answering this question will be crucial for getting clear about the proper interpretation of (1b). As for Q2, I refer to it as the realism question, since it asks about the rationale for Anselm's realism about the comparison of good things. Answering this question will be crucial for getting clear about the proper interpretation of (1c). Determining the answers to these two questions-and hence the proper interpretation of the proof's central premises-turns out to be far from easy. Indeed, most of the discussion that follows will be devoted to exploring some initially promising answers, as well as the different interpretations to which they give rise, before settling on what I take to be the correct answers for each. But before examining the central premises of Anselm's proof, I want to say something about its starting point, since this will help to clarify Anselm's conception of goodness, which is something that will become important later on. 2. Starting Point of Anselm's Proof Recall the starting point of Anselm's proof: (1a) There is a class of good things, which contains members of many different types. I think it is clear from Anselm's own statement of the proof that he intends this premise to highlight something non-controversial or obvious (cf. again lines 1–2). Even so, there is a question about how to interpret (1a), due to ambiguity on the different "types" of good thing to which it refers. For when Anselm speaks of such types, he might have in mind different specific ontological types (or subcategories of being)-such as plants, animals, or human beings, which are all specific types of substance. On the other hand, when Anselm speaks of good thing as being of Two Questions Q1 What sort of fact does Anselm mean to be calling our attention to when he says that the members of the class of good things can be "compared" to one another? [= Comparison question] Q2 What is Anselm's rationale for thinking that a Platonic form or universal is required to account for the comparison of good things? [= Realism question] JEFFREY E. BROWER 8 different "types", he might have in mind different fundamental ontological types (or categories of being)-including not only substances, but also accidents, and perhaps even artifacts, events, and states of affairs, if there are such things. This ambiguity on 'types' gives rise to two different readings of (1a), one weaker and one stronger: Two readings of (1a) (1a*) There is a class of good things, which contains members of many different specific ontological types. [= Weaker reading] (1a†) There is a class of good things, which contains members of many different fundamental ontological types. [= Stronger reading] I call (1a*) the weaker reading because it leaves open the possibility that the members of the class of good things all belong to the same fundamental ontological type or category. I call (1a†) the stronger reading because it explicitly excludes this possibility. Although it is possible to interpret the starting point of Anselm's proof in terms of either reading, I think (1a) is best understood in terms of the stronger reading at (1a†)-and this for three reasons. First, the stronger reading fits best with Anselm's emphasis on the "great diversity" (multam diversitatem) to be found among good things. In medieval philosophical Latin, it is common to refer to things as "diverse" (diversae) when they belong to different Aristotelian categories of being, and as "different" (differentes) when they merely belong to different specific kinds within the same Aristotelian category. 8 Anselm's choice of terminology, therefore, is suggestive of (1a†). Second, the stronger reading fits best with the Augustinian basis for Anselm's proof. Indeed, Anselm's talk of "great diversity" seems calculated to put us in mind of Augustine's own list of familiar goods, which itself ranges over things belonging to fundamentally different ontological types or categories of being-including substances, accidents, and artifacts. Finally, the stronger reading seems to fit best with the ultimate goal of Anselm's argument, which is to show that there is a supreme good-that is, something that can be regarded as best in relation to all other existing things, regardless of their specific ontological type or category. The proper interpretation of (1a) turns out to be relevant for understanding the conception of goodness at work in Anselm's proof as a whole. Indeed, understanding (1a) in terms of the stronger reading at (1a†) helps us to see that the relevant conception is not to be understood in terms of any sort of kind-relative 8 Cf. Aquinas, ST 1.3.8 ad 3 and 90.1 ad 33, which cites Aristotle's Metaphysics as clarifying the strict and proper use of these terms (or their Greek equivalents). PLATONISM ABOUT GOODNESS 9 goodness but must rather be understood in terms of absolute goodness, where these two conceptions of value can be contrasted as follows: If what I've just said is correct, then the conception of goodness that Anselm is working with in his proof must be distinguished from a familiar Aristotelian conception, according to which goodness is to be understood in terms of what is required to be fully developed (i.e., completely actualized) as a member of some kind.9 On this conception, there might or might not be any forms or standards of goodness. But to the extent that there are, such standards will presumably be different for different kinds of things. Indeed, we might think of such kind-relative standards along the following lines: It is precisely because Anselm is working with an absolute (as opposed to any sort of kind-relative) conception of goodness that he takes his proof of Platonism about goodness to establish the existence of a single standard for all good things (rather than different standards for different types of good). Indeed, as we shall eventually see, Anselm's proof is intended to establish the existence of an absolute 9 For an introduction to this conception, as well as the place it occupies in medieval philosophy, see MacDonald (1990b). Two Conceptions of Goodness • Kind-relative goodness =def A type of value that applies only to the members of a specific ontological type or category of being (i.e., goodness for some type of thing). • Absolute goodness =def A type of value that is not kind-relative but can in principle be applied to any existing thing, regardless of its specific ontological type or category (i.e., goodness as such). Kind-Relative Standards of Goodness • There are different (perhaps