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What it is to be located

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What it is to be located

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Content WHAT IT IS TO BE LOCATED MATT LEONARD USC GRADUATE SCHOOL DOCTOR OF PHILOSOPHY (PHILOSOPHY) UNIVERSITY OF SOUTHERN CALIFORNIA DECEMBER 2019 TABLE OF CONTENTS Chapter 1: Introduction 1 Chapter 2: What is it to Be Located? 8 Chapter 3: On the Contingency and Vagueness of Where I Am 41 Chapter 4: Explaining Harmony 68 Chapter 5: Concluding Remarks 99 References 101 WHAT IT IS TO BE LOCATED 1 CHAPTER ONE: Introduction Consider two boring facts about myself. First, I have a certain size and shape. And second, I am composed of a number of elementary particles. Upon re ection, however, these facts become a bit more interesting. Though there are innitely many regions of spacetime, one of them is a lot like me my location. It, too, has my exact size and shape. And for each particlep that is a part of me,p's location is a part of my location. In fact, for any of my parts whatsoever, its location is a part of my location. Moreover, these facts are necessarily true. But this is a bit puzzling. Why are they true? This puzzle leads to the central question of this dissertation: what it is for a material object to be located at a particular region of spacetime? In my second chapter, \What is it to be Located?", I begin by sketching the two main theories of location suggested by the literature. According to the more popular theory, location is a primitive relation holding between material objects and regions of spacetime. Those who go for this theory think that material objects are distinct from (and share no parts with) regions of spacetime. I call this the primitivist theory of location. According to the rival theory, being located at a 2 MATT LEONARD region just is being identical to that region. On this view, material ob- jects are not entities in addition to spacetime regions: they are identical to spacetime regions. I call this the identity theory of location. Both theories are vulnerable to well-known problems. The primitivist theory seems to make the above necessary truths spooky and mysteri- ous. It is impossible for a square object to be located at a triangular region. But why is that impossible? If the location relation is primi- tive, and there is nothing to be said about what it is, then it seems at the very least a bit strange that square objects never exactly t into triangular regions. Relatedly, consider your arm and its location. It is impossible for your arm to be a part of your body while your arm's location is not a part of your body's location. Why is it that whenever x is a part of y, x's location must be a part of y's location? For the primitivist, this all seems to be a remarkable coincidence. Enter the identity theory. If being located at a region just is being identical to that region, and thus, if located material objects just are re- gions of spacetime, then the spooky coincidence disappears. If a square material object just is a square region of spacetime, then of course it is impossible for the one to have a dierent shape than the other: after all, they are identical! The same goes for the problem above about my arm, my body, and their locations. However, the identity theory has problems of its own. First, the identity theory is inconsistent with the extremely plausible judgment that it is contingent where I am located: WHAT IT IS TO BE LOCATED 3 that is, with the judgment that though I am located at a particular re- gion, I could have been located somewhere else. I am now in the United States. But if the identity theory is true, then it follows that I could not now have been in Europe. And that seems obviously false. Second, some philosophers (including myself) take seriously the view that dis- tinct material objects can both be located at the same region. There are a number of arguments for why philosophers take this possibility se- riously. For instance, physicists tell us that it is possible for two bosons to be exactly co-located at a region of space. Moreover, Leibniz's Law seems to straightforwardly imply that a statue is distinct from the clay from which it is made, given their dierent modal properties. However, though distinct, they are both located at the same region. This looks to be straightforwardly inconsistent with the identity theory. A good theory of location ought to have a number of features. This dissertation focuses on four. A good theory of locationT ought to have the following features: 1. T is consistent with contingent location. 2. T is consistent with two material objects having the same location. 3. T implies that necessarily, a material object has the same size and shape as its location. 4 MATT LEONARD 4. T implies that necessarily, if x is located at R, y is located at S, and x is a part of y, then R is a part of S. I propose a new theory of location the mereogeometrical theory (or the `MG theory' for short) where to be located at a region just is to be geometrically congruent with the region (to have the same size and shape) and mereologically coincident with the region (to overlap all and only the same things). I then argue that the MG theory has all four features. This, I take it, is a powerful reason to accept the view. I then explore how the view relates to a couple of debates and puzzles in contemporary metaphysics. The remainder of the dissertation can be thought of as a pair of case studies. The rst case study (Chapter 3) explores in more detail feature (1), as well as a related feature concerning vagueness. The second case study (Chapter 4) explores in more detail feature (4). In my third chapter, \On the Contingency and Vagueness of Where I Am", I develop two arguments against the identity theory of location. I am located at a particular region of spacetime. This region has the exact same size and shape as I do, and it bears the same distance relations to things that I do. However, it is contingent where I am located. I take it that I could have had a dierent size or shape. And I could have been ve feet to the left, failing to bear the same distance relations to things as my location. Thus, I could WHAT IT IS TO BE LOCATED 5 have been located somewhere else. Moreover, it is vague where I am located. I take it that I do not have a perfectly precise shape. There are a number of very similarly shaped regions in my vicinity, but its vague which of them I have the same shape as. Thus, there's no region that I am denitely located at. It is contingent where I am located and it is vague where I am located. A good theory of location, all things considered, should be consistent with the contingency and vagueness of location. In this paper, I argue that the identity theory is inconsistent with these theses. It is no surprise that the identity theory is in tension with the thesis of contingent location. And given the formal similarities between con- tingency and vagueness, one might think that the tension straightfor- wardly carries over to the thesis of vague location. Thus, it is tempting to suppose that it is not really worth developing in detail the ways in which the identity theory is inconsistent with the theses of contingent and vague location. But as I hope to convince you, this is mistaken. It is mistaken for two reasons. First, though familiar, the objection from contingent location is quite undeveloped. In fact, there is actually a family of versions of the argument from contingent location which might be made against the identity theory. And as we will see, a num- ber of them are not compelling. In this chapter, I will defend what I take to be the strongest version of the argument. Second, as it turns out, developing a strong argument from contingent location does not 6 MATT LEONARD generate a strong argument from vague location. Though I will eventu- ally develop a strong argument from each, perhaps surprisingly, they are structurally dierent arguments. This is because even though necessity and deniteness are formally similar, there are some key philosophical dierences between them. I will highlight two one in section two, af- ter defending an argument from contingent location, and one in section three, after defending an argument from vague location. These philo- sophical dierences are not only intrinsically interesting; they require us to develop these arguments with care. While the central goal of the chapter is to defend a pair of arguments against the identity theory, a subsidiary goal is to explore along the way some interesting, though perhaps previously unnoticed, dierences between contingent location and vague location. In my fourth chapter, \Explaining Harmony", I explore in more detail the interaction of parthood and location. Specically, I look in more detail at the fourth feature of a good theory of location: T implies that necessarily, if x is located at R, y is located at S, and x is a part of y, then R is a part of S. Philosophers are great at constructing exotica, but it's hard to even imagine how this principle could be false. But why is it true? What explanation could one possibly have for this modal mystery? The main WHAT IT IS TO BE LOCATED 7 purpose of this chapter is to explore ways one might explain this prin- ciple of harmony. The way forward, I propose, is to develop analyses of parthood and location that can explain the above modal mystery. In this nal chapter, I sketch three analyses, each of which can oer a satisfying explanation for it is the case that necessarily, ifx is located atR,y is located atS, andx is a part ofy, thenR is a part ofS. This principle is a principle of mereological harmony roughly, the view that the mereological relations among material objects mirrors and is mirrored by the mereological relations among those objects' locations. This suggests a natural question: if these three analyses imply this principle of harmony, do they imply other principles of harmony? If so, which ones? The middle part of this chapter explores which principles of harmony follow from the three analyses. Since each of the three analyses can explain why this principle of harmony is true, we should take them seriously. But which of them is best? In the nal section of the chapter, I oer a couple of tentative arguments for why I prefer one of the analyses to its two rivals. 8 MATT LEONARD CHAPTER TWO: What is it to be Located? What is it for a material objectO to be located at a region of space- timeR? 1 In searching for an answer, I am not just interested in search- ing for a necessary biconditional of the form: necessarily, O is located at R if and only if O bears some relation ' to R. I am looking for a metaphysical analysis of location; I am asking what location is. 2 Philosophers who have written on questions involving location largely fall within two broad camps, and these camps suggest two natural re- sponses to our question. Some take material objects to be distinct from (and share no parts with) regions of spacetime and thus take location to be a primitive relation. This suggests a rst response according to which the question has no answer. That is, there is no informative metaphysical analysis of what it is for O to be located at R. I will call this the primitivist theory of location. 3 Others identify material objects 1 I am asking about exact location (also sometimes called occupation). See Casati and Varzi (1999), Parsons (2007) and Gilmore (2018). 