Realists about universals face a question about grounding. Are things how they are because they

Transcription

1 T. Scott Dixon Ashoka University Upward Grounding 1 Forthcoming in Philosophy and Phenomenological Research ABSTRACT: Realists about universals face a question about grounding. Are things how they are because they instantiate the universals they do? Or do they instantiate those universals because they are how they are? Take Ebenezer Scrooge. You can say that (i) Scrooge is greedy because he instantiates greediness, or you can say that (ii) Scrooge instantiates greediness because he is greedy. I argue that there is reason to prefer the latter to the former. I develop two arguments for the view. I also respond to some concerns one might have about the view defended. I close by showing that analogous views regarding the truth of propositions (that if the proposition that p is true, then it is true because p) and the existence of facts (that if the fact that p exists, then it exists because p) are supported by analogs of one of these arguments. 1. Introduction Realists about universals face a question about grounding. Are things how they are because they instantiate the universals they do? Or do they instantiate those universals because they are how they are? Take Ebenezer Scrooge. You can say that (i) Scrooge is greedy because he instantiates greediness, or you can say that (ii) Scrooge instantiates greediness because he is greedy. In what follows, I argue that there is reason to prefer the latter to the former. I develop two arguments for the view. The first is an argument from analogy first mentioned by Ted Sider (2006), in which instantiation is taken to be relevantly similar to truth. The second is a regress argument first mentioned by Jacek Brzozowski (2008), Ross Cameron (2008), and Paul Audi (2012). Roughly, the idea is that the view that claims about how things are are grounded in instantiation claims results in a vicious regress, while the opposing view does not. While each of these arguments has been mentioned in the literature, neither has been developed in detail. In addition to further developing these arguments, I respond to some concerns one might have about the view defended. I close by 1 Many thanks to Michael Bertrand, Christopher Buckels, David Copp, Cody Gilmore, Daniel Nolan, William Robinson, Adam Sennet, and audiences at the 2012 APA Eastern Division Meeting and the University of Delhi Department of Philosophy, for providing helpful comments and discussion. Thanks also to several anonymous referees and an editor for very helpful suggestions. 1

2 showing that analogous views regarding the truth of propositions (that if the proposition that p is true, then it is true because p) and the existence of facts (that if the fact that p exists, then it exists because p) are supported by analogs of one of these arguments. I emphasize at the outset that I am only arguing that, between two choices, a person ought to prefer one to the other. There are alternatives to the two views outlined above. One might reject both (i) and (ii). One might be a skeptic about grounding. Or even if one thinks that grounding is an intelligible concept, one might reject both (i) and (ii) because, for example, Scrooge s being greedy just is Scrooge s instantiating greediness. Either way, neither grounding claim would hold. This view is certainly worthy of consideration. Nevertheless, I will not directly argue against it here. Given constraints on space, a complete discussion of this alternative could only come at the cost of discussing interesting issues that arise when one restricts one s attention to the two views I have put on the table. 2. Grounding Grounding has been characterized as metaphysical dependence, determination, or priority (see Cameron 2008: 4, Schaffer 2008 and 2009, Rosen 2010: 109, and Audi 2012a: 102), and also as metaphysical explanation (see Fine 2001: 15, 2012: 37, and 2012: 1, Schnieder 2006: 31 32, and Correia 2011: 1). Grounding claims are expressed with phrases like grounds or is grounded in/by, in virtue of, explains or is explained by, and because. Grounding has proven to be very resistant to analysis, and is often taken as primitive (as in Schaffer 2009: 373). The notion that is taken as primitive is usually full grounding, as opposed to partial grounding. When x is fully grounded by some things, they provide a complete explanation of the existence of x. This may not be the case when x is partially grounded by some things. Those things might provide only an incomplete explanation of it. In what follows, I take the notion of full grounding as primitive, and define partial grounding in terms of it. There are at least two ways to understand the fundamental notion of grounding. Some philosophers (Cameron (2008: 4), Schaffer (2008: 17 and 2009: 364), Rosen (2010), and Audi (2012: 103 4), for example) believe that the most fundamental concept of grounding is a dyadic relation, and is expressed by a two-place predicate which takes terms as arguments. These terms are often taken to denote facts (see, for example, Rosen 2010: 114, Schaffer 2012: 123, and Audi 2012a: 2