even incommensurable) standards of goodness for different kinds of things (since what is required for complete development or actuality varies across such kinds); and • The standard of goodness for a given kind is determined by (perhaps even reducible to) some nature shared in common by the members of that kind. JEFFREY E. BROWER 10 (as opposed to a kind-relative) standard of goodness, to be understood along the following lines: As we shall also eventually see, it is precisely because such an absolute standard cannot be determined by (much less reduced to) any single common nature that it must be regarded as a Platonic (rather than an Aristotelian) form or universal-that is, as one "separate from" (rather than "immanent to") its possessors.10 So much for the starting point of Anselm's proof, and what it reveals about the conception of goodness with which he is working. Let us now turn to the proof's central premises and the main interpretive challenge they pose. 3. Central Premises of Anselm's Proof As already noted, the main challenge or difficulty posed by Anselm's proof has to do with the proper interpretation of its central premises: Central premises (1b) The members of the class of good things can be compared to one another. 10 It is important to note that, although Anselm is not working with the familiar Aristotelian conception of goodness in his proof, his own preferred conception is not strictly incompatible with it. For there is nothing in Anselm's proof that prevents him from admitting kind-relative goodness in addition to absolute goodness. Indeed, for all his proof says, it may be that he is thinking of both types of goodness in terms of what is required for completeness (or full actuality). For just as we can compare things with respect to their completeness or actuality as members of some kind, so too it would seem that we can compare them with respect to their completeness (or full actuality) as such. For example, we might judge a substance to be more complete (or fully actual) than an accident, insofar as substances lack the dependency associated with accidents; likewise, we might judge a quantity or a quality to be more complete (or fully actual) than a relation, insofar as quantities and qualities depend on a single subject, whereas relations depend on multiple subjects. Absolute Standard of Goodness • There is a single standard (or common measure) of goodness for all good things, regardless of their specific ontological type or category; and • This standard cannot be determined by (much less reduced to) any nature, since there is no nature shared in common by the members of different fundamental ontological types or categories. PLATONISM ABOUT GOODNESS 11 (1c) If the members of the class of good things can be compared to one another, then there must be some one thing through which they are good-namely, a Platonic form of goodness. In what follows, I examine two initially promising interpretations of these premises, as well as some slight modifications of them, before settling on what I take to be their correct interpretation. 3.1. First interpretation Recall the two questions that I suggested earlier must be answered if we are to get clear about the proper interpretation of (1b) and (1c): We can, I think, begin to make progress in answering these two questions if we start by considering how standard realist arguments for universals proceed. As noted earlier, such arguments often attempt to identify some important fact about the members of a given class-one that distinguishes their class from other classes and requires explanation in terms of a universal. In the case of such arguments, the relevant fact is often understood in terms of objective resemblance. For as realists often point out, when the members of some class objectively resemble one another in some respect, they seem to form a special sort of class-what are often called natural classes or kinds: Unlike mere class membership, which can result from any arbitrary grouping or Two Questions Q1 What sort of fact does Anselm mean to be calling our attention to when he says that the members of the class of good things can be "compared" to one another? [= Comparison question] Q2 What is Anselm's rationale for thinking that a Platonic form or universal is required to explain the comparison of good things? [= Realism question] Kinds • A kind, F things =def A class whose members objectively resemble one another F-wise. JEFFREY E. BROWER 12 collection, kind membership is something that seems to cry out for explanation. For insofar as the members of some class objectively resemble one another, and hence form a kind, they seem to possess a sort of special sort unity-one that is both lacking in the case of mere classes and is naturally explained in terms of a universal. Indeed, it is precisely for this reason, that realists often insist that there is universal shared in common by the members of the class of blue or green things, but not by the members of Goodman's famous class of grue things (i.e., things which are either green before some future time t or blue thereafter). For unlike blue or green things, grue things do not form a kind. Although Anselm doesn't speak of objective resemblance, his notion of comparison seems to be closely connected to it. For just as things that resemble one another must resemble in a certain respect, so too things that are compared to one another must be compared in a certain respect. Strictly speaking, Anselm takes comparison to be something subjective (or mind-dependent)-namely, a mental act or judgment. Even so, his appeal to comparison in the context of his proof is calculated to call our attention to its objective (or extramental) basis. For when Anselm speaks of our ability to compare good things in respect of their goodness, he is calling attention to our ability to do so truly or in a way that tracks extramental reality. But, of course, if that is right, then to say that two or more things can be (truly) compared in some respect just appears to be another way of saying that they objectively resemble in that respect, and hence form a kind (as opposed to a mere class). All of this suggests a first solution to our interpretive challenge. For if we understand comparison in terms of objective resemblance, we have an answer to our comparison question. The fact that Anselm is highlighting when he speaks of such comparison is kind membership. What is more, this understanding of comparison provides us with a straightforward rationale for