2 Two important notes. First, I will assume substantivalism about spacetime, but I will ignore complications arising from relativity theory. Second, I am not just look- ing for a conceptual analysis of location. I'm following a general kind of framework laid out in Dorr (2016) and Rayo (2015). In Dorr's ideology, I am looking for an informative answer of the form: to be located is to be F . And in Rayo's ideology: for it to be the case that O is located at R just is for it to be the case that '. Of course, analogous questions can be asked about location in other frameworks. For instance, Markosian (2014), p. 75, asks: in virtue of what does an object occupy the region of space that it occupies? In this paper, however, I am interested in asking what location is. 3 In the same spirit, some dene exact location in terms of some other locative primitive; see Parsons (2007), for instance. The main points I make in this paper WHAT IT IS TO BE LOCATED 9 with regions of spacetime and thus have said things like: necessarily,O is located at R if and only if O is identical to R. This suggests an an- swer according to which being located at a region just is being identical to that region. I will call this the identity theory of location. But each theory is vulnerable to well-known problems (which we will develop more carefully in due course). The identity theory is in tension with the plausible thesis that it is contingent where I am located. It is also in tension with the plausible thesis that a statue is distinct from the clay from which it is made, while sharing the same location. The primitivist theory, on the other hand, is consistent with these theses. However, the primitivist theory makes certain facts seem mysterious. Why is it that my body and its location must have the same size and shape? And why is it that if my arm is a part of my body, then my arm's location must be a part of my body's location? Unlike the prim- itivist, the identity theorist has a straightforward explanation of these facts: they are implied by the thesis that material objects are identical to their locations. It would be nice to have a theory of location T with the following features: 1. T is consistent with contingent location. 2. T is consistent with two material objects having the same location. about primitivism will also apply to taking other locative relations as primitive, as well. 10 MATT LEONARD 3. T implies that necessarily, a material object has the same size and shape as its location. 4. T implies that necessarily, if x is located at R, y is located at S, and x is a part of y, then R is a part of S. In this paper, I explore a new theory of location and argue that it has all four features. Unlike the identity theory, the theory I explore implies that material objects are distinct from their locations. But unlike the primitivist theory, it implies that material objects and their locations share parts. Consider a statue made out of a lump of clay. The statue and the clay seem to have dierent properties and so are distinct, by Leibniz's Law. Though distinct, the statue is very intimately related to the clay. Of course, the precise nature of this relationship (often called `constitution') is a matter of controversy, and is not an issue I will get into here. 4 However, here are two natural claims. First, the statue and the clay are the same size and shape. And second, the statue and the clay share parts with all and only the same things. Though the theory of location I develop does not imply that the statue is located at the clay, it takes the intimacy between the statue and the clay to be analogous to the intimacy between an object and its location. Being located at a region just is (what I will call) being mereogeometrically equivalent with the region. 4 Though see Wasserman (2017) for some details. WHAT IT IS TO BE LOCATED 11 Since the theory to be explored has some advantages over the prim- itivist and identity theories, I argue that it should be taken seriously. However, given the speculative nature of the enterprise, I am not claim- ing that the theory is completely cost-free. Along the way, I will also sketch some close rivals and argue that they too have a number of ad- vantages. Though I will defend a particular answer to the question of what it is to be located, my larger goal in the paper is to raise an inter- esting question and explore the problems and merits of some dierent metaphysical analyses of location. 5 In what follows, I brie y adapt some standard arguments to show that the identity theory fails to have features (1) and (2), and that the primitivist theory fails to have features (3) and (4). I then develop the new theory - the mereogeometrical theory - and argue that it has all four. In the last parts of the paper, I explore some connections the theory has with a current debate about coincidence and a question about material objects and their shapes. 5 A quick note on multilocation. There is a debate in the metaphysics of persistence literature about whether material objects are located at many dierent instanta- neous regions, or if they are only located at their four-dimensional paths. See Gilmore (2018) for a discussion. For the sake of space, I will mostly ignore the issue of the possibility of multilocation. Though the background mereology assumed later in the paper is inconsistent with standard views of multilocation, it can be easily tweaked in order to be consistent with multilocation. With these tweaks in place, the view I will defend can be made consistent with multilocation and thus a major version of endurantism. Details will be sketched in footnote 26. 12 MATT LEONARD 1. The Identity Theory In this section, I will brie y adapt some standard arguments to show that the identity theory fails to have features (1) and (2); that is, to show that it is not consistent with it being contingent where I am located and that it is not consistent with two material objects having the same location. The identity theory says that for O to be located at R just is for R to be a region such that O is identical to R. This analysis implies the following necessary biconditional: Necessarily, O is located at R i R is a region such that O is identical to R. (LI) The identity theory does not have features (1) or (2). Though this might look straightforward, it is worth considering a pair of arguments in a little detail. The identity theory identies being located with be- ing identical to a region. The arguments below rely on a higher-order version of Leibniz's Law which says that if being located just is being identical to a region, `being located' and `being identical to a region' are intersubstitutable. 6 Following Dorr (2016), let `' ' express the higher-order identication that for it to be the case that ' is for it to 6 It's worth noting that the arguments I'll present against the identity theory have close relatives which would work similarly against the mere necessary biconditional: necessarily, O is located at R i R is a region such that O =R. WHAT IT IS TO BE LOCATED 13 be the case that . Higher-order Leibniz's Law states: ' ! ['=p]![ =p] where ['=p] is a sentence which is just like except that ' is substi- tuted for free occurrences of the propositional variable p in . 7 I will begin with the argument from contingent location. Suppose that I am located at R. The argument runs as follows: 5. There is a region R 0 distinct from R such that it is possible that I be located at R 0 . 6. But if the identity theory is true, then no region R 0 distinct from R is such that it is possible that I be located at R 0 . 7. Therefore, the identity theory is false. Though I do not have a knock down argument for (5), I think it's very plausible. It's not a metaphysically necessary fact that I be located where I am in fact located. I could have been located elsewhere: there 7 See Dorr (2016), pgs. 48-49, for a couple of dierent versions of this principle. And see pgs. 50-51 for a discussion of how these principles relate to the issue of opaque contexts, generated for instance by attitude ascription reports, such as \Lois believes Superman can y." 14 MATT LEONARD are a number of regions distinct from my location that I could have been located at. 8 Next, (6) follows from higher-order Leibniz's Law and the necessity of distinctness, which says that if x and y are distinct, then they are necessarily distinct. And the necessity of distinctness follows from the necessity of identity 9 and a very plausible schema for the logic of meta- physical necessity: If p, then it's necessarily possible that p. (B) (B) is widely accepted. As Williamson (2013) notes, \If something is so, then how could it have been metaphysically impossible?" (p. 44). 10 The argument for (6) is as follows. Since I am located at R, the identity theory and higher-order Leibniz's Law tell us thatR is a region that I am identical to. But for any distinct region R 0 , I am necessarily distinct from R 0 , by the necessity of distinctness. That is, no distinct R 0 is such that I am possibly identical to R 0 . But since according to the identity theory, being identical to R 0 is part of what it is to be located at R 0 , it is not possible to be located at R 0 . Therefore, if the identity theory is true, then no region R 0 distinct from R is such that 8 This is something that should sound incredibly compelling to both endurantists who think that my locations are instantaneous regions and perdurantists who think that my location is my four-dimensional spacetime path. 9 On which, see Marcus (1946) and Kripke (1971). 10 See, however, Garson (2014) for some hesitation regarding (B). Also see Bacon (forthcoming b). WHAT IT IS TO BE LOCATED 15 it is possible that I be located at R 0 . Thus, given the truth of (5), it follows that the identity theory is false. 11 Now for the argument from co-location. The argument runs as fol- lows: 8. The statue and the clay are distinct and have the same loca- tion. 9. But if the identity theory is true, then no distinct things have the same location. 10. Therefore, the identity theory of location is false. Famously, the statue and the clay seem to have dierent properties; they seem to dier modally, aesthetically, and so on. Therefore, by Leibniz's Law, the statue and the clay are distinct. 12 But even though they have dierent properties, they are both composed of the same 11 See Skow (2005) and Schaer (2009) for the familiar counterpart theoretic re- sponse to the argument from contingent location, which rejects (5) though note that they might instead reject (6). I happen to not nd the standard counterpart theoretic response convincing, but this is not the place to defend this point. One problem worth mentioning, however, is that this line of reply requires rejecting sub- stitutional Leibniz's Law: a = b! '[a=x]! '[b=x]. But substitutional Leibniz's Law follows from a weak version of Leibniz's Law, (a = b! Fa! Fb), together with the plausible principle of -conversion: (x:')a$'[a=x]. 12 This view has been developed and/or endorsed in Thomson (1983), (1998), Cot- noir (2010), (2013), (2016), Cotnoir and Bacon (2012), Hovda (2013), Walters (forthcoming), Gilmore (forthcoming), and Goodman (ms). 16 MATT LEONARD particles and so are both located at the same region. 13 And so (8) is true. Next, (9) follows from higher-order Leibniz's Law and the symmetry and transitivity of identity. Given the identity theory, if it is the case that distinct thingsx andy are both located at some regionR, then it is the case that distinct things x and y are both identical to region R, contrary to the symmetry and transitivity of identity. So if the identity theory is true, then no distinct things have the same location. Thus, given the truth of (8), it follows that the identity theory of location is false. 2. The Primitivist Theory In this section, I will brie y sketch some standard arguments against primitivism which stem from the fact that primitivism fails to have features (3) and (4); that is, from the fact that it fails to imply that necessarily, my body and its location have the same shape, and that nec- essarily, my parts' locations are a part of my body's location. The prim- itivist thinks that no informative analysis of location can be given. 14 13 I have in mind the particular case where the statue and the clay are permanently coincident, otherwise the identity theorist could opt for the standard temporal parts friendly view that the statue and the clay do not share the same location. Of course, see Sider (2001) for the familiar counterpart theoretic response to the case of permanent coincidence, which rejects the distinctness of the statue and clay. 14 It's worth noting that there are a number of possible answers to our question in this paper which are not primitivist about location, but about some other relation; for instance, the relation an object bears to conguration space; see Albert (1996). But in this paper, I am just interested in the location relation. WHAT IT IS TO BE LOCATED 17 This is subject to the following family of complaints: \Some material objects are spherical. Some regions of space are spherical. And it is necessary that every spherical material object is located at a spherical region of space. But this can't just be a magical, mysterious necessity, a necessity that must be unex- plained." (Skow 2007, p. 116) \Why should there be this perfect correspondence between the mereological structure of things, and the mereological structure of the largest region they ll?" (Eagle 2016b) Upon re ection, it looks like there is a tight correspondence between the geometrical and mereological structure of material objects and the structure of their locations. But to what extent are the structures aligned? This is a controversial question. Some think that the struc- tures are perfectly aligned. 