3 101 and 2012b: 1). I call this view predicationalism. 2 Others (Fine (2001: 16), Schnieder (2006: 31 32), and Correia (2010: and 2011: 2)) believe that the fundamental notion of grounding is an operation, and is expressed by a (hyperintensional) binary sentence-forming operator. The operator chosen for this has typically been because. I call this view operationalism. In either case, full and partial grounding claims must be distinguished from one another. I propose the following schemas. Predicationalism Operationalism Full Grounding x is fully grounded by Γ ϕ Because G Ψ Partial Grounding x is partially grounded by Γ ϕ because G Ψ In two schemas on the left, x is a singular schematic variable and Γ is a plural one, each of which range over facts or whatever entities that are taken to stand in the grounding relation. In the right two, ϕ is a singular schematic variable and Ψ is a plural one, each of which range over sentences. I allow a single thing to be assigned to any plural term. Note that upper-case Because expresses full grounding in an operationalist framework, while lower-case because expresses partial grounding. I will drop G in what follows, as context will serve to make clear when I intend to use those operators to express grounding claims and when I don t. In what follows, I argue that, assuming realism about universals, one ought to think that things instantiate the universals they do Because they are how they are, and not vice versa. Technically, operationalism is not essential to secure the desired conclusion. But the view strikes me as more attractive when it is presented in that framework. Officially, then, in what follows, grounding claims will be expressed with the operators Because and because. For example, given that Scrooge is greedy, it is a consequence of my view that Scrooge instantiates greediness Because Scrooge is greedy. It will, however, be convenient to have at hand a non-fundamental notion of grounding expressed by a predicate, which takes names of sentences as arguments. (Def G S ) x is fully grounded S by y = df Σϕ x = ϕ and Σψ y = ψ and ϕ Because ψ. 3 2 For an alternative version of this view, according to which the grounding relation can hold of things other than facts, see Schaffer 2009: In these definitions, Σ is the existential substitutional quantifier. Generally, where α is a variable and ϕ is a formula, Σα ϕ is true if and only if, for some individual constant β, the formula that results from replacing every occurrence of α in ϕ with β is true. It is necessary to employ substitutional quantifiers rather than objectual quantifiers in this definition and those that immediately follow since ϕ and ψ occur in sentence position rather than in object position. See Hill 2002: for more discussion of substitutional quantification. 3

4 Now one can say that the sentence Scrooge instantiates greediness is fully grounded S by the sentence Scrooge is greedy. Whether one thinks that grounding is fundamentally a relation or an operation, it must behave according to certain structural principles. It is usually assumed that grounding is asymmetric (and hence irreflexive) and transitive. 4 I will assume this also in what follows. 3. Realism and Its Attendant Choice I now characterize more perspicuously the two views introduced at the outset: (i) things are how they are Because they instantiate the universals they do, and (ii) things instantiate the universals they do Because they are how they are. These two views will be of interest only to those who endorse realism about universals (hereafter simply realism ) the view that there are such things as universals. But realism is, at its heart, an answer merely to the question of what exists. Schaffer (2009) distinguishes between two important questions in metaphysics. One is about what exists. This is what Schaffer might call the Quinean question. The other is about what grounds what. This is what he might call the Aristotelian question. While Schaffer places more emphasis on the second question, I think they are of roughly equal importance. Formulations of certain metaphysical views ought to provide answers to both of these questions. Realists about universals have been clear about an answer to the Quinean question. They say that universals exist. But not nearly as much attention has been given to the Aristotelian question. Throughout the history of realism, since Plato at least, the view that things are how they are Because they instantiate the universals they do has been taken for granted, and has enjoyed universal acceptance among realists, while the alternative view has been largely ignored. In Phaedo, for example, Plato says the following. It seems to me that whatever else is beautiful apart from absolute beauty is beautiful because it partakes of that absolute beauty, and for no other reason. (Hamilton and Cairns 1961: 100c, italics added) And, in Parmenides, he says 4 I ll introduce these principles only as necessary. For characterizations of grounding as having these formal properties, i.e., as a strict partial order, in both predicationalist and operationalist frameworks, see Cameron 2008: 3, Correia 2010: 262 and 2011: 3 4, Fine 2010: 100, Rosen 2010: , Schaffer 2010: 37, Schnieder 2011: 451, and Raven 2012: 689 and It should be noted that there are some who disagree with the claim that grounding has these properties. See Schaffer 2012 for arguments against the claim that grounding is transitive, and Jenkins 2011 for arguments against the claim that it is irreflexive. See Barnes manuscript for arguments against the claim that a similar notion, ontological dependence, is asymmetric. 4

5 [T]here exist certain forms of which these other things come to partake and so to be called after their names; by coming to partake of likeness or largeness or beauty or justice, they become like or large or beautiful or just[.] (Hamilton and Cairns 1961: 130e 131a) The torch was carried into the twentieth century by Russell, who says, If we believe that there is such a universal [as whiteness], we shall say that things are white because they have the quality of whiteness. (Russell 1912: 149, italics added) Brian Garrett and Gonzalo Rodriguez-Pereyra provide more recent formulations of realism that show that Plato s view about dependence continues to be regarded as an integral part of the thesis. Since Plato, many philosophers have held that properties and relations are universals... If my billiard ball is red, that is because redness (the universal) inheres in the ball (the particular). (Garrett 2006: 38, italics added) [W]hat makes a square thing square? For the realist about universals if something is square, this is in virtue of the thing instantiating the universal squareness. (Rodriguez-Pereyra 2011, 4.1, italics added) The view expressed in the passages above is captured by the following principle. (DG) For any x 1,..., x n, if R(x 1,..., x n ) and there is such a thing as the universal being R, then R(x 1,..., x n ) Because x 1,..., x n instantiate being R in that order. 5 Suppose that Scrooge is greedy, and that there is such a thing as the universal greediness. Then, according to (DG), Scrooge is greedy Because he instantiates greediness. Or suppose that Scrooge is an uncle of Fred, and that there is such a thing as the universal being an uncle of. Then, according to (DG), Scrooge is an uncle of Fred Because Scrooge and Fred instantiate being an uncle of. I ll refer to the conjunction of realism and (DG) as downward grounding, since, according to this view, how particulars are related to universals up in Platonic heaven grounds how things are down here in the everyday world. 5 Several remarks are in order. First, to allow for complete generality, the view is officially expressed as follows. For any x 1,..., x n, if ϕ(x 1,..., x n) and there is such a thing as the universal Λy 1... y n ϕ(y 1,..., y n), then ϕ(x 1,..., x n) Because x 1,..., x n instantiate Λy 1... y n ϕ(y 1,..., y n) in that order, where Λy 1... y n ϕ is syntactically a term that denotes the property being a y 1,..., and y n such that ϕ. (I follow Fine 2012a: in my use of the term-forming operator Λ.) Second, the extra condition that the universal must exist is important. I do not want to force the downward grounder to be committed to the view that every predicate expresses a universal. I want to allow them to adopt a non-abundant view of universals if they so choose. For examples of sparse views of universals, and motivation for them, see Armstrong 1978a: 113, 1978: 19 29, 1989: 75 88, and 1997: and 38 43, and Lewis Even Plato seems reluctant to adopt a maximally abundant view of the Forms in Parmenides 130b d. Finally, to avoid repetitiveness, I will henceforth discontinue the use of the phrase in that order except when clarity demands it. 5