his realism about comparison, and hence an answer to our realism question. For universals are often taken by realists to be required to explain kind membership. In short, these answers provide us with our first interpretation of the central premises of Anselm's proof: Central premises-First interpretation (1b') The members of the class of good things form a kind, good things. (1c') If good things is a kind, then there must be a universal through which its members are good-namely, a universal of goodness. PLATONISM ABOUT GOODNESS 13 3.2. Slight modification of the first interpretation Despite the initial attraction of this interpretation, I don't think it can be correct as it stands. In particular, the sort of realism that it attributes to Anselm doesn't quite support the conclusion that I described as the target of his proof-namely, that there is a Platonic form or universal of goodness. For even if all Platonic forms are universals, it is not the case that all universals are Platonic forms. On the contrary, as hinted earlier, Platonic forms are universals of a specific type-they are "separate from" their possessors, and hence must be regarded as transcendent (as opposed to immanent) universals. Although it is possible to understand the distinction between transcendent vs. immanent universals in different ways, for our purposes we can understand it as follows: Insofar as Anselm ultimately wants to identify the form or universal of goodness with God, I think it is clear that he must be thinking of it as a transcendent (as opposed to an immanent) universal. But if that's right, then the realist principle specified at (1c') is too broad. There might seem to be an easy solution to this problem. Just slightly modify our understanding of the realist principle at work in Anselm's proof. That is to say, instead of thinking of Anselm's rationale for realism about comparison merely in terms of a universal, think of it instead in terms of a specifically transcendent universal. In that case, the problem with our first interpretation can be avoided by slightly modifying our statement of the principle articulated in (1c'): Central premises-First interpretation (slightly modified) (1b') The members of the class of good things form a kind, good things. (1c'') If good things is a kind, then there must be a transcendent universal through which its members are good-namely, a Platonic form of goodness. Two Types of Universal • Transcendent universal =def A universal that is wholly distinct from its possessors. • Immanent universal =def A universal that is a constituent part of its possessors. JEFFREY E. BROWER 14 Although this slight modification certainly gets us to the right conclusion, it does so at the cost of spoiling much of the force of Anselm's overall argument. For even if one grants that kind membership must be explained in terms of universals of some sort, it is not at all clear that it must be explained in terms of transcendent universals. On the contrary, realists about universals standardly assume (almost to a person) that insofar as kind membership is to be explained in terms of universals at all, it must be explained in terms of immanent (rather transcendent) universals.11 Despite weakening the philosophical force of Anselm's argument, this slight modification might appear to have a lot going for it as an interpretation of the sort of realism at work in his proof. Indeed, insofar as Anselm is often described as a Platonist about universals, we might well expect him to endorse such Platonism about kind membership.12 In previous work I interpreted Anselm's views along just such Platonistic lines.13 But I'm now convinced that such an interpretation is mistaken for two reasons. First, I've become convinced, on independent grounds, that in the case of most kinds- including those associated with the ten Aristotelian categories-Anselm not only rejects Platonic realism about universals, but actually accepts something much more like Aristotelian (or immanent) realism.14 Second, I think there are good textual grounds, even in the context of Anselm's proof in Passage A, for thinking that his Platonism is restricted to kind membership of a very specific sort. I cannot pursue the first reason any further here. But fortunately there is no need to. For the force of the second reason will become clear almost as soon as we recall some of the things that Anselm explicitly says in his proof about comparison. And this, in turn, will lead us to a second interpretation of (1b) and (1c). 11 No doubt this is due to the influence of David Armstrong (esp. 1978 and 1989) and David Lewis (esp. 1983). But see Audi (forthcoming), which argues that it is only kind membership based on objective resemblance in some intrinsic respect that requires explanation in terms of an immanent (as opposed to a transcendent) universal. 12 This common description of Anselm is not surprising given the influence of Augustine on his thought. For an attempt to locate Anselm's Platonism about universals within the context of a broader Platonic-Augustinian metaphysics and epistemology, see Rogers (1997). 13 See again Brower (2004). 14 There is, in fact, a growing consensus among scholars that Anselm's theory of universals must be understood at least partly in terms of the type of immanent realism that was common during his time (roughly, the 11th and 12th centuries). For a detailed analysis of the history and sources of early medieval theories of universals, see Erismann (2011). For a briefer overview, with special reference to Anselm and the figures active around his time, see Erismann (2003) and (2007). See also the chapters by Marenbon and King in Brower and Guilfoy (1994). PLATONISM ABOUT GOODNESS 15 3.3. Second interpretation There can, I think, be little doubt that Anselm's notion of comparison is closely connected to objective resemblance. Indeed, I suggested earlier that Anselm thinks of the connection in terms of something like the following principle: But if we return to what Anselm specifically says about the comparison of good things in Passage A, we can see that he has more than mere objective resemblance in mind. For he doesn't say merely that good things can be (truly) compared to one another insofar as they are good, but rather than they can be (truly) compared to one another insofar as they are "more or less or equally" good (cf. esp. lines 4, 9, 11). The qualification here is significant. For it suggests that comparison is to be understood not merely in terms of objective