15 And some think that the structures can come apart; for instance, some think that extended simples are possible, that material objects can interpenetrate, that co-location or multiloca- tion is possible, and so forth. 16 However, here are two facts: 15 See Schaer (2009). 16 See Gilmore (2018) for a thorough exposition of relevant views. 18 MATT LEONARD Necessarily, ifx is located atR, thenx is the same size and shape as R. (G-Harmony) Necessarily, ifx is located atR,y is located atS, andx is a part of y, then R is a part of S. (P-Harmony) (G-Harmony) is a principle of geometrical harmony and(P-Harmony) is among the weakest principles of mereological harmony discussed in the literature. 17 (G-Harmony) and (P-Harmony) seem to cry out for an expla- nation. Why are they true? For the primitivist, this looks like a remarkable coincidence. On the other hand, the identity theorist has an explanation;(G-Harmony) and(P-Harmony) follow straightfor- wardly from the identity theory and Leibniz's Law. 18 But at rst glance, it looks like the primitivist has no explanation for (G-Harmony) and 17 Mereological harmony is the rough idea that the mereological structures of ma- terial objects and their locations match. As far as I know, it was rst discussed in Casati and Varzi (1999), pg. 122, though in terms of mereotopological structure (they call (P-Harmony) \(L.3)"). Varzi (2007) develops a number of mirroring principles and the idea is developed in Uzquiano (2006) and (2011). Saucedo (2011) develops an argument against the idea. See Leonard (2016) for a taxonomy of sys- tems of harmony. 18 Supersubstantivalism is the view that material objects are identical to regions of spacetime. It is sometimes claimed that supersubstantivalism can explain these necessary truths. But it is worth noting that it is the identity theory of location that explains them (which, of course, is a very natural theory of location for the supersubstantivalist). WHAT IT IS TO BE LOCATED 19 (P-Harmony). 19 Primitivism does not have features (3) and (4). Call that the argument from explanation. 20 A more precise and more controversial way of running the argument is sometimes defended in the literature. Some philosophers are attracted to the view that fundamental relations obey combinatorial principles. Though a number of combinatorial principles have been defended in the literature, 21 they all aim to capture the thought that the world contains a distribution of fundamental properties and relations, and that these properties and relations can be freely recombined to represent dierent possible worlds. On this view, if location is primitive, any pattern of location should be metaphysically possible. In particular, there are pat- terns of location violating (G-Harmony) and (P-Harmony). Here is a quick argument: 11. If location is primitive, then any pattern of location is meta- physically possible. 12. If any pattern of location is metaphysically possible, then (G-Harmony) is false. 13. (G-Harmony) is true. 19 This complaint is also made in Schaer 2009, p. 138, Arntzenius 2012, p. 146, and Nolan 2014, p. 98. 20 Though there are a number of arguments against primitivism (see Schaer (2009)), I focus on what I take to be the strongest. The other two main arguments are based on considerations of parsimony and modern physics. On the former, see Quine (1981), Lewis (1986), Sider (2001), Hawthorne (2006, Chapter 6), Nolan (2014), and Schaer (2009). On the latter, see Schaer (2009) and Lehmkuhl (2016). 21 See McDaniel (2007) and Saucedo (2011) for relevant principles of recombination. 20 MATT LEONARD 14. Therefore, location is not primitive. A parallel argument can be given in terms of (P-Harmony). In fact, Eagle (2016b) develops this sort of argument against locational en- durantism (the view that material objects persist in virtue of being ex- actly located at many dierent instantaneous regions of spacetime). 22 Eagle notes that this combinatorial argument also threatens perduran- tism and suggests that the perdurantist identify material objects with their locations. Though I have not gone into the details in this paper for how philosophers have defended or rejected this argument, I will develop a theory of location which is compatible with the soundness of this argument. 23 However, it's worth stressing that one need not be tempted by controversial Humean principles of recombination to feel the pull of the argument from explanation: the primitivist theory lacks an explanation of a number of necessary truths, and the identity theory has an explanation. All things considered, this clearly seems to count against the primitivist theory. 22 Eagle argues that the endurantist must take the location relation to be primitive in order to adequately respond to Sider (2001)'s argument from vagueness, but then argues that since the endurantist must take location to be primitive, she faces the combinatorial argument. 23 The combinatorial argument is developed in depth in McDaniel (2007) and Saucedo (2011). Of course, my formulation of the argument was very quick. Prim- itivists have rejected premise (12) in a number of ways. McDaniel (2007), for instance, claims that material objects have their shapes in virtue of their locations' shapes. We will come back to this idea in the last section. And Markosian (2014), for instance, reduces material parthood to subregionhood and claims that the mere- ological structure of objects is determined by the structure of their locations. WHAT IT IS TO BE LOCATED 21 3. The Mereogeometrical Theory Consider an arbitrary material object O and its location R. As we have seen, there are competing reasons for thinking that there are two things there and for thinking that there is really only one thing there. We have a modal reason to think that there are two things there: if O and R are identical, then O couldn't have been located somewhere else and that seems false. And yet,O andR instantiate many of the same geometrical and mereological properties, and so we are pressured to say that there is really only one thing there. A similar situation arises when we think about the statue and the lump of clay. We have a modal reason to think that there are two things there: since the statue and the lump dier in their de re modal proles, Leibniz's Law implies that they are distinct. And yet, they share many of the same geometrical and mereological properties, and so there is some pressure for thinking that there is really only one thing there. The theory to be explored takes seriously the idea that the statue and the clay are distinct. Of course, one dicult question for those who take this idea seriously is: what exactly is the relationship between the statue and the clay? Without attempting to answer this question here, I want to return to the two straightforward observations I made at the beginning of the paper. First, they are the same size and shape. Second, they share parts with all and only the same things. According 22 MATT LEONARD to the theory of location to be explored, these two observations are also true of material objects and their locations. On this view, to be located is to be mereogeometrically equivalent with a region, where x andy are mereogeometrically equivalent (\MG- equivalent" for short) if and only if (i) x and y are congruent and (ii) x and y are mereologically coincident. For any O and R, what it is for O to be located at R just is for R to be a region such that O is congruent and coincident withR. Call this the mereogeometrical theory of location (\the MG theory" for short). Let me brie y dene these terms. First, say that x and y are con- gruent just in case x and y have the same shape. 24 Part of what it is to be located at a region is to be congruent with that region. Second, say that x and y are mereologically coincident if and only if x and y overlap the same things (where x and y overlap if and only if x and y have a part in common); that is, for all z, z overlaps x if and only if z overlaps y. Part of what it is to be located at a region is to overlap all and only the same things as the region. 25 The MG theory implies the 24 By \shape" I really mean size and shape. What is it to have a certain size and shape? Though not much turns on this, here is a rst stab, supposing that space is Euclidean. It is natural to take shapes to be those properties that can be analyzed in terms of the metric function d. For instance, what it is for x;y and z to be three vertices of an equilateral triangle is for there to be an r such that d(x;y) =d(y;z) =d(z;x) =r: See Skow (2007). 25 Related accounts of location have been mentioned in the literature. In fact, Hawthorne (2006, pg. 118) suggests a view on which location is reducible to mereo- logical coincidence (see also Parsons (2007), f.n. 14). Gilmore (2014) develops this thought to show that there are a number of ways of accepting both multilocational endurantism and monistic substantivalism, the view that only spacetime regions are fundamental. And Kleinschmidt (2015) argues that part of what it is to be WHAT IT IS TO BE LOCATED 23 following necessary biconditional: Necessarily, O is located at R i R is a region such that O is MG-equivalent to R. (LM) Though there are many regions of spacetime, one of them my location is a lot like me: it is the one with which I am MG-equivalent. 26 The MG theory has a number of striking features. The rst, again, is that material objects and regions share parts. The second is that located material objects are fusions of spacetime regions. SupposeO is located atR and thatR is a fusion of some regionsxx. SinceO andR are coincident, whatever overlaps O overlaps at least one of xx. 27 The located somewhere is to have the same size and shape as that region. The view I am exploring is a combination of these ideas; later I will discuss why I think both components are necessary. 26 As mentioned in footnote 5, some think that persisting material objects are lo- cated at many dierent instantaneous regions of spacetime. Given the background mereology I have implicitly assumed (where parthood is absolute and 2-place), the MG theory is inconsistent with an object being located at two disjoint regions (if I am located at distinct regions R and S, then I am coincident with R and coinci- dent with S; since the relation being coincident with is symmetric and transitive, it follows that R and S are not disjoint). According to standard views of multilo- cational endurantism, though objects can gain and lose parts, they are nonetheless exactly located at many dierent disjoint regions. It is no news that such views are in tension with absolute 2-place parthood. As a result, endurantists standardly relativize parthood to times or regions. With this sort of tweak in place, an en- durantist friendly version of the MG theory would say that to be located at a region R (at time t, or at region R) is to be congruent (at t or R) with R and coincident (at t or R) with R. The MG theory is consistent with multilocation so long as MG-equivalence is relativized: that is, so long as geometrical predicates and the parthood relation are relativized in the standard way. 27 The notion of fusion I am using here is what Hovda (2009) calls a Type-1 Fusion, where A is a fusion of xx = df for any y, y overlaps A i there exists an x among xx and y overlaps x. 24 MATT LEONARD third is that it relies on non-extensional mereology. I will assume that parthood is re exive and transitive; and when restricting the quantiers to range over regions, I will assume that classical extensional mereol- ogy is necessarily true. 28 But when taking the quantiers to range over both material objects and regions, I will reject the following principle: If A and B are fusions of xx, then A =B. (Uniqueness) (A is a fusion of xx just in case each x among xx is a part of A and every part ofA overlaps somex amongxx. 