6 Platonic Heaven grounds Spatiotemporal Realm Figure 1. Downward Grounding Downward grounding, then, is just the version of realism about universals according to which things are how they are Because they instantiate the universals they do what has, as a matter of historical fact, been the overwhelmingly dominant view among realists. 6 It is important to realize, however, that endorsing realism about universals does not force one to adopt (DG). The two views are logically independent. I formulate the alternative to (DG) that things instantiate the universals they do Because they are how they are as follows. (UG) For any x 1,..., x n, if x 1,..., x n instantiate the universal of being R, then x 1,..., x n instantiate the universal being R Because R(x 1,..., x n ). 7 Suppose, for example, that Scrooge instantiates the universal of greediness. Then, according to (UG), Scrooge instantiates greediness Because he is greedy. Or suppose that Scrooge and Fred instantiate the universal being an uncle of. Then, according to (UG), they do so Because Scrooge is an uncle of Fred. I ll refer to the conjunction of realism and (UG) as upward grounding, since, according to this view, how things are down here in the everyday world grounds how particulars are related to universals up in Platonic heaven. 6 In a recent paper, Chad Carmichael endorses a view he calls deep platonism, which is somewhat in the spirit of downward grounding, though consistent with the falsity of it. According to deep platonism, all the facts about particulars are grounded in facts purely about universals (Carmichael 2015: 1). 7 Officially, For any x 1,..., x n, if x 1,..., x n instantiate Λy 1... y n ϕ(y 1,..., y n), then x 1,..., x n instantiate Λy 1... y n ϕ(y 1,..., y n) Because ϕ(x 1,..., x n). 6

7 Platonic Heaven grounds Spatiotemporal Realm Figure 2. Upward Grounding The reader is now in a position to fully understand my thesis, which is just the claim that realists ought to prefer upward grounding to downward grounding. Perhaps the earliest formulation of upward grounding (and an early foray into the investigation of the concept of grounding more generally) can be found in Van Cleve 1994: In this article, Van Cleve develops arguments against realism (among other things). Responding to his criticism of realism will clarify another respect in which upward grounding should be taken to differ from more traditional versions of realism. Van Cleve aims to reject realism by arguing that because (step 1) instantiation is a supervenient relation, i.e., because it could not cease to hold... without some change in the intrinsic natures of its relata (580), (step 2) the fact that, for example, Scrooge instantiates greediness must be grounded by his being greedy. There is, Van Cleve maintains, simply nothing else about the intrinsic nature of Scrooge that could adequately explain why he instantiates greediness. But, he continues (step 3, ), this is in tension with the realist claim that every subject-predicate statement of the form R(x 1,..., x n ) is analyzed by the corresponding statement of the form x 1,..., x n instantiate the universal of being R. After all, it makes no sense to say that an analysandum explains why its analysans holds. In Van Cleve s words, [n]o fact can be the ground of its own analysis (585). I have two things to say in response to this argument. First, instantiation is not supervenient. (I take the notion of a supervenient relation to the same as that of an internal relation. 8 ) This is so because it can hold between some thing(s) that stand in a non-internal relation. Suppose Scrooge is ten feet away from Marley. Then, for the realist, Scrooge and Marley instantiate being ten feet away from. That these two things instantiate that universal does not supervene on their intrinsic natures (it could cease to hold without a change in the intrinsic natures of its relata). 9 8 See Lewis 1986: 62 and Bricker 1993: 292, en Here I follow Armstrong 1989:

8 But even if the realist is happy to take steps 1 and 2 of Van Cleve s argument, she need not take step 3. Van Cleve saddles realism with the analytical claim that subject-predicate sentences have property-theoretic analyses. This is not surprising, as it had been common to do so (explicit in, for example, Armstrong 1978a: 64 and Hochberg 1978: 306, and perhaps implicit in Russell 1912: 140). But note that upward grounding is immune to Van Cleve s problem, as long as it is not taken to provide property-theoretic analyses of subject-predicate statements. Nor should we expect it to; it is a thesis about grounding. And the realist need not commit herself to such analyses. The upward grounder might simply decline to answer the analytical question, or perhaps answer it in the opposite way, providing non-property theoretic analyses of property-theoretic statements. 10 In more recent years, other metaphysicians have been toying with the idea of upward grounding. Correia and Schnieder (2012: 27), Fine (2012: 68), and Rosen (2015: 204), for example, provide statements of the view and/or speak to its plausibility, though they provide no arguments to prefer it to the Platonic alternative. Others have gestured at arguments in favor of the view. In the next three sections, I develop two of them. 4. An Analogy between Truth and Instantiation The first argument that realists ought to prefer upward grounding to downward grounding was originally suggested by Ted Sider in the following passage. Warmup argument: when I am sitting, the proposition that I am sitting is true. But: is the proposition true because I am sitting, or am I sitting because the proposition is true? Obviously the former: I, not the proposition, wear the metaphysical pants. Now for the argument that tempts me. When I am sitting, am I sitting because I instantiate the property of sitting, or do I instantiate the property because I am sitting? Again, I want to answer: the latter. Particulars, not properties, wear the pants. (Sider 2006: 4 5) The argument in this passage is most charitably understood as an argument from analogy. In general, if the proposition that ϕ is true, then it is true Because ϕ, and not vice versa. And instantiation is to properties and relations what being true is to propositions (or is at least relevantly similar). So things instantiate the universals they do Because they are how they are, and not vice versa. Sider does not provide any reason to think that either of these premises is true. For now, I will not say much in support of the first premise that, in general, if the proposition that ϕ is 10 For more discussion on this topic, see the last part of section 7. 8

9 true, then it is true Because ϕ and not vice versa. Later (section 8), I ll present an argument for it. But for now, I ll say just a couple of things in its favor. First of all, it is, at first look, very plausible. To suggest the reverse seems to be getting the order of explanation (assuming there is one) backwards. Indeed, the claim has enjoyed adherents as far back as Aristotle, and contemporary ones as well, some of whom apparently do not feel the need to provide an argument for it. 11 Secondly, compelling arguments have been put forth in favor of the claim, as in Rodriguez- Pereyra 2005: and Hornsby 2005: 42 45, who provide (very different) arguments for it. (Due to limitations on space, I simply direct the reader to those arguments.) I will say much more in favor of the second premise that truth is to propositions what instantiation is to properties and relations, or, at least, that they are similar in a way that is relevant to the argument. To begin, Peter van Inwagen provides a very helpful way of distinguishing properties and relations from other kinds of entities, which speaks in favor of the strength of the analogy. According to his characterization, properties are things that can be said of something (2004: , 2006: 472, and 2006: 27) and relations are things that can be said of [some] things (2006: 472). He notes that there are alternatives to the schema x can be said of y, including x is either true or false of y (2004: 132, 2006: 28). 12 This locution illustrates particularly well the close relationship between truth and instantiation that falls out of his view. Redness and being both round and square are both properties on van Inwagen s account, because each is either true or false of something. Redness is true of the stop sign, and redness is instantiated by the stop sign. Being both round and square is false of the stop sign, and is not instantiated by the stop sign. My refrigerator, in contrast, is neither true nor false of the stop sign, nor indeed of anything. On van Inwagen s account, it is not a property. Nor is it instantiated by anything. Thus it would appear that, at the very least, x is true of y if and only if y instantiates x. Now this does not by itself show that truth is to propositions what instantiation is to properties and relations, or even that they are similar in a way that is relevant to the analogy. But it is suggestive of these views. And there are other things that are said of propositions on the one hand, and properties and relations on the other, that add to the case. To begin with, it is natural 11 See Aristotle Categories 12 (e.g. in Barnes 1984). See also Prior 1962: , Hornsby 2005: 42 45, Rodriguez- Pereyra 2005: 26, Merricks 2007: xiii, and Horwich 2009: Similar characterizations of properties can be found in McTaggart 1920: 61, Moore 1962: , and McGrath 2012,