resemblance, but rather in terms of objective resemblance in some respect that admits of variation in degree. But this introduces a substantive restriction on the sorts of kinds at issue in Anselm's proof. For not all kinds have members whose respect of resemblance admits of such variation. Consider, for example, the kind human beings. Although the members of this kind objectively resemble in a certain respect-namely, species-wise-their respect of resemblance does not admit of variation in degree. On the contrary, it is an all or nothing affair. That is to say, things are not more or less human; they're simply human or not (full stop).15 What Anselm says about good things suggests that he takes their respect of resemblance to differ in this regard-and rightly so. For resemblance in respect of goodness, or value-wise, isn't an all or nothing affair (or at least it needn't be). On the contrary, things can be (and often are) more or less good. But if that's right, then when Anselm says that good things can be compared to one another, he is not merely highlighting the fact that they form a kind, but rather they form a kind of a distinctive sort-what we might call a degreed kind: 15 Here I am taking for granted the broadly Aristotelian conception of human beings that Anselm would have been familiar with from the Categories. See in particular Cat. 5, 3b32–4a10, where Aristotle argues that substances do not admit of variation in degree. Comparison as Objective Resemblance • Things can be (truly) compared to one another insofar as they are F iff they objectively resemble one another F-wise. JEFFREY E. BROWER 16 All of this suggests another interpretation of the central premises of Anselm's proof-one in which his talk of comparison is specifically associated with membership in degreed kinds and his Platonism about universals is specifically restricted to membership in these same kinds: Central premises-Second interpretation (1b*)The members of the class of good things form a degreed kind, good things. (1c*) If good things is a degreed kind, then there must be a transcendent universal through which its members are good-namely, a Platonic form of goodness. I think this interpretation does a much better job than previous ones of capturing the notion of comparison at work in Anselm's proof. What is more, the restricted form of Platonism it attributes to Anselm-namely, Platonism about membership in degreed kinds-seems much more plausible than Platonism about kind membership in general. To see why, consider first a non-degreed kind such as human beings. One could insist that, in order to explain the objective resemblance or unity associated with this kind, we must appeal to a transcendent universal of humanity. But that seems implausible-or at very least, easy to resist. For we could appeal instead to an immanent universal of humanity, which would seem equally, if not better, suited to explaining the relevant sort of unity. Or perhaps we could appeal to particular properties or tropes of humanity, which might go some distance toward explaining such unity, even if this appeal requires taking resemblance among tropes to be brute or primitive. Or perhaps we could simply insist that species-wise resemblance among human beings is brute or primitive, and hence deny that there is any explanation at all to be given of the unity associated with their kind.16 16 Armstrong (1978, vol. I, 16) calls those who refuse to offer an explanation or analysis of objective resemblance 'ostrich nominalists' in order to suggest that they are putting their head in the sand. But compare Lewis (1983, 352): "The ostrich that will not look at it [sc. the demand for an explanation where none is required] is a wise bird indeed". It is worth noting that even some realists about universals deny that objective resemblance can be explained, and hence qualify as "ostriches" in Degreed Kinds • A degreed kind, F things =def A class whose members objectively resemble one another F-wise, where this respect of resemblance admits of variation in degree. PLATONISM ABOUT GOODNESS 17 By contrast, consider now a degreed kind such as flat things-that is, the class of things that are more or less or equally flat. What must be accounted for in the case of a kind such as this one is not merely the fact that its members objectively resemble in a certain respect (say, flat-wise)-but also the fact that their resemblance in this respect admits of variation in degree. And unlike mere objective resemblance, it is difficult to see how to account for such degreed resemblance without appealing to something like a Platonic form. For such resemblance seems to involve a relation to a common standard or ideal-in this case, of perfect flatness. Indeed, it is hard to see what it could even mean to say that one thing is, say, flatter than another if not that it lives up to the standard or ideal of perfect flatness more fully than the other.17 But if that is right, then it would seem that resemblance in respect of flatness, or any other degreed respect of resemblance, just is to be analyzed or explained in terms of a relation to something like a Platonic form. It is worth pausing briefly to note that, on this second interpretation, Anselm's proof provides us with a distinctive line of argument for realism about universals- one whose virtues have yet to be appreciated in the context of contemporary debates. As just noted, the standard realist demand for explanation of mere objective resemblance will have no force against those who either deny that such resemblance requires explanation or else think that it can be explained (at least partly) in terms of something other than universals (e.g., tropes). But even such thinkers may still feel the force of Anselm's realist demand for explanation of degreed resemblance. For as was also just noted, it is hard to see how to make sense of degreed resemblance without appealing to universals. In fact, as our discussion of flat things helps to make clear, to say that two things objectively resemble each other in some respect that admits of variation in degree is apparently just to say that there exists some third thing, a common standard or ideal, that the original two things approximate to some degree. But, if that is right, then unlike mere objective resemblance, degreed Armstrong's sense. See, e.g., van Inwagen (2011), who refers to his own preferred realism as ostrich Platonism (since it postulates transcendent universals not to explain kind membership but rather to serve as the meanings of predicate expressions or abstract singular terms). 