29 ) If (Uniqueness) is true, then the MG theory implies that location and identity are necessarily coextensive; that is, it implies (LI), and thus has no advantages over the identity theory. These striking features deserve a pause. One might be worried: isn't the MG theory counterintuitive? Why go for a theory that implies, for instance, that I share parts with a spacetime region? While conforming to intuition may be a theoretical virtue, it is by no means an overriding criterion for the evaluation of a theory. Other theoretical features of the theory such as, importantly, explanatory fruitfulness should be considered as well. A theory of location should be evaluated with respect to how well it does with respect to both of these criteria. If we took conforming to our intuitions to be an overriding 28 On classical mereology, see Simons (2000), Hovda (2009), and Varzi (2016). 29 Here I am relying on what Hovda (2009) calls a Type-2 Fusion, framed in standard plural logic. WHAT IT IS TO BE LOCATED 25 criterion for acceptability, then we could discard the identity theory right away: after all, it implies that I am identical to a spacetime region! However, it would be misguided to reject the identity theory just because it is inconsistent with our everyday judgments. For the identity theory has other theoretical pay-os, such as, for example, explanatory fruitfulness: it has a straightforward explanation for (G- Harmony) and (P-Harmony). And though the primitivist theory is, obviously, not at odds with our everyday judgments (since it says nothing at all about what location is), it is explanatorily barren. The MG theory certainly has drawbacks, as does every theory of location. However, the MG theory is, all things considered, a better theory than its two main rivals. Or so I will now argue. 30 The MG theory has the four features with which we began: 1. T is consistent with contingent location. 2. T is consistent with two material objects having the same location. 30 Here is one more objection. The MG theory implies that a located material object is a fusion of spacetime regions and that its location is a fusion of the same regions. Unlike material objects, regions, one might think, are not contingently coincident with themselves. How can a material object be contingently coincident with some regions while its location not be contingently coincident with those regions, if both are fusions of the same regions? This is a fair objection. However, this result is not entirely dierent in structure from the thesis that though the statue and the clay are fusions of the same particles, while one has its parts contingently (the statue), the other doesn't (the clay). Since we are already taking seriously the distinctness of the statue and the clay, there doesn't seem to be anything distinctively worrying about the above objection. 26 MATT LEONARD 3. T implies that necessarily, a material object has the same size and shape as its location. 4. T implies that necessarily, if x is located at R, y is located at S, and x is a part of y, then R is a part of S. The rst three are straightforward. The MG theory clearly has feature (1), since it is consistent with the theses that it is contingent what a material object's shape is, and what parts it has, and thus that it might have been MG-equivalent with a dierent region than it is. 31 Moreover, it clearly has feature (2). It is consistent with the statue and the lump both being congruent with a regionR and overlapping all and only the same things asR (and in this case, all three would be MG-equivalent). It also has feature (3). Part of what it is to be located at a region is to be congruent with the region. And so the MG theory implies (G-Harmony). Moreover, the MG theory has feature (4). Suppose my arm (call it \Arm") is a part of my body (call it \Body"), that Arm is located at R arm and that Body is located at R body . Say that xvy i everything that overlaps x also overlaps y. Since Arm is a part of Body, we know that Armv Body. Since R arm and Arm are mereologically coincident, and since R body and Body are mereologically coincident, we know that R arm v Arm and that BodyvR body ; thus, we know that R arm v Arm 31 Indeed, the MG theory has an explanation for the contingency of location. It is contingent where I am located because it is contingent what my shape is and contingent what my parts are. WHAT IT IS TO BE LOCATED 27 v Bodyv R body . By classical extensional mereology for regions, it follows that R arm is a part of R body . 32 Thus, (P-Harmony) is true. Before I move on to some connections the MG theory has with some current debates, I want to brie y respond to a few natural questions one might have about the theory. First, in explaining (G-Harmony) and (P-Harmony), one might be tempted to think that we have just pushed back the problem by introducing another brute necessity. Here, however, it is important to note that the MG theory is not simply the necessary biconditional (LM): necessarily, O is located at R i R is a region such that O is MG-equivalent to R. The MG theory is a meta- physical analysis of what it is to be located. The analysis identies being located with being MG-equivalent to a region and, as stressed by Rayo (2015) and Dorr (2016), identications are a natural stopping place for explanation: \Suppose it is agreed on all sides that Hesperus (and Phosphorus) exist. Someone says: `I can see as clearly as can be that Hesperus is Phosphorus; what I want to understand is why.' It is not just that one wouldn't know how to comply with such a request one nds oneself unable to make sense of it." (Rayo 2015, p. 54) 32 In particular, xv y implies that x is a part of y, by (Strong Supplementa- tion): if x is not a part of y, then there is some part of x which does not overlap y. 28 MATT LEONARD (G-Harmony), (P-Harmony), and (LM) are all explained by what location is. And since the MG-theory is an identication, it is a natural stopping place for explanation. Note that a similar concern arises for those who think that mate- rial objects are identical to regions and who merely adopt the neces- sary biconditional (LI) that is, necessarily, O is located at R if and only if R is a region such that O is identical to R rather than the identity theory itself. 33 Though (LI) and Leibniz's Law jointly imply (G-Harmony) and (P-Harmony), one might wonder: do we really have a satisfying explanation of (G-Harmony) and (P-Harmony)? What explanation do we have for (LI)? However, one virtue of the framework in which I am working is that the MG theory and the iden- tity theory yield satisfying explanations for truths like (G-Harmony) and(P-Harmony), and do not themselves cry out for an explanation. Second question. It is natural to suppose that coinciding objects have the same mass; moreover, it is natural to think that the statue inherits its mass from the clay. Since regions and material objects are coincident, do regions of spacetime have mass? This is an open question and the MG theory is non-committal, though I am inclined to think that the more natural answer is that material objects inherit their masses (and other magnitudes associated with elds) from the regions 33 Principles similar to (LI) have been discussed in the literature. Parsons (2007) calls the following principle `the identity theory of location': x is located at r if and only if x is identical to r. And Gilmore (2018) calls the following principle `Supersubstantivalism+': necessarily, if x is located at r, then x is identical to r. WHAT IT IS TO BE LOCATED 29 with which they are coincident. 34 In this respect, the MG theory is a lot like the identity theory in that it pins down fundamental magnitudes on to regions themselves. Third, the MG theory has two parts. Do we need both? Why not go for a simpler view according to which being located at a region just is being congruent with the region? This would lead to a distinct view where material objects would be massively multilocated at every region with which they are congruent, a view which I do not endorse. 35 Or why not go for the simpler view according to which being located at a region just is being coincident with the region? This is a more complicated question, and one which I will postpone until the nal section of the paper. Fourth, the MG theory says that being located at a region just is being MG-equivalent with the region. But this raises an interesting question. Can something be a location if it is not a region? Con- sider a simpler view (call this \the Simple MG theory") according to which being located just is being MG-equivalent with something. MG- equivalence is an equivalence relation that partitions objects into equiv- alence classes; thus, the statue, the clay, and a particular region R are all members of an equivalence class. According to the Simple MG the- ory, every member in a particular equivalence class is located at every 34 This idea is in Hawthorne (2006: pg. 188, fn. 18), Sattig (2006), Eagle (2010), Eagle (2016b: pp. 522-523), and is what Simon (forthcoming) calls outsourcing accounts of property possession. 35 Though see Bacon (forthcoming a). 30 MATT LEONARD other member; in particular, R itself is located at both the statue and the clay, and the statue and the clay are located at each other. Why not go for the Simple MG theory? 36 One might rst appeal to the fact that we do not normally speak as if material objects are locations. But it is not clear how much weight we should give this consideration given that we are engaged in speculative metaphysics. Here are two more substantive arguments against the Simple MG theory. The rst is that, depending on some details, the Simple MG theory might not imply (P-Harmony) and thus fail to have feature (4). As before, suppose Arm is located at R arm and Body is located atR body . The Simple MG theory implies thatR arm is located at Arm andR body is located at Body, as well. Now suppose thatR arm is part ofR body . From the fact that (i)R arm is located at Arm, (ii)R body is located at Body, and (iii) R arm is a part of R body , (P-Harmony) tells us that Arm is a part of Body. But in order to conclude that Arm is a part of Body, we need to rely on (Strong Supplementation). Though we assumed that this was necessarily true of regions, this is a principle that some philosophers deny when it comes to material objects (this will come up in the next section in more detail). Here is a second argument. For any regionR,R could not have been located somewhere other than where it is located. But there might 36 Note that a similar question arises for the identity theory. Say that the simple identity theory is the view that what it is to be located just is to be identical to something. On this view, the number seven is located at itself. One might ask: why go for the identity theory rather than the simple identity theory? WHAT IT IS TO BE LOCATED 31 have been a material object O located at R even though O is not in fact located atR. If the Simple MG theory is true, thenR would have been located somewhere else (namely, at O). Therefore, the Simple MG theory is false. Though these two worries seem to me to count in favor of the MG theory, I take the Simple MG theory to be a view on the same team, rather than a competing rival. 37 4. Coincidentalism In this section, I explore some connections the MG theory has with a current debate about mereological coincidence. Say that coincidental- ism is the view that some material objects are distinct and yet overlap the same things, and thus that (Uniqueness) is false. The theory of location we have been exploring takes seriously the view that the statue and the clay are distinct and yet overlap the same things. There are two rival versions of coincidentalism in the literature, corresponding to two sets of mereological principles. Both types of coincidentalists agree that the statue and the clay overlap the same things. After all, the statue and the clay are composed of the very same particles; and if 37 It's also worth brie y thinking about other location relations. Following some ideas by Casati and Varzi (1999) and Parsons (2007), we can say that x is weakly located atR ix is exactly located at a regionS which overlapsR. In other words, x is congruent and coincident with a region S andR overlapsS. As Parsons notes, dening other relations in terms of exact location implies that anything with a weak location has an exact location, which some might nd problematic. While I am aware that dening other relations in terms of exact location generates this worry, I do not attempt to resolve this dicult issue in this paper. For a taste of location theories, see Casati and Varzi (1999), Hudson (2001), Gilmore (2006), Sider (2007), Parsons (2007), Eagle (2010), (2016a), (2016b), and Kleinschmidt (2016). 