10 to take properties, relations, and propositions to constitute a single ontological category. Edward Zalta (1988: 7), for example, says, we shall regard properties and propositions as one-place and zero-place relations, respectively (henceforth the term relations is frequently used to encompass both properties and relations). Van Inwagen himself says, A proper presentation of this theory [of properties] would treat properties as a special kind of relation.... And it would treat propositions as a special kind of relation: it would treat properties as monadic relations and propositions as 0-adic relations. (Inwagen 2004: 131) George Bealer (1983: 1) collectively refers to these entities as PRPs. For this reason, I ll call this view PRP-ism. 13,14 In addition, it is natural to accept the existence of argument places in relations. In particular, one can say that a relation is n-adic if and only if there are exactly n argument places in it. Following Gilmore (2013), I ll call this view slot theory. 15 PRP-ism and slot theory together result in a picture according to which it is very natural to take truth to be nothing other than instantiation, or, depending on how instantiation is conceived, the monadic version 13 For other endorsements of PRP-ism, see Zalta 1983: 6, 59, 61 and 1988: 57, Menzel 1993: 66 67, and Inwagen 2006a: 454 ff. and 2006: 27. Swoyer (1998: 303 and 322 en. 4) identifies properties with monadic relations, and, while he does not himself identify propositions with 0-adic relations, his reasons for this seem to be entirely programmatic (see 1998: 297). He is always careful to mention how his system can be modified in order to accommodate this further thesis (see 1998: 303 and 322 en. 4). 14 While this is not the place to go into a detailed defense of PRP-ism, it is worth noting a consideration in favor of the view. PRP-ism is strongly suggested by van Inwagen s very natural characterizations of propositions as things that can be said (2004: , and 2006: 27), properties as things that can be said of something (2004: , 2006: 472, and 2006: 27, italics added), and relations as things that can be said of [some] things (2006: 472, italics added). The naturalness and usefulness of these characterizations suggest that these are all the same type of entity, which he calls assertibles propositions are fully saturated assertibles (2004: 136, 2006: 472, and 2006: 30 31), properties are singly unsaturated assertibles (2004: 132, 2006: 472, and 2006: 27), and relations are multiply unsaturated assertibles (2006: 472). It is also worth taking a moment to respond to King s (2007: 136) argument against the thesis that propositions may be categorized with properties and relations. Following Jubien (2001), he says that propositions are importantly different than properties and relations in that the former can, on their own, represent other things, while the latter cannot. While this is not the place for a detailed discussion, I will just say that the PRP-ist can make sense of this difference by appealing to a remaining difference between propositions on the one hand and properties and relations on the other: the former are 0-adic, while the latter are not. The former are completely saturated, while the latter are not. It is plausible that the degree of saturation of a thing is relevant to whether or not it can, on its own, represent other things. Thus, while the PRP-ist does believe that propositions are the same type of thing as properties and relations at one level, she is free to say that they are a different type of thing at a lower level (same genus, different species), and that there are differences between propositions on the one hand and properties and relations on the other that explain why the former can, on their own, represent other things, while the latter cannot. 15 Fine (2000: 16), Dorr (2004: 175), and King (2007: 123) recognize the naturalness of a version of slot theory (which they call positionalism ), though Dorr and Fine reject the view in the end. More recently, Gilmore (2013) provides a sustained defense of the view. For some of the more explicit endorsements of slot theory, see Armstrong 1997: , Yi 1999: 168 ff., and King 2007: 123. In addition to such endorsements, the literature is replete with implicit endorsements of the view, wherein authors refer to specific argument places (or positions) in properties and relations (using phrases like those of the form the nth argument place of the relation being R ) or quantify over them (using phrases like those of the form the relation being R has n argument places ). For examples of these phenomena, see Zalta 1983: 21, 23 24, 32, 174 ch. 1 en. 6 and 1988: 28, 49, 52, 57 58, 79, , 218, Williamson 1985: 251 ff., Menzel 1993: 68 ff., Swoyer 1998: 303, Newman 2002: 68 ff., McKay 2006: 8 ff., and King 2007: 20 ff. For references to other endorsements of slot theory, explicit and implicit, see Gilmore 2013: fn. 3 and 2014: fn

11 of it. Either way, the analogy between truth and instantiation would be a strong one. I ll now explain why identity, or at least conspecificity, of truth and instantiation would naturally follow from this combination of views. For simplicity, I ll refer to the conjunction of these two views as slot PRP-ism, and to those endorse this view as slot PRP-ists. Suppose that a is taller than b. Most realists would agree that the universal being taller than is a dyadic relation. A PRP-ist would say further that it is a dyadic PRP. In a slot-theoretic framework, the relation may be represented as follows. being taller than Figure 3. the universal being taller than Realists would also agree that, given that a is taller than b, a and b instantiate being taller than (at least, as long as they agree that there is such a thing as the universal being taller than). In a slot-theoretic framework, this may be represented as follows. a b being taller than 3-place instantiation Figure 4. a and b instantiate the universal being taller than Downward and upward grounders would agree because, as realists about properties and relations, they would likely agree that if the property or relation being R exists, then Rxy if and only if x and y instantiate it. Many realists would also agree that a instantiates being taller than b, given that a is taller than b and that there is such a thing as the universal being taller than b. This is because they would likely agree that for any x and y, x instantiates R-ing y if and only if x and y instantiate being R (at least as long as they agree that being R and R-ing y exist). The slot PRP-ist has a metaphysical 11