17 It's worth noting an ambiguity on the notion of a standard or ideal of perfect F-ness. In the case of flatness, for example, we might think of the relevant standard or ideal in terms of the shape of a plane in Euclidean space, even if it turns out to be impossible for any object to occupy a region with such a shape. On the other hand, we might think of the relevant standard or ideal instead in terms of the shape of the flattest possible object, regardless of whether this shape corresponds to that of a plane in Euclidean space. In some contexts, the distinction between these two notions-what we might call the in principle vs. individual-relative notions of a standard or ideal-is important. But for the purposes of Anselm's proof we can ignore it, since establishing the existence of a standard or ideal of either type will be sufficient to guarantee some form of Platonism. JEFFREY E. BROWER 18 resemblance is something that must be explained in terms of a relation to a universal-indeed, a Platonic form or universal. 3.4. Slight modification of the second interpretation Despite the initial promise of this second interpretation, I think it can't be right as it stands either. In fact, the realist principle that it attributes to Anselm is still too broad to be acceptable as an interpretation of his proof. To see why, we must first appreciate that, although there appear to be different types of degreed kinds, the realist principle specified at (1c*) is plausible only for one of them. Consider again the kind flat things. This kind appears to be not only a degreed kind, but what we might call a maximal degreed kind-or a maximal kind for short. And this is because the respect in which flat things resemble not only admits of variation in degree but also possesses an intrinsic limit or maximum-namely, a degree of flatness than which nothing can be flatter. Indeed, it is precisely this intrinsic limit or maximum that is specified by the standard or ideal of perfect flatness. Thus, when we say that one thing is flatter than another, we are in effect just saying that one thing more closely approximates the intrinsic maximum of flatness than the other.18 It is not obvious that all degreed kinds possess an intrinsic maximum. Consider, for example, kinds such as white things or hot things. These are clearly degreed kinds, but it is not obvious that either possesses an intrinsic maximum. For no matter how white or how hot something is, it would seem that it could always be a little whiter or a little hotter (even if our sensory faculties are incapable of registering increases in whiteness or heat beyond a certain point). But, of course, if that's right, then it would make no sense to speak of standards or ideals that specify intrinsic maxima for these kinds. And likewise for other merely degreed kinds. Anselm doesn't show any awareness of the distinction between maximal vs. merely degreed kinds, either in Passage A or in any other texts that I am aware of. This may well be because he doesn't think there are any degreed kinds that lack intrinsic maxima. I say this because, at least in the case of white things and hot things it was standard, among medieval philosophers, to think that they possess intrinsic maxima.19 It wouldn't be surprising, therefore, if Anselm himself accepted this line 18 Since intrinsic maxima are specified by standards or ideals of perfection, the same ambiguity will arise for them that arose in the case of standards or ideals. That is to say, we can distinguish in principle vs. individual-relative intrinsic maxima, depending on the specific type of standard or ideal of perfection they are associated with. See again previous note. 19 See, e.g., Aquinas's discussion of hot things in ST 1.2.3 ("the hotter something is the closer it approximates that which is maximally hot"). See also his discussion in, e.g., SCG 1.43 of the neoPLATONISM ABOUT GOODNESS 19 of thought, taking it to apply to degreed kinds in general. But even if that is true, it seems most likely that the fact about degreed kinds that Anselm's talk of comparison is meant to highlight is their maximality, not their mere variation in degree. For it is only their maximality that seems to require explanation in terms of a Platonic form or standard, and hence makes it plausible to suppose (as Anselm does) that comparative judgments about such kinds must make reference to, and hence quantify over, a Platonic a form or standard. All of this suggests a slight modification to our second interpretation, one in which we understand the central premises of Anselm's proof in terms of maximal rather than merely degreed kinds. For the sake of completeness, we can understand maximal kinds as follows: And then, relying on this characterization, we can state the relevant modification of our second interpretation as follows: Central premises-Second interpretation (slightly modified) (1b**) The members of the class of good things form a maximal kind, good things. (1c**) If good things is a maximal kind, then there must be a transcendent universal through which its members are good-namely, a Platonic form of goodness. With this interpretation we have nearly arrived at the proper understanding of the notion of comparison at work in Anselm's proof, as well as the realist principle on which this same proof relies. I say 'nearly arrived' because, as it happens, even the restricted form of Platonism to which (1c**) commits Anselm turns out too broad to capture his actual views. Platonic axiom "whatever is received is received according to the mode of the receiver", which he takes to govern the degree to which things of various kinds are capable of participating in certain qualities (including whiteness and heat). For a detailed treatment of Aquinas's understanding of this axiom, see Wippel (2007). Maximal Kinds • A maximal kind, F things =def A class whose members objectively resemble one another F-wise, where this respect of resemblance not only admits of variation in degree but also possesses an intrinsic maximum. JEFFREY E. BROWER 20 To see what the problem is here, as well as to prepare the way for the final interpretation of his proof, consider first what the truth of (1c**) would require- namely, there exists a Platonic form or standard for every maximal kind. But that's not obviously true, even in the case of flat things. Indeed, for reasons I shall now explain, it would seem that even if membership in a maximal kind requires