32 MATT LEONARD A and B are both fusions of the same set of particles, then A and B overlap the same things. But whereas one group holds that the statue and the clay are both parts of each other, the other holds that it is only the clay that is a part of the statue (and not vice versa). This debate about the statue and the clay can be carried over to an object and its location: one group might say that an object and its location are parts of each other, and the other might say a region is a part of an object located at it (but not vice versa). Which view is right? I don't know; the MG theory is non-committal. But below I note some interesting consequences for each version of coincidentalism. To clarify what these groups are saying, let's look at what principles they accept and deny. Notice that(Strong Supplementation) and (Anti-Symmetry) jointly imply(Uniqueness), and so coincidental- ists must reject at least one of them: If x is not a part of y, then there is some part of x which does not overlap y. (Strong Supplementation) If x and y are parts of each other, then x =y. (Anti-Symmetry) The rst camp rejects(Anti-symmetry) and accepts(Strong Sup- plementation). Following Gilmore (forthcoming), let S-Coincidentalism WHAT IT IS TO BE LOCATED 33 be the conjunction of (Strong Supplementation) and the negation of (Uniqueness). If the statue and the clay overlap the same things, it follows from(Strong Supplementation) that the statue and the clay are parts of each other; that is, the statue and the clay are mutual parts. 38 In the spirit of S-Coincidentalism, suppose that we also take part- hood to obey(StrongSupplementation). From this it follows that objects and their locations are mutual parts. As we saw in the previ- ous section, the MG theory of location implies (P-Harmony). But without assuming stronger mereological principles like (Strong Sup- plementation), it does not imply a number of further principles of mereological harmony. 39 But in the presence of (StrongSupplemen- tation), the MG theory (like the identity theory) implies a particu- larly strong version of mereological harmony. From this it follows, for instance, that it is impossible for there to be extended simples (simple objects located at composite regions), for objects and their locations to dier in how many parts they have, and for gunky material objects (objects all of whose parts have proper parts) to be located in atom- istic space. 40 This all follows straightforwardly given the transitivity 38 This view has been developed and/or endorsed in Thomson (1983), (1998), Cot- noir (2010), (2013), (2016), Cotnoir and Bacon (2012), Hovda (2013), and Gilmore (forthcoming). 39 Though note that it straightforwardly implies the following principle of harmony: if x is located at R and y is located at S, then x overlaps y i R overlaps S. 40 See Cotnoir (2013) for a discussion on how to dene proper parthood for S- Coincidentalism. 34 MATT LEONARD of parthood and the fact that objects and their locations are mutual parts, and thus stand in all the same parthood relations. The other version of coincidentalism, A-Coincidentalism, is the con- junction of (Anti-Symmetry) and the negation of (Uniqueness). From A-Coincidentalism it follows that the statue and the clay are not mutual parts. But A-Coincidentalists typically make an additional mereological claim: the clay is a part of the statue (and thus not vice versa). 41 In the spirit of A-Coincidentalism, what happens if we take parthood to obey (Anti-Symmetry) and maintain that the location of an object is a part of the object (though not vice versa)? Given these assumptions, it turns out that the MG theory is inconsistent with a number of principles of harmony. For instance, it is inconsistent with: ifx's location is a part ofy's location, thenx is a part ofy. The statue and clay are both located at some region R, and R is a part of itself, but the statue is not a part of the clay. 42 It is also inconsistent with: ifx is a proper part ofy, thenx's location is a proper part ofy's location. The statue and the clay are both located at R, the clay is a proper part of the statue butR is not a proper part of itself. Moreover, it is inconsistent with: ifx's location is simple, then so isx. No simple object is located at a simple region. 43 41 This view is defended or discussed in Goodman (ms), Walters (forthcoming), and Gilmore (forthcoming). 42 This is rst observed in Cotnoir (2013), who argues that this is one reason to prefer S-Coincidentalism to A-Coincidentalism; though see Walters (forthcoming) for a response to this argument. 43 Setting aside theS- vs. A-coincidentalism debate, the MG view itself implies that if a point particle p is located at a point-sized region R, then they overlap (by the WHAT IT IS TO BE LOCATED 35 5. Material Objects and Their Shapes In this nal section, I return to a question posed earlier: why not go for a simpler theory the mereological theory (or the \M theory" for short) where to be located at R is to be coincident with R and for R to be a region? Why prefer the MG theory rather than the M theory? The main reason is that the M theory does not seem to have an explanation for (G-Harmony). If I am located at R, and location is mere coincidence with a region, why is it that necessarily, my location and I have the same shape? Unlike the M theory, the MG theory has a straightforward explanation being congruent with a region is part of what it is to being located at that region. There is, however, an interesting route from the M theory to (G- Harmony), by way of the following thought: perhaps the geometrical re exivity of parthood). That is, p and R must have a part in common. Question: what is that part? Though the MG theory itself doesn't tell us, I will make three remarks about this case. First, I do not think we should identify the particle with its location, otherwise the worries raised earlier about contingent location and co- location would re-emerge for particles. Second, though the MG view itself doesn't tell us what part is common to both p and R, both S-coincidentalism and A- coincidentalism do tell us. The S-coincidentalist will tell us that p and R are both parts of each other; they are mutual parts. The A-coincidentalist will tell us that R is a part ofp, butp is not a part of R. Third, note that while A-coincidentalism implies that p is not simple, the standard way of thinking about S-coincidentalism does not. As Cotnoir (2013) notes, there are independent reasons for which the S-coincidentalist ought to dene proper parthood as x < y = df x y^y6 x, rather than the more standard x < y = df x y^x6= y. Notice that with this alternative denition, even thoughp andR are mutual parts, they are actually both simples, since neither has a proper part, that is, a part of which it is not a part. And so theS-coincidentalist can maintain that the MG view is consistent with the thesis that simple particles are located at simple regions (though, admittedly, for a kind of strange denitional reason). 36 MATT LEONARD properties of material objects are reducible to the geometrical proper- ties of their locations. That is, perhaps the geometrical properties of material objects are inherited from the shapes of their locations. 44 There are a couple of ways to make this idea more precise. First, an M theorist might adopt the following principle: Necessarily, x has a material shape S i x is located at some R and R has shape S. (Inheritance) It might be tempting to appeal to (Inheritance) in order to ex- plain (G-Harmony); after all, it implies (G-Harmony). However, (Inheritance) stands no less in need of an explanation than does (G-Harmony) itself both are necessary biconditionals. A more promising suggestion is to formulate the idea as an identication: (x has a material shape S) 9R(x is located at R and R has shape S) (Inheritance) This says that for a material object to have a shape is for it to have a location with that shape. Unlike the necessary biconditional above, this principle implies (G-Harmony) and stands in no need of an ex- planation, since it claims that the property of having a material shape 44 Again, see Hawthorne (2006: pg. 188, fn. 18), Sattig (2006), Eagle (2010), Eagle (2016b: pp. 522-523), and Simon (forthcoming). WHAT IT IS TO BE LOCATED 37 S is identical to the property of having a location with shape S. The M theorist, therefore, is free to accept (Inheritance) in order to explain (G-Harmony). 45 Moreover, it's worth noting that the M theory helps solve a little puzzle about geometrical inheritance principles. Consider the following plausible principle: The shape of a material object is an intrinsic property. (Intrinsic) Shapes are paradigmatic examples of intrinsic properties. 46 Of course, it is notoriously dicult to say just what it is for a property to be intrinsic. However, one very rough plausible idea is that my intrinsic properties are the ones which do not depend on what other things are like, other than me and my parts. In the same spirit, Skow (2007) writes: \A shape is intrinsic, then, just in case it can be completely analyzed in terms of the fundamental spatial relations among the parts of things that instantiate it" (pg. 115). (Intrinsic) is plausible because it seems like what shape I am does not depend on what other things are like. 45 Indeed, the main argument philosophers like McDaniel (2007) and Skow (2007) have given for these sorts of inheritance principles is that they explain why it is that necessarily material objects and their locations have the same shape; that is, it explains(G-Harmony). It's worth noting that though I formulate the principle in the framework of saying what it is for a material object to have a shape, McDaniel and Skow formulate the principle in terms of an object having a shape in virtue of being located at a region with that shape. 46 See Lewis (1983), (1986), (1988). 38 MATT LEONARD The puzzle is that (Intrinsic) seems to be inconsistent with geo- metrical inheritance principles. (Intrinsic) says that what shape I have does not depend on what other things are like, other than me and my parts. And inheritance principles say that what shape I have does depend on other things (in particular, my location). For this rea- son, McDaniel (2007) and Skow (2007) argue that one ought to reject (Intrinsic). 47 However, the M theory solves the puzzle. Note that the M theory to- gether with either S-Coincidentalism or (the spirit of) A-Coincidentalism implies: If O is located at R, then R is a part of O. (Region Parts) If (Region Parts) is true, then (Intrinsic) and (Inheritance) are not inconsistent after all. If a material object is spherical, then one can consistently maintain that what it is for an object to be spherical is for it to be located at a spherical region, and that the object's being spherical is an intrinsic property of the object (since its location is a part of it). 48 47 That is, assuming that material objects and their locations are distinct. 48 There is another version of coincidentalism in the literature according to which if x constitutes y, neither is a part of the other; see Lowe (2003). If we say that an object and its location are coincident and neither is a part of the other, the M theorist of course no longer has the resources to solve the puzzle, since she does not have (Region Parts). WHAT IT IS TO BE LOCATED 39 Is (Inheritance) true? I have a certain geometrical property: having a material shape S. Is the property of having a material shape S identical to the property of being located at a region R with shape S? I don't know. It is natural to suppose that material objects instan- tiate the very same geometrical properties that regions do. Indeed, it is natural to suppose that what it is for a material object to in- stantiate a geometrical property has nothing to do with its location. 49 Moreover, the main reason to go for(Inheritance) is that it implies (G-Harmony). But as we've seen, the MG theory already implies(G- Harmony), so it's not clear what is to be gained by(Inheritance) . Furthermore, the MG theory and (Inheritance) are circular in a certain respect: the MG theory says that part of what it is for O to be located atR is to be congruent with R (that is, have the same shape as R), and (Inheritance) says that what it is for a material object to have a shapeS is to be located at a region with shape S. Presumably, these sorts of circularities are to be avoided. 50 Those who nd them problematic and who accept(Inheritance) can adopt the M theory. 