12 explanation for this biconditional ready to hand. It is natural for her to recognize a structural relationship between n-adic PRPs (where n 2) and certain m-adic PRPs (where 1 m < n). In the case at hand, the slot PRP-ist will take the monadic PRP (property) being taller than b to be the result of saturating one of the two argument places (the second one) of the dyadic PRP being taller than with an object (b in this case). 16 being taller than b Figure 5. the universal being taller than b The structural relationship between being taller than and being taller than b guarantees that the biconditional holds of the two PRPs, and metaphysically explains why it does. It is the structural relationship between the relation being taller than and the property being taller than b that guarantees (and metaphysically explains) the biconditional. 17 This will ensure that a instantiates being taller than b. 16 The general relationship may be expressed as follows. (SAT-1) For any x, if there are such things as the PRPs Λy 1... y n ϕ(y 1,..., y n) and Λy 1... y n 1 ϕ(y 1,..., y i 1, x, y i,..., y n 1), then Λy 1... y n 1 ϕ(y 1,..., y i 1, x, y i,..., y n 1) is the result of saturating the ith argument place of Λy 1... y n ϕ(y 1,..., y n) with x. This principle expresses the relationship amongst the PRPs generated by Swoyer s (1998: 303) family of operations Plug i. Note that (SAT-1) only says that a PRP x is the result of saturating another PRP y when x is the result of saturating a single argument place of y with a single entity. So (SAT-1) guarantees that the PRP being an x such that x is between a and b is the result of saturating a single argument place of being an x and y such that x is between y and b with the single entity a. And it guarantees that the latter PRP is the result of saturating a single argument place of being an x, y, and z such that x is between y and z with the single entity b. But it does not guarantee that being an x such that x is between a and b is the result of saturating two argument places of being an x, y, and z such that x is between y and z with the two entities a and b. It seems just as natural, however, for the slot PRP-ist to hold that this is the case as well. This can be guaranteed by generalizing (SAT-1), or more easily by adding to it the following transitivity principle. (SAT-T) For any x, y, z, w 1,..., w n, and v 1,..., v n, if x is the result of saturating the i 1th,..., i nth argument places of y with w 1,..., w n, respectively, and y is the result of saturating the j 1th,..., j mth argument places of z with v 1,..., v m, respectively, then x is the result of saturating the i 1th,..., i nth, j 1th,..., j mth argument places of z with w 1,..., w n, v 1,..., v m, respectively. 17 The slot PRP-ist can actually provide an explanation of the stronger claim that it is essential to two appropriately related PRPs that the relevant biconditional holds of them. 12

13 a being taller than b 2-place instantiation Figure 6. a instantiates the universal being taller than b There are at least two ways of characterizing the relationship between n- and m-adic instantiation, where n m (as in the example above). First, one can characterize it as numerical identity. On this view, instantiation is variably polyadic. It can take more than one number of arguments on different occasions. Usually, it is assumed that variably polyadic instantiation can take two or more arguments. Second, one can say that n- and m-adic instantiation are versions of the same type of relation, viz. instantiation. On this view, there are a number of (at least denumerably many) instantiation relations, each with a fixed adicity. Usually, it is assumed that there is a unique n-adic instantiation relation for each n 2. Whichever way one understands instantiation, there is reason to think that truth is to propositions what being instantiated is to properties and relations, and so that the analogy between truth and instantiation is a strong one. Now I will explain why. I have shown that it is natural for the slot PRP-ist to recognize a relationship between n-adic PRPs (where n 2) and certain m-adic PRPs (where 1 m < n). It seems to me that it is just as natural for the PRP-ist to take an analogous relationship to hold between n-adic PRPs (where n 1) and 0-adic PRPs (i.e., propositions). In the case at hand, the slot PRP-ist will take the proposition that a is taller than b, to be the result of saturating each of the two argument places of the dyadic PRP being taller than with an object (a and b, respectively). (Equivalently, the slot PRP-ist can also take it to be the result of saturating each of the argument places (the only argument place) of the monadic PRP being taller than b with a.) The general relationship may be expressed as follows. (SAT-2) For any x 1,..., x n, if there are such things as the PRPs Λy 1... y n ϕ(y 1,..., y n) and the proposition that ϕ(x 1,..., x n), then the proposition that ϕ(x 1,..., x n) is the result of saturating the saturable argument places of Λy 1... y n ϕ(y 1,..., y n) with x 1,..., x n, respectively. Unlike (SAT-1), it is relatively easy to formulate this principle in a general way, which will not require the use of something like (SAT-T) (see fn. 16). 13

14 a being taller than b Figure 7. the proposition that a is taller than b Now most (realist or not) would agree that the proposition that a is taller than b is true, given that a is taller than b and that there is such a thing as the proposition that a is taller than b. a being taller than b being true (1-place instantiation) Figure 8. the proposition that a is taller than b is true This is because they would agree that for any x, the proposition that x R s b is true if and only if x instantiates R-ing b. As before, the slot PRP-ist has a metaphysical explanation for this biconditional ready to hand. It is the structural relationship between the property being taller than b and the proposition that a is taller than b that guarantees (and metaphysically explains) the biconditional. 19 It should now be clear that, given slot PRP-ism, it is very natural to take truth to be to propositions what instantiation is to properties and relations, whichever way instantiation is conceived. If instantiation is conceived as a variably polyadic relation, then it is natural to take the (apparently fixedly) monadic PRP being true to be nothing other than the variably polyadic PRP instantiation. It is just that this is the guise instantiation takes when it takes only a single argument (a proposition). The only change to this conception of instantiation is that, instead of its minimal adicity being 2, its minimal adicity is 1. If, on the other hand, it is assumed that there is a class 19 As before (see fn. 17), the slot PRP-ist can actually provide an explanation of a stronger claim in this case, that it is essential to an appropriately related proposition and property or relation that the relevant biconditional holds of them. I actually suspect an Armstrongian argument for slot PRP-ism has been lurking. See Armstrong 1978a: 49. Just as the realist is in an excellent position to explain the formal properties of resemblance such as symmetry, while others must take them as brute, the slot PRP-ist is in an excellent position to explain various biconditionals which hold (essentially) among certain claims about truth and instantiation, while others must take them as brute. 14