the existence of some sort of standard or ideal, we needn't think of it as a specifically Platonic form or universal. To see why, consider a class of things whose members are good in some kindrelative sense-say, the class of things that are more or less good as members of the kind human beings. The members of this class surely resemble one another objectively, and in a respect that both admits of variation in degree and possesses an intrinsic maximum. Indeed, it is hard to see what it could mean to say that one person is better than another as a human being if not that such a person lives up to the standard or ideal of perfection for human beings more fully than the other. But as we noted in our discussion of the starting point of Anselm's proof, in cases of kind-relative goodness, it is not implausible to think that the standards associated with them are always determined by (perhaps even reducible to) some nature that is shared in common by the members of the relevant kind. Thus, in the specific case of human beings, we might say that the standard of perfection is determined by (perhaps even reducible to) human nature itself.20 And presumably we could say something similar in the case of the standard of flatness, which would seem to be determined by (perhaps even reducible to) some nature common to all flat things (say, being a material object, which entails the possession of certain geometrical properties).21 And so on in the case of the standards associated with many other maximal kinds. But, of course, if this is right, then we needn't think of the standards associated with every maximal kind in terms of a Platonic form or universal-that is, one "separate from" or "transcendent to" its possessors. On the contrary, in the case of many such kinds we can think of their associated standards in terms of an Aristotelian form or universal-that is, one that is just as "immanent to" its possessors as their nature is. 20 This will be especially plausible if we follow medieval Aristotelians in conceiving of human nature in terms of a distinctive set of powers or capacities, whose perfection just consists in their manifestation or actualization. See MacDonald (1990a) for helpful discussion of this conception of human nature in Aquinas. 21 It might be suggested that when we make comparative judgements about flat things, we are doing so merely with respect to somethings that exists in the mind-say, an idea or concept of perfect flatness. But note that this suggestion is inconsistent with the commonsense assumption, taken for granted throughout our discussion, that flat things resemble one another objectively, and hence that our comparative judgements about them do not depend for their truth on the existence of any minddependent beings such as ideas or concepts. PLATONISM ABOUT GOODNESS 21 In short, even if (true) comparative judgments about flat things, or the members of other maximal kinds, appear to make reference to (and hence quantify over) a specifically Platonic form or ideal, such reference (or quantification) case can often be paraphrased away-at least in terms of something like an Aristotelian form or universal. 3.5. Final interpretation So far so good. But how is all of this relevant to the proper interpretation of the central premises? To answer this question, we must return to the starting point of Anselm's proof. In particular, we must recall that the conception of goodness with which this proof is operating is an absolute (as opposed to a kind-relative) one. As we have seen, when Anselm speaks of comparison in the context of good things, he is not thinking of a class whose members all belong to the same specific ontological type or category. On the contrary, he is thinking of a class whose members belong to fundamentally different ontological types or categories of being. That is to say, when Anselm speaks of the comparison of good things, he is thinking of them as members of what we might call a category-neutral kind: But if Anselm is thinking of good things as forming a kind of this sort, then the central premises of his proof must be interpreted accordingly-which brings us to our final interpretation: Central premises-Final interpretation (1b†) The members of the class of good things form a category-neutral maximal kind, good things. (1c†) If good things is a category-neutral maximal kind, then there must be a transcendent universal through which its members are good-namely, a Platonic form of goodness.22 22 Like other medieval philosophers, Anselm assumes that there is a necessary connection between being and goodness, and hence that good things does not merely qualify as a category-neutral kind but also as what we might call a transcendental kind-that is, one whose members include every Category-Neutral Kinds • A category-neutral kind =def A kind whose members belong to different fundamental ontological types or categories of being. JEFFREY E. BROWER 22 With this interpretation, we now finally have what I take to be the proper understanding of the central premises of Anselm's proof, including both the notion of comparison at work in premise (1b), as well as the realist principle at work in (1c). What is more, with this understanding of the central premises, we can finally appreciate what's really motivating Anselm's Platonism about goodness. As we have seen, standards of some sort appear to be required to account for membership in maximal kinds. For comparative judgments about the members of such kinds appear to make reference to standards, and hence quantify over them. But as we have also seen, at least in the case of certain maximal kinds, these standards can often be explained in terms of some nature (or immanent property) shared in common by their members-and hence in terms of an Aristotelian (rather a Platonic) form or universal. But the same cannot be said in the case of categoryneutral maximal kinds. For insofar as their members belong to different fundamental ontological types, there will no common nature (or immanent properties) for the relevant standards to be explained in terms of. Indeed, like other philosophers of his time, Anselm thinks that it is only things belonging to the same fundamental ontological type or category-indeed, to one of Aristotle's ten categories of being-that can be said to share a common nature (or immanent property).23 As Anselm sees it, therefore, the standards associated with categoryneutral maximal kinds must be regarded as wholly distinct from their members, and hence any apparent