49 Again, this paper is not the place to develop a metaphysics of shapes, but, presum- ably, the single analysis of what it is for a complex object (whether it be material or spatiotemporal) to have a shape S is to be spelled out in terms of the distance relations between its parts. 50 See Dorr (2016), pgs 70-79, who explores a number of No-Circularity principles. Dorr attempts to formulate a principle that distinguishes between benign and vi- cious circles and it is worth noting that the circularity between the MG theory and (Inheritance) violates the principle he settles for. This principle (which he calls Only Logical Circles) roughly states: \the case where the term on one side of a true identication occurs as a proper constituent of the term on the other side can arise only if all of the other expressions on the more complex side ... are or are equivalent to logical terms (p. 74)." 40 MATT LEONARD I happen to be agnostic about the truth of (Inheritance) and thus have developed the MG theory which can have both (G-Harmony) and (Intrinsic) without (Inheritance) . 6. Conclusion Though the primitivist and identity theories have a number of virtues, they also suer from a number of problems. They are either inconsistent with the contingency of location and the possibility of co-location, or they leave a number of modal mysteries unexplained. The MG theory, on the other hand, has straightforward explanations for these modal facts. It's also consistent with the contingency of location and the pos- sibility of co-location. Given these virtues, the MG theory is a better overall theory (as are some of the MG theory's close rivals, such as the Simple MG Theory and the M theory), and thus should be taken seriously as an answer to the question of what it is to be located. WHAT IT IS TO BE LOCATED 41 CHAPTER THREE: On the Contingency and Vagueness of Where I Am I am located at a particular region of spacetime. This region my location has the exact same size and shape as I do. 51 However, it is contingent where I am located. I take it that I could have had a dierent size or shape. Thus, I could have been located somewhere else. Moreover, it is vague where I am located. I take it that I do not have a perfectly precise shape. There are a number of very similarly shaped regions in my vicinity, but it's vague which of them I have the same shape as. Thus, there's no region that I am denitely located at. It is contingent where I am located and it is vague where I am located. A good theory of location should be consistent with the contingency and vagueness of location. In this paper, I look at an emerging theory of location and argue that it is inconsistent with these theses. According to this theory of location which I will call the identity theory of location to be located at a region just is to be identical to that region. This theory of location naturally falls out of supersubstantivalism, the view that material objects are just identical to regions of spacetime, 51 By location I mean exact location (which is sometimes also called the region that I occupy). In addition to having my exact size and shape, my exact location also bears the same distance relations to things that I do. See Casati and Varzi (1999), Parsons (2007) and Gilmore (2018). 42 MATT LEONARD which has been defended in Sider (2001), Skow (2005), Schaer (2009), Nolan (2014), and Eagle (2016). 52 It is no news that the identity theory is in tension with the thesis of contingent location. 53 And given the formal similarities between con- tingency and vagueness, one might think that the tension straightfor- wardly carries over to the thesis of vague location. Thus, it is tempting to suppose that it is not really worth exploring the ways in which the identity theory is inconsistent with the theses of contingent and vague location. But as I hope to convince you, this is mistaken. It is mistaken for two reasons. First, though familiar, the objection from contingent location is quite undeveloped. In fact, there is actually a family of versions of the argument from contingent location which might be made against the identity theory. As we will see, a number of them are not compelling. In this paper, I will defend what I take to be the strongest version of the argument. Second, as it turns out, developing a strong argument from contingent location does not gen- erate a strong argument from vague location. Though I will eventually develop a strong argument from each, perhaps surprisingly, they are structurally dierent arguments. This is because even though necessity and deniteness are formally similar, there are some key philosophical dierences between them. I will highlight two. 52 The name comes from Sklar (1974), pg. 214. 53 Supersubstantivalists have responded here: Sider (2001), Skow (2005), and Schaf- fer (2009). WHAT IT IS TO BE LOCATED 43 The paper is structured in the following way. In the second sec- tion, I will defend an argument from contingent location, and then I will highlight the rst dierence. In the third section, I will defend an argument from vague location, and then I will highlight the second dierence. These philosophical dierences are not only intrinsically in- teresting; they require us to develop these arguments with care. While the central goal of the paper is to defend a pair of arguments against the identity theory, a subsidiary goal is to explore along the way some interesting, though unfortunately neglected dierences between con- tingent location and vague location. I begin with some background assumptions. 7. Preliminaries The Identity Theory of Location. There are a number of frame- works for thinking about theories of location. Sometimes the identity theory is sketched as a mere material biconditional (x is located aty i x =y), 54 and sometimes as a necessary material conditional (necessar- ily, x is located at y only if x =y). 55 However, I am going to work in a dierent framework. Though the above biconditional and necessary conditional might be true, a theory of location ought to say more; it ought to say what it is to be located. One of the strongest arguments in favor of the identity theory is that it can provide an explanation for 54 See Parsons (2007), pg. 227, who calls this material biconditional `the identity theory of location'. 55 Gilmore (2018), who calls this `Supersubstantivalism+'. 44 MATT LEONARD a number of necessary truths which seem to cry out for an explanation; for instance, for why it is the case that necessarily, my location and I have the same shape, and for why necessarily, my location contains the locations of my parts. This is because I am identical to my location. 56 But if the identity theory is formulated merely as a biconditional or a necessary conditional, then it's not obvious that it does explain these necessities in a satisfying way. The biconditional and the necessary conditional stand no less in need of explanation than the things they were invoked to explain. However, as Rayo (2015) and Dorr (2016) note, identications are excellent stopping places for explanation. Though we can sensibly ask what explanation one has for the necessary conditional, we cannot sen- sibly ask what explanation one has for the identity theory if the theory says that location just is identity. For the explanatory argument to be plausible, the identity theorist must oer more than a mere bicon- ditional or necessary conditional. One way forward is by oering an identication of what location is: if location just is identity, then we have an awesome explanation for the above necessities, which doesn't itself cry out for an explanation. Therefore, I propose that we take the identity theory to be oering a metaphysical analysis of location. 57 Following the framework laid out 56 See Schaer (2009), Nolan (2014), and Eagle (2016) for this line of argument. 57 It's worth noting that the arguments I'll present against the identity theory have close relatives which would work similarly against the conditionals. WHAT IT IS TO BE LOCATED 45 in Dorr (2016) and Rayo (2015), let's begin with a simple analysis. Ac- cording to what I will call the simple identity theory, what it is forx to be located at y is for x to be identical to y. Following Dorr (2016), we will write `' ' to express the higher-order identication that for it to be the case that ' is for it to be the case that . We can formulate the simple theory as: (x is located at y) (x =y) (Simple-Identity) This says that to be located is to be identical. 58 However, this simple formulation of the identity theory is not very plausible. I can imagine two ways in which one might want to amend the simple identity theory. First, note that one consequence of the simple theory is that everything is located at itself: for instance, the number seven! But there are plenty of things that seem to not be lo- cated anywhere at all. This suggests a second theory, where forx to be located at y is for y to be a region such that x is identical to y: 58 Or with the Rayo gloss: being located just is being identical. Ocially, Dorr would formulate the thesis as: x is located aty x;y x is identical toy, where the subscripts indicate that x andy are both bound by `'. For simplicity of notation, I will not include the subscripts throughout the paper, but will leave them implicit. I am not just taking these identities to hold case by case; rather, I take this to say that the relations are identical. Moreover, I've formulated the theory in what Dorr calls the sentential style. But we can also formulate (Simple-Identity) in the predicative style as follows: (x;y)(x is located aty)(x;y)(x is identical to y). Or more brie y, location is identity. 46 MATT LEONARD (x is located at y) (y is a region and x =y) (R-Identity) This says that to be located is to be identical to a region. That is, being a region is part of what it is to being something's location. An alternative amendment to the simple theory is natural for those who accept (i) a restricted version of supersubstantivalism according to which only some (and not all) regions count as material objects, and (ii) the thesis that location is only a relation between material objects and regions. Some supersubstantivalists Schaer (2009), for instance not only claim that all material objects are regions, but also that all regions are material objects. 59 But others Nolan (2014), for instance deny this and claim that only some regions of space- time count as material objects. 60 The problem is that (i) and (ii) are inconsistent with both (Simple-Identity) and (R-Identity), since both (Simple-Identity) and (R-Identity) imply that every region has a location, and thus (if only material objects have locations) then every region is a material object. But restricted supersubstantivalism is consistent with another natural formulation of the identity theory: 59 One natural question for those supersubstantivalists who also think that all regions of spacetime are material objects is whether this is necessarily the case. If so, one natural way of stating the view would be as a metaphysical analysis: Mx Ry, which says that forx to be a material object is for x to be a spacetime region (and equivalently, given that `' is symmetric, to be a region is to be a material object). See Dorr (2016), pg. 43. 60 Which regions count as material objects? Nolan discusses a number of possible answers to this question. See pg. 95. WHAT IT IS TO BE LOCATED 47 (x is located at y) (x is a material object and x =y) (M-Identity) Given supersubstantivalism, this theory of location says that to be lo- cated is to be a material object that is identical to a region. That is, being a material object is part of what it is to being located. As we've just seen, the supersubstantivalist has a number of candi- date identity theories of location. As we press on, when I refer to the identity theory without qualication, I will be referring to the disjunc- tion of these three theories (keeping in mind that (R-Identity) and (M-Identity) are the more plausible of the three). Modality and Vagueness. I am going to make a series of fairly uncontroversial assumptions about modality and vagueness. And then I am going to make an assumption which is, admittedly, more contro- versial, though one that I happen to nd quite plausible. When it comes to issues concerning modality, I will introduce an op- erator for necessity `'' which means `it is necessary that ''. We can then dene possibility in the usual way for it to be possible that' is for it to not be necessary that not-'. I will assume some fairly standard and uncontroversial axiom schemas governing necessity. I will assume that it obeys the modal logic T, and thus that it obeys the following two schemas: 48 MATT LEONARD '!' (T ) ('! )! ('! ) (K ) I am going to treat issues concerning vagueness very much like I treat issues concerning modality. 