15 of denumerably many instantiation relations, each with a fixed adicity, then it is natural to take the monadic PRP being true to be nothing other than the unique monadic member of that class of relations. The only change to this conception of instantiation is that, instead of there being an n-adic instantiation relation in this class for each n 2, there is one for each n 1. Not only are these moves natural, they are encouraged by Ockham s razor. On the first conception of instantiation, doing otherwise would leave one with two universals when only one is needed, while on the second conception of instantiation, doing otherwise would leave one with at least two types of universal when only one is needed. 20 Given slot PRP-ism, then, there is good reason to take truth to be nothing other than instantiation, or, depending on how instantiation is conceived, the monadic version of it. Either way, truth is to propositions what instantiation is to properties and relations. Moreover, while it is outside the scope of this project to provide a sustained defense of PRP-ism and slot theory, it is important to note that they are well-motivated (see fns. 14 and 15). As a result, there is good reason to think that the analogy between truth and instantiation is a strong one; just as the proposition that Sider is sitting is true because Sider is sitting and not vice versa, Sider instantiates the property of sitting because he is sitting, and not vice versa. Before moving on to the second argument for upward grounding, it is worth addressing a concern the reader might have about my defense of the analogy between truth and instantiation just given. In particular, one might be concerned that I am muddling the distinction between representational entities (such as propositions and concepts) and their non-representational correlates (states of affairs and properties, respectively). 21 This concern can take a more specific form as follows: Surely, one may believe that Hesperus is shining while failing to believe that Phosphorus is shining. But if things work as I have outlined them in this section so far, then each of (i) the proposition that Hesperus is shining and (ii) the proposition that Phosphorus is shining is the result of saturating the property shining with Venus. On that view, these propositions must be numerically identical, and so it is impossible for anyone to believe one but not the other. 22 I have two things to say in response to this objection. First, it is not obvious that the systematic 20 Menzel (1993: 65 and 86 en. 27) hints at the insight that has played a crucial role in my argument that truth is just a monadic version of instantiation. 21 Thanks to an anonymous referee for this characterization of the objection. 22 Thanks to Gabriel Rabin (personal communication) for this characterization of the objection. 15

16 relationship between truth and instantiation outlined in this section so far cannot be modified to accommodate this distinction. In particular, while the structural relationship between n-adic and m-adic PRPs (where n 2 and 1 m < n) may be retained, that between n-adic PRPs (where n 1) and propositions can be reconceived to be sensitive to something like Fregean senses. On this view, the proposition would not be the result of saturating the property or relation with the referent(s) of the names involved, but rather with its (their) sense(s). This would ensure that the proposition that Hesperus is shining and the proposition that Phosphorus is shining are distinct, as long as Hesperus and Phosphorus have different senses. The second thing I have to say in response to this objection is that there is still an argument in the vicinity of Sider s (and which is, I think, no less compelling) that respects the representational/non-representational divide. So, for the moment, I ll grant that a fix like the one just discussed will not effectively solve the problem. But one may replace Sider s talk of propositions and truth with states of affairs and obtaining (the non-representational analog of truth), and run the argument from analogy from a different starting point. The argument would run as follows: In general, the state of affairs of x 1 s,..., x n s being R obtains Because R(x 1,..., x n ), and not vice versa. And instantiation is to properties and relations what obtaining is to states of affairs (or is at least relevantly similar). So things instantiate the universals they do Because they are how they are, and not vice versa. The argument I gave for identifying truth and instantiation can be changed in a quite straightforward manner into one for identifying obtaining and instantiation. On this view, what would result from saturating all the argument places of a property or relation is a state of affairs rather than a proposition, and this would provide a metaphysical explanation for the systematic relationship between a state of affair s obtaining and its non-predicative constituents instantiating its predicative constituent. And, importantly, the claim that, in general, the state of affairs of x 1 s,..., x n s being R obtains Because R(x 1,..., x n ), and not vice versa, is no less plausible than corresponding claim about truth on which Sider relies. Moreover, the argument to come in section 8 for Sider s premise can be easily translated into one for this new premise. 5. The Regress Argument There is a regress that some have thought poses a problem for downward grounding (see Brzozowski 2008: , Cameron 2008: 3, and Audi 2012a: ). In this section, I will fully 16

17 develop this argument, incorporating recent work regarding the viciousness of regresses, particularly those involving grounding. I will argue that, under the assumption that grounding is well-founded, the regress does in fact pose a problem for downward grounding, though not for upward grounding. Thus realists who think that grounding is well-founded have a second reason to prefer the latter to the former. In the next section, I will reply to a tempting objection to this argument. Recall that the long and short of (DG) is that sentences of the form R(x 1,..., x n ) are fully grounded S by sentences of the form x 1,..., x n instantiate the universal being R, and consider again the following sentence. (1) Scrooge is greedy. (1) is of the form F (x), and so, according to (DG), it must be fully grounded S by a sentence of the form x instantiates F -ness, viz., (2) Scrooge instantiates greediness. This full grounding S claim can be expressed in terms of Because as follows. (1B2) Scrooge is greedy Because Scrooge instantiates greediness. Now the problem begins to reveal itself. Because (2) is of the form R(x 1, x 2 ), it too must be fully grounded S by an instantiation statement, viz., (3) Scrooge and greediness instantiate instantiation. 23 This full grounding S claim can be expressed in terms of Because as follows. (2B3) Scrooge instantiates greediness Because Scrooge and greediness instantiate instantiation. 23 On many presentations of this regress, a distinct instantiation relation is introduced at each step, each having a different adicity. (See, for example, Brzozowski 2008: , and Cameron 2008: 2. Nolan (2008: 178) and Schnieder (2004: 227) also do this, though the instantiation regresses they discuss do not involve grounding.) For my purposes, it does not matter whether there is an infinite number of instantiation relations or just one. The viciousness of the regress under consideration has nothing to do with this. As a result, I simplify matters and work under the assumption that there is a single instantiation relation, and that it is variably polyadic. 17