reference to (or quantification over) such standards must be explained in terms of a specifically Platonic form or transcendent universal.24 existing thing. I will briefly return to this common medieval assumption in §4.1. But here it is important to note two things. First, the assumption that good things form a transcendental kind seems perfectly coherent. For, in principle, there is nothing to prevent all things from objectively resembling in some respect (say, good-wise or even being-wise) and hence forming a natural class or kind that includes every existing thing (say, good things or beings). Second, even if the assumption that good things form a transcendent kind were incoherent, that would be irrelevant from the perspective of Anselm's proof. For this assumption plays no role in the proof. On the contrary, Anselm relies in Mono 1 only on the commonsense assumption about the diversity of good things mentioned in (1a), and doesn't even begin developing the more general connection between being and goodness until Mono 3. 23 Indeed, it precisely because God does not share any nature (or immanent property) in common with any creaturely substances or accident that Anselm argues in Mono 26 that God cannot belong to any of the ten Aristotelian categories, but must instead belong to a fundamental ontological type of his own. 24 Although Anselm's appeal to Platonism about category-neutral kinds in the Monologion is initially restricted to good things (ch. 1), he immediately proceeds to extend this same appeal to other kinds-in particular to great things (chs. 1–2) and existing things or beings (chs. 3–4). Thus, his proof of PLATONISM ABOUT GOODNESS 23 4. A Start at Evaluating Anselm's Proof If what I have said about Anselm's proof to this point is correct, we can state the proof itself, in its final form, as follows: Proof of (1)-Final statement (1a†) There is a class of good things, which contains members of many different fundamental ontological types. (1b†) The members of the class of good things form a category-neutral maximal kind, good things. (1c†) If good things is a category-neutral maximal kind, then there must be a transcendent universal through which its members are good-namely, a Platonic form of goodness. ∴ (1) There is a Platonic form of goodness. It should be clear from this statement that Anselm's proof of Platonism about goodness is valid. When it comes to evaluating its soundness, therefore, the real question has to do with the truth of its premises. Let us briefly consider what can be said on behalf of these premises, beginning with the first two. 4.1. Premises (1a†) and (1b†) The first two premises of Anselm's proof are interesting, in part, because they serve to highlight a series of questions (five in total), each of which corresponds to an important decision point regarding the nature of goodness. The first two questions are the following: Platonism about goodness turns out to be important for understanding his Platonism about certain other forms or universals as well. Two questions about goodness (i) Is there a class of good things? (ii) Assuming there is a class of good things, do the members of this class form a kind? JEFFREY E. BROWER 24 In premises (1a†) and (1b†), Anselm offers affirmative answers to each of these two questions, thereby committing himself to a plausible form of realism about goodness. Anti-realists of various sorts will, of course, want to challenge these answers. Thus, eliminativists about value will want to challenge Anselm's answer to (i), and subjectivists about value will want to challenge his answer to (ii). Even so, both of Anselm's answers here strike me as compelling, if not utterly secure. In addition to (i) and (ii), the first two premises of Anselm's proof also serve to highlight three further questions about the nature of goodness: Here again, in premises (1a†) and (1b†), Anselm offers affirmative answers to each of these three questions, thereby committing himself to the further view that goodness is not only a degreed property possessing an intrinsic maximum, but also one that is distributed across fundamental ontological types or categories. Anselm's answers to questions (iii)–(v) are much more controversial than his answers to questions (i) and (ii). Indeed, his answer to question (iii) is, perhaps, the most controversial of all. For to insist that good things is a degreed kind is, in effect, to insist that that there is some common scale by which the degree of goodness of all good things can be measured. But this, in turn, just seems to imply that the goodness of all good things is commensurable-a thesis which is a highly controversial.25 Anselm's answer to question (iv) is also controversial, insofar as it takes for granted that goodness is a property possessing an intrinsic maximum. This view is not obviously false, but neither is it obviously true.26 Indeed, one might well think of goodness as property that comes in measurable units (such as utils or hedons), and hence think of comparative judgments about good things as making claims 25 See Hsieh (2016), as well as the references cited therein, for an introduction to some of the controversy surrounding this thesis. See also Nagasawa (2017, ch. 2) for discussion of how this controversy specifically bears on questions about divine goodness. 26 See again Nagasawa (2017, ch. 2), and the references cited therein, for some discussion of the controversy surrounding the maximality of goodness, especially as it bears on the project of perfect being theology. Three further questions about goodness (iii) Assuming there is a kind, good things, is it degreed? (iv) Assuming there is a degreed kind, good things, is it maximal? (v) Assuming there is a maximal kind, good things, is it categoryneutral? PLATONISM ABOUT GOODNESS 25 about the number of such units they possess (where there is, in principle, no upper limit on this number). As for Anselm's answer to question (v), insofar as it presupposes affirmative answers to both (iii) and (iv), it will of course inherit whatever is controversial about them. But insofar as we think of Anselm's answer to this question as insisting only on the category-neutrality of goodness, it does not strike me as particularly controversial. Indeed, as we noted at the outset of the article, the idea that the class of good things includes members from fundamentally different ontological types or categories appears to be a part of commonsense. As far as I know, Anselm never explicitly addresses the possibility