61 Before I describe the formal way I treat vagueness, I should mention that it is sometimes thought that the phenomenon of vagueness shows that we should reject classical laws of logic like bivalence and excluded middle. But I think that this is mistaken. I will assume classical logic. 62 My formal treatment of vagueness will employ a pair of sentential operators, where `'' means `it is denite that '', and where `r'' means `it is vague whether '' (I will use `vague' and `borderline' in- terchangeably). And I will assume that we can sensibly quantify into these operators. As usual, we can take either operator and dene the other in terms of it: for it to be vague whether' is for it to be neither denite that', nor denite that not-'; and for it to be denite that' is for it to be the case that ', but not vague whether '. 63 61 My way of setting up issues concerning vagueness follows the setup in [redacted]. 62 Obviously, this is not the place to defend this point in any detail. However, see Williamson (1994), pgs 187-198, for an argument for why the temptation to reject classical logic for vagueness related reasons is mistaken. Moreover, see Bacon (2018), Chapter 1, for some additional problems with weakening classical logic for vagueness related reasons. Also note that supervaluationists standardly accept the directly relevant principles of classical propositional logic, like the law of excluded middle, as do the ontic views in Barnes and Williams (2009) and Wilson (2017). 63 Though it doesn't matter for our purposes which we take as primitive, I will assume that the following linking-biconditionals denitely hold: r' $ (:'^::') WHAT IT IS TO BE LOCATED 49 As with necessity, I will assume that deniteness obeys the modal logic T: '!' (T ) ('! )! ('! ) (K ) Moreover, I will assume that (T ) and (K ) are both denitely, de- nitely, ..., denitely true. Now, more controversially, I am going to assume the legitimacy of substitutional Leibniz's Law: a =b!'[a=x]!'[b=x] where'[a=x] is the formula which is just like' except thata is substi- tuted for free occurrences ofx in'. Famously, substitutional Leibniz's Law implies the necessity of identity. 64 One instance of substitutional Leibniz's Law is: a =b!(a =a)!(a =b) ' $ ('^:r') 64 See Marcus (1947) and Kripke (1971). 50 MATT LEONARD Since we know that(a =a), 65 it follows that a =b!(a =b). Similarly, substitutional Leibniz's Law implies the deniteness of identity. 66 An analogous instance of it is: a =b! (a =a)! (a =b) But since we know that (a =a), 67 it follows that a =b! (a =b). Evans (1978) gave a much-discussed proof of a related conclusion: r(a =b)!a6=b. In other words, if it is vague whether a is identical to b, then a is distinct from b. However, he thought that his argument showed too much. Evans seems to think that he provides a reductio on the assumption thatr(a = b), thereby showing that it can never be vague whether a is identical to b. However, he never actually de- rives a contradiction, 68 and thus fails to show that there can never be cases of borderline identity. He did, however, correctly show something interesting: borderline identity implies distinctness. Note that we should not confuse the necessity and deniteness of identity with the necessity and deniteness of distinctness: 65 One can either take this as an assumption (which is what I ocially will do), or one can add the Rule of Necessitation as an additional assumption, and then derive (a =a) from a =a and Necessitation. 66 See Evans (1978) and Salmon (1982) for related arguments. 67 Again, I'll take this as an assumption. But one can also derive it from Necessita- tion and from a =a. 68 Without S5, that is, which is totally implausible for . This point is made in Heck (1998) and McGee (1997). WHAT IT IS TO BE LOCATED 51 a6=b!(a6=b) (Necessity of Distinctness) a6=b! (a6=b) (Definiteness of Distinctness) There is no similarly straightforward argument from substitutional Leib- niz's Law for the necessity and deniteness of distinctness (I will return to this important point in section 2, where I will argue that we should accept one of them, but not the other). Before moving on, it is worth pausing to note why some might reject substitutional Leibniz's Law. First, a number of philosophers have de- fended certain versions of counterpart theory. Indeed, some have relied on counterpart theory to defend contingent identity. And some super- substantivalists have relied on counterpart theory in response to the argument from contingent location. 69 Substitutional Leibniz's Law (in a modal language) rules out these ideas. Second, it is worth noting that a certain kind of supervaluationist might reject substitutional Leibniz's Law and claim that it is only valid when the names `a' and `b' are not vague. 70 Though this paper is not the place to develop a full defense of sub- stitutional Leibniz's Law, I should at least mention in passing that it is 69 See Sider (2001), Skow (2005), and Schaer (2009) for the standard counterpart theoretic reply to the objection that the identity theory is inconsistent with contin- gent location. 70 See Thomason (1982), Lewis (1988), and McGee (1997). Relatedly, if one thought that names were not modally rigid (which was the standard pre-Kripkean view), then one might deny the parallel argument for the necessity of identity by rejecting substitutional Leibniz's Law. 52 MATT LEONARD jointly implied by two incredibly plausible principles: (1) a weak ver- sion of Leibniz's Law, (a = b! Fa! Fb), and (2) the principle of -conversion: (x:')a$ '[a=x]. 71 I, therefore, will assume substitu- tional Leibniz's Law. It's worth brie y mentioning one nal objection. One might take `r'' to be mere shorthand for a metalinguistic claim like `the sen- tencep'q is indeterminate'. 72 And so one might deny the legitimacy of quantifying into vague contexts. However, it is worth mentioning that historically, some philosophers also thought that quantifying into modal contexts was illegitimate. Such philosophers thought that claims like `necessarily, a =b' were implicitly quotational, and that modality was purely metalinguistic. 73 I, however, will assume that we can quantify into both modal and vague contexts. The purpose of this paper, of course, is not to convince the reader that counterpart theory is false or that metalinguistic approaches to vagueness are false. The paper can be read conditionally: if, like me, one adopts substitutional Leibniz's Law, if one takes modality seriously, 71 Note that some supervaluationists reject -conversion for precisely this reason: see Thomason (1982), Lewis (1988), McGee (1997). 72 See, for instance, McGee (1997), pg. 152, who writes \Just to make sense of the attachment of the word `determinately' to an open sentence containing free variables is a bit of a stretch, since we primarily think of determinacy as an attribute of sentences." 73 See, for instance, Quine (1953). WHAT IT IS TO BE LOCATED 53 and one treats vagueness like modality in certain ways, then we'll see that the identity theory is false. 74 8. The Argument from Contingent Location In this section, I will develop an argument to show that the iden- tity theory is inconsistent with the plausible thesis that it is contingent where I am located. The version of the argument I will defend relies on a higher-order version of Leibniz's Law (as does the argument from vague location I defend in the next section). This principle states that given a higher-order identication, the terms anking `' are intersub- stitutable: ' ! ['=p]![ =p] where ['=p] is a sentence which is just like except that ' is substi- tuted for free occurrences of the propositional variable p in . 75 74 Though some further problems for metalinguistic approaches to vagueness are worth mentioning. First, there are well-known issues concerning higher-order vague- ness discussed in Williamson (1994). Second, many take Montague's Theorem to show a problem with metalinguistic approaches to modality. But as Bacon (2018) notes, the same problem arises for metalinguistic approaches to vagueness which take vague expressions to be implicitly quotational, so long as obeys the modal principles K and T. Also see Bacon (2018; chapter 2) for additional problems with supervaluationist approaches to vagueness. 75 See Dorr (2016), pgs. 48-49, for a couple of dierent versions of this principle. And see pgs. 50-51 for a discussion of how these principles relate to the issue of opaque contexts, generated for instance by attitude ascription reports, such as \Lois believes Superman can y." 54 MATT LEONARD The argument begins with what I take to be an incredibly plausible judgment: it is contingent where I am located. The claim that it is contingent where I am located has a number of possible readings. But the one I will highlight is the following: if I am located at R, I could have possibly been located somewhere else. It's not a metaphysically necessary fact that I be located where I am in fact located. I could have been located elsewhere: there are a number of regions distinct from my location at which I could have been located. Suppose that I am located at R. The argument runs as follows: (1 c ) There is a region R 0 distinct from R such that it is possible that I be located at R 0 . (2 c ) But if the identity theory is true, then no region R 0 distinct from R is such that it is possible that I be located at R 0 . (3 c ) Therefore, the identity theory is false. (2 c ) follows from higher-order Leibniz's Law and the necessity of dis- tinctness, which, again, says that if x and y are distinct, then they are necessarily distinct. And the necessity of distinctness follows from our background assumptions (in particular, the necessity of identity), together with a very plausible additional schema for the logic of meta- physical necessity: WHAT IT IS TO BE LOCATED 55 If not-p, then it's necessarily not necessary that p. (B ) Or, equivalently: if p, then it's necessarily possible that p. (B ) is widely accepted. As Williamson (2013) notes, \If something is so, then how could it have been metaphysically impossible?" (p. 44). 76 The argument for (2 c ) is as follows. Since I am located at R, the identity theory and higher-order Leibniz's Law jointly imply that I am identical toR. But for any distinct regionR 0 , I am necessarily distinct fromR 0 , by the necessity of distinctness. That is, no distinctR 0 is such that I am possibly identical to R 0 . But since according to the identity theory, being identical toR 0 is part of what it is to be located at R 0 , it is not possible to be located at R 0 . Therefore, if the identity theory is true, then no region R 0 distinct from R is such that it is possible that I be located at R 0 . Thus, given the truth of (1 c ), it follows that the identity theory is false. This leads us to the rst important dierence between necessity and deniteness: Necessary Distinctness, but not Denite Distinctness: If x6=y, it follows that necessarily x6=y. But if x6=y, it does not follow that it is denite that x6=y. 76 See, however, Garson (2014) for some hesitation regarding (B). Also see Bacon (forthcoming). 56 MATT LEONARD Though (B ) is plausible and very widely accepted (and thus, so is the necessity of distinctness), the corresponding schema for deniteness is not plausible: If not-p, then it's denitely not denite that p. (B ) Consider a sorites series of shades of color from red to orange. And now consider a particular shade s in the series which is not red, but which is a borderline case of being red. We know thats is not denitely red, otherwise, by (T ), s would be red. (B ) tells us that since s is not red, it is denite that s is not denitely red. But why think that? It seems as if it could be vague whether s is not denitely red. The problem with (B ) is that it rules out higher-order vagueness, which is totally implausible. Not only can it be vague whether a particular shade is red, but it can be vague whether it's vague whether it is red. 77 This highlights an interesting dialectical point. As I mentioned in the Introduction, one might have thought that developing a strong version of the argument from contingent location would automatically generate an analogously strong argument from vague location. But this is false. Say that it may be that ' just in case it's not denitely not-' (`it may be that'' is formally analogous to `it is possible that''). Consider the argument which runs parallel to the argument from contingent location 77 This point is made in Williamson (1994) and McGee (1997). Also, see Bacon (ms a), who argues that the B principle is what accounts for many of the paradoxes of higher-order vagueness, and thus should be rejected. WHAT IT IS TO BE LOCATED 57 defended above. Assume that I am located at R: (1 v ) There is a region R 0 distinct from R such that I may be located at R 0 . (2 v ) But if the identity theory is true, then no region R 0 distinct from R is such that I may be located at R 0 . (3 v ) Therefore, the identity theory is false. Premise (1 v ) is plausible: there are a number of regions that I am not denitely not located at. However, we have no reason to buy (2 v ). Without (B ), we are not assured of the deniteness of distinctness. If I am located at R, the identity theory implies that I am identical to R. So I am not identical to (and thus not located at) a distinct region R 0 . But it doesn't immediately follow that I am denitely distinct from R 0 , and thus it doesn't immediately follow that I am denitely not located at R 0 . Though necessity and deniteness share a number of formal similarities, they are philosophically quite dierent. Therefore, our argument from vague location against the identity theory will need to be a structurally dierent argument. 9. The Argument from Vague Location The argument I will defend is as follows. (4 v ) Denitely, I am located somewhere. 58 MATT LEONARD (5 v ) For any region R, if I am located at R, then it is vague whether I am located at R. (6 v ) If (4 v ), and the identity theory is true, then for any region R, if I am located atR, then it is not vague whether I am located at R. (7 v ) Therefore, the identity theory is false. Premises (4 v ) and (5 v ) are both plausible. Premise (4 v ) says that de- nitely, there is some R such that I am located at R. But, even though it is denite that there is some R such that I am located at R, no R is such that I am denitely located at it. Of course, I am not saying that I don't have a location. I am located at some region, even though I can't pick it out precisely. But whichever region I am located at, that region is merely a borderline case of being my location. That is, there is no region which I am denitely located at. And so premise (5 v ) is plausible. Now for (6 v ). It is straightforward to see that if (4 v ) is true and (Simple-Identity) is true, then for any region R, if I am located at R, then it is not vague whether I am located atR. For any region R, if I am located at R, then by the simple theory and higher-order Leibniz's Law, I am identical toR. By the deniteness of identity, I am denitely identical to R. And by one more application of higher-order Leibniz's WHAT IT IS TO BE LOCATED 59 Law, I am denitely located at R. Thus, it is not vague whether I am located at R. But again, (Simple-Identity) is not very plausible. Let's turn to the more promising formulations of the identity theory: (R-Identity) and (M-Identity). Let's start with (R-Identity), which, to remind the reader, says that to be located is to be identical to a region. It will be helpful to rst prove a lemma: If denitely, I am located somewhere, and (R-Identity) is true, then I am denitely a region. (Lemma 1) If denitely, there is someR that I am located at, then by higher-order Leibniz's Law and (R-Identity), denitely, there is some R that is a region and that I am identical to. From this it follows that I am denitely a region, since ('^ ) implies both ' and . I will now show that if (4 v ) is true and (R-Identity) is true, then for any region R, if I am located at R, then it is not vague whether I am located atR. Suppose I am located atR. By higher-order Leibniz's Law, it follows thatR is a region and I am identical toR. By (Lemma 1) and the deniteness of identity, it follows that denitelyR is a region and denitely I am identical to R. From this it follows that denitely, R is a region and I am identical toR (because ' and jointly imply ('^ )). And by higher-order Leibniz's Law and (R-Identity), it 60 MATT LEONARD follows that denitely, I am located at R. Therefore, it is not vague whether I am located at R. Parallel reasoning shows that if (4 v ) is true and (M-Identity) is true, then for any region R, if I am located at R, then it is not vague whether I am located atR. As before, this argument relies on an anal- ogous lemma: If denitely, I am located somewhere, and (M-Identity) is true, then I am denitely a material object. (Lemma 2) If denitely, there is someR that I am located at, then by higher-order Leibniz's Law and (M-Identity), denitely, there is someR such that I am a material object and identical to R. From this it follows that I am denitely a material object. Suppose I am located atR. By higher-order Leibniz's Law, it follows that I am a material object and am identical toR. By (Lemma 2) and the deniteness of identity, it follows that denitely I am a material object and denitely I am identical to R. From this it follows that denitely, I am a material object and am identical toR. And by higher- order Leibniz's Law and (M-Identity), it follows that denitely, I am located at R. Therefore, it is not vague whether I am located at R. So if either (Simple-Identity), (R-Identity) or (M-Identity) are true, and denitely I am located somewhere, then for any R, if I WHAT IT IS TO BE LOCATED 61 am located at R, it is not vague whether I am located at R. So (6 v ) is true. And from (4 v )(6 v ), it follows that the identity theory is false. 78 One response to this argument might be to reject the premise that it is vague where I am located. One might still try to salvage some other claims in the vicinity: for example, one might explore the alternative view that it is vague what my parts are and, thus, vague in a sense what my boundary is. But this is just to deny the highly plausible judgment with which we began the paper: no region is such that I am denitely located there. However, this response does raise an interesting issue. It is not immediately clear how the thesis that I am denitely located somewhere interacts with the mereological thesis that it's vague what my parts are. In general, it's not clear at all how parthood and location interact once vagueness is introduced. 79 78 A ddly issue arises: might the identity theory be only vaguely true? If I am located at R, and it is vague whether the identity theory is true, then it would be vague whether I am located at R. So one might think that the identity theory is consistent with it being vague whether I am located at R after all. But this would also show that the identity theory is false. By paralleling the argument for the deniteness of identity, we can use higher-order Leibniz's Law to show that the identity theory implies the deniteness of the identity theory (and thus the vagueness of the identity theory implies the falseness of the identity theory). 79 Unfortunately, this paper is not the place to explore these interesting issues. How- ever, here is one relevant observation. I've been talking about my exact location, but other authors discuss other locative relations. It's interesting to note that even if we assume that I have a denite exact location, very little follows about the def- initeness of other locative relations (dened mereologically) I bear to regions. For instance, following Parsons (2007), say that x pervades y just in case there is some region R such that x is located at R and y is a part of R. Even if I am denitely located atR, it doesn't follow that I denitely pervade every part of R (in particu- lar, I do not denitely pervade the particle shaped region y on the outskirts of my body where is it vague whether y is part of R). 62 MATT LEONARD This brings us to the second dierence between deniteness and ne- cessity: Denite Location, but not Necessary Location: Though I am denitely located somewhere, it's not the case that I am necessarily located somewhere. To see this, note that either contingentism is true or necessitism is true. The contingentist thinks that it is contingent whether I exist. There are possible worlds where I do not exist, and in those worlds I am not located. Therefore, it is not necessary that I am located somewhere. The necessitist, on the other hand, thinks that necessarily, everything is necessarily something. But even she should think that it's not the case that I am necessarily located. 80 This is because even though I exist in all possible worlds, there are worlds where I exist but am not concrete. And if I am not concrete in a world, then, presumably, I am not located in that world as Williamson (2013: pg. 13) says: \Where is the merely possible coin actually? Nowhere." So if either contingentism or necessitism is true, it is not necessary that I am located. This is enough to show that the following argument which is analo- gous to the argument from vague location defended above is unsound: (4 c ) Necessarily, I am located somewhere. 80 For defenses of necessitism, see Williamson (2013) and Goodman (2016). WHAT IT IS TO BE LOCATED 63 (5 c ) For any region R, if I am located at R, then it is contingent whether I am located at R. (6 c ) If (4 c ), and the identity theory is true, then for any region R, if I am located at R, then it is not contingent whether I am located at R. (7 c ) Therefore, the identity theory is false. This argument is unsound because (4 c ) is clearly false. The version of the argument from contingent location I defended in the previous section highlighted the fact that the identity theory is in- consistent with the judgment that I could have been located somewhere other than where I am located. It might also be tempting to think that the identity theory is inconsistent with the claim that if I am located at R, then I am contingently located at R. And so it might be tempting to think that if the identity theory is true, then if I am located at R, then it's not the case that I am contingently located at R (and thus that the identity theory is false). While this style of argument works against (Simple-Identity), it does not work against (R-Identity) and (M-Identity), which are both consistent with it being the case that if I am located at R, then I am contingently located at R. To see this, note that (M-Identity) is consistent with it being con- tingent whether I am a material object. And if (M-Identity) is con- sistent with it being contingent whether I am a material object, then 64 MATT LEONARD (M-Identity) is consistent with it being contingent whether I am lo- cated at R, since being located at R just is being a material object that is identical to R. Similarly, (R-Identity) is consistent with it being contingent whether I am a region. In fact, if the identity theory is true, there are reasons to think that it is contingent whether I am a region. It is plausible to think that for every particular region of spacetime, it is contingent whether that region is a region. Not only is it plausible to think that spacetime itself might have not existed, but general relativity suggests that the structure of spacetime could have been dierent (in which case, the regions that actually exist might not have existed). Since (R-Identity) is consistent with it being contin- gent whether I am a region, (R-Identity) is consistent with it being contingent whether I am located atR, since being located just is being identical to a region. A related observation can be made about the identity theory and vague location. We've just seen that (M-Identity) and (R-Identity) do not imply that if I am located at R, I am not contingently located at R. This is because (M-Identity) and (R-Identity) are both consistent with it being contingent whether I am a material object and contingent whether I am a region. Analogously, (M-Identity) and (R-Identity) do not imply that if something is located at R, then it is not vague whether it is located R. This is because (M-Identity) WHAT IT IS TO BE LOCATED 65 and (R-Identity) are consistent with there being borderline material objects and borderline regions. Could there be borderline material objects, or borderline regions? I don't know. But here are a couple of examples that are worthy of consideration. First, suppose that a restricted version of supersubstan- tivalism is true, and that only some regions count as material objects. And suppose that what it is for a region to