18 But (3) is of the form R(x 1, x 2, x 3 ), and so it too must be fully grounded S by an instantiation statement. And so on. It is obvious that an end of this grounding chain will never be reached. I ll call this regress the ground-theoretic instantiation regress, or the instantiation regress for short. 24 People disagree about whether the instantiation regress is problematic, and those who think it is disagree about just how problematic it is. Some find it objectionable because, they say, if there is no bottom, it is hard to see how the whole chain, or each step in it, comes into being in the first place (see for example Brzozowski 2008: ). Schaffer says, Being would be infinitely deferred, never achieved (2010: 62), about a regress concerning the metaphysical dependence of wholes on parts. Another way to put the concern is that, according to downward grounding, there exist series of grounding claims that entail that grounding is not well-founded. Those who adopt views like these can be called metaphysical foundationalists. In other work (Dixon 2016b), I argue that the most plausible well-foundedness axiom for grounding is the following. 25 (FS) Every non-fundamental fact is fully grounded by some fundamental facts, 26 where a fundamental fact is a fact that is not grounded (fully or partially) by any fact, and a non-fundamental fact is just a fact which is not fundamental. 27 Since talk of well-foundedness is (I think) most easily understood with the help of entities, I will temporarily drop my operationalist predilections, and frame the regress in predicationalist terms. (All that is said in this section and the next can be translated into an operationalist framework if one allows quantification into sentence position.) I ll also suppose that only facts stand in the grounding relation, and abbreviate the fact that ϕ as [ϕ]. In such a fact-based predicationalist framework, downward grounding would be formulated as follows. (DG-P) For any x 1,..., x n, if R(x 1,..., x n ) and there is such a thing as the universal being R, then [R(x 1,..., x n )] is fully grounded by [x 1,..., x n instantiate being R]. 24 It is worth noting that the ground-theoretic instantiation regress need not be understood as Bradley s regress, as the notion of grounding need not make appearance in the latter. It is also worth distinguishing the instantiation regress from what might be called the grounding grounding regress, which appears in one form or another in Bennett 2011, derosset 2013: 19 25, and Rabin and Rabern forthcoming, and which is a grounding regress of grounding claims (rather than one of instantiation claims). 25 Much of the following discussion can be found in Dixon 2016b: 6. It serves a very different aim there, however. And since it is central to my current aim, it is worth presenting here in some detail as well. 26 Rabin and Rabern (forthcoming: 15 16) also suggest that this is the most plausible way to understand the claim that grounding is well-founded. 27 Adapted from Schaffer 2009:

19 Given that F a (for example, Scrooge is greedy), (DG-P) results in an infinite number of grounding claims, which are suggested by the following elliptical list. (1B2-P) (2B3-P) (3B4-P) [F a] is fully grounded by [a instantiates F -ness]. [a instantiates F -ness] is fully grounded by [a and F -ness instantiate instantiation]. [a and F -ness instantiate instantiation] is fully grounded by [a, F -ness, and instantiation instantiate instantiation].. So conceived, the regress may be depicted as follows.. [F a] [a instantiates F -ness] [a and F -ness instantiate instantiation] [a, F -ness, and instantiation instantiate instantiation] Figure 9. The Naïve Version of the Ground-Theoretic Instantiation Regress In this diagram and the one to follow, each node represents an individual fact, and a solid line running in a downward direction from a node x to another node y (which may run through multiple other nodes) indicates that x is fully grounded by y. As a matter of convention, I assume that, in any given diagram, the only grounding claims that hold are those that are either depicted or implied by those depicted along with the definition of partial grounding (see below) and full transitivity (see Rosen 2010: 116). So whether the instantiation regress conflicts with the claim that grounding is well-founded, properly understood, and so should be seen by the foundationalist as constituting a genuine problem for downward grounding, depends on whether each non-fundamental fact in this non-terminating grounding chain is fully grounded by some fundamental facts or others. 28 Is this the case? Well, [F a] may be partially grounded by [a exists] and [F -ness exists] (collectively), where Partial Grounding-P. x is partially grounded by Γ = df for some, x is fully grounded by and Γ are among I rely on the reader s intuitive understanding of the concept of a non-terminating grounding chain. See Dixon 2016b: 443 and Rabin and Rabern forthcoming: 9 for formal characterizations. Paseau 2010: 172 defines a notion similar to a grounding chain, which he calls a path. 29 See Rosen 2010: 115, Audi 2012b: 698, Fine 2012a: 50, and Raven 2013: 194 for endorsements of this definition. 19