that good things might be incommensurable or that goodness might lack an intrinsic maximum. Perhaps this is because, as I noted previously, he thinks of all degreed properties as possessing an intrinsic maximum. But perhaps it is also due to the fact that he is operating in a historical context in which each of the following is generally taken for granted: (a) there is a necessary connection between being and goodness (so that everything that exists is good); (b) all beings can be ranked in terms of their goodness (so that reality can be thought of as forming a great chain of being); and (c) there is at least the possibility of a supreme being (so that, in principle, there could be a being which is best in relation to all other things). Obviously medieval views about degreed kinds and (a)–(c) can no longer be taken for granted.27 Even so, there can be little doubt that questions about the commensurability and maximality of goodness remain very much alive. For the same reason, I think it's fair to say that the position Anselm takes on the nature of goodness in premises (1a†) and (1b†) represents a perfectly respectable position, even if not an irresistible one. 4.2. Premise (1c†) Just as the first two premises of Anselm's proof are interesting for the light they shed on certain questions regarding the nature of goodness, so too the third premise of Anselm's proof is interesting for the light it sheds certain metaphysical principles (again five in total) regarding the nature of kinds: 27 But see MacDonald (1990b) for some contemporary defenses of (a), and see Nagasawa (2017) for an extended contemporary defense of (b) and (c). JEFFREY E. BROWER 26 Strictly speaking, in premise (1c†), Anselm commits himself only to the truth of P5. Even so, in the course of discussing this premise, we have had occasion to touch on all five principles. Thus, in the case of P1 and P2, we have seen that the first can be resisted and the second should be resisted. For contrary to what P1 suggest, there can be nominalist accounts of kind membership. And contrary to what P2 suggests, all that membership in a degreed kind requires is a common scale or measure, but we needn't think of such a scale or measure in terms of a universal standard or ideal (as opposed, say, to a unit of value). In the case of P3–P5, by contrast, we have seen each of them is quite plausible. Thus, in line with what P3 suggests, membership in a maximal kind would seem to require the existence of a universal standard or ideal of some sort-namely, one specified by the intrinsic maximum associated with this kind. Again, in line with what P4 suggests, it is plausible to suppose that members of different fundamental ontological types or categories are wholly distinct, and hence do not share any nature or immanent properties in common. As for P5, it is a straightforward consequence of P4 and P5. Indeed, as I have suggested, it is precisely because Anselm thinks both (a) that the maximality of goodness requires a universal standard, and (b) the category-neutrality of goodness excludes the possibility of such a standard's being an immanent form or property, that he appeals to a version of P5 to justify his Platonism about goodness. Assuming I'm right about this, Anselm's commitment to the truth of P5 in premise (1c†) would appear to have a lot going for it. There is, of course, much more that could be said about the premises of Anselm's proof-and much more that would need to be said, if we were going to provide Five metaphysical principles about kinds (P1) If the members of some class form a kind, F things, then there must be a universal F-ness. (P2) If the members of some class form a degreed kind, F things, then there must be a universal standard of F-ness. (P3) If the members of some class form a maximal kind, F things, then there must be a universal standard of F-ness. (P4) If the members of some class form a category-neutral kind, F things, then there cannot be an immanent universal of F-ness. (P5) If the members of some class form a category-neutral maximal kind, F things, then there must be a transcendent universal standard of F-ness. PLATONISM ABOUT GOODNESS 27 anything like a complete evaluation of them. But I hope that what I have already said is enough to emphasize the intrinsic philosophical interest of the proof itself, as well as to reveal that it has much more going for it than commentators have generally recognized.28 Bibliography Armstrong, D. M. 1978. A Theory of Universals. 2 vols. Cambridge: Cambridge University Press. Armstrong, D. M. 1989. Universals: An Opinionated Introduction. Boulder: Westview. Audi, Paul. R. Forthcoming. "Partial Resemblance and Property Immanence." Noûs. Brower, Jeffrey E. 2004. "Anselm's Ethics." In The Cambridge Companion to Anselm, eds. Brian Davies and Brian Leftow (Cambridge University Press, 2004), 222–56. https://doi.org/10.1017/CCOL0521807468.011. Brower, Jeffrey E. and Kevin Guilfoy, eds. 2004. The Cambridge Companion to Abelard. Cambridge, UK; New York: Cambridge University Press. https://doi.org/10.1017/CCOL0521772478. Erismann, Christophe. 2011. L'homme commun: La genèse du réalisme ontologique durant le haut Moyen Âge. Vrin. Erismann, Christophe. 2007. "Immanent Realism: A Reconstruction of an Early Medieval Solution to the Problem of Universals." Documenti E Studi Sulla Tradizione Filosofica Medievale 18: 211–29. Erismann, Christophe. 2003. "Collectio proprietatum. Anselme de Canterbury et le problème de l'individuation," Mediaevalia: Textos e estudos 22: 55–71. Fine, Gail. 1984. "Separation." Oxford Studies in Ancient Philosophy 2: 31–87. 28 Work on this article was supported by the Classical Theism Project, funded by the John Templeton Foundation. Earlier drafts were presented at the 2016 Classical Theism workshop (University of St. Thomas, Minnesota); a session of the 2016 SLUCHI reading group (online); the 2017 California Metaphysics Conference (University of Southern California); the 2018 Analytic Theology Workshop on God's Nature (Paris, France); a 2018 workshop in metaphysics and philosophy of religion (Bend, Oregon); a session of the 2018 Rutgers Center for Philosophy of Religion reading group (Rutgers University); and a 2019 Medieval Philosophy Network Meeting (Warburg Institute, UK). I'm grateful to audiences on all of these occasions for stimulating comments, criticism, and discussion, and in particular to James Arcadi, who served as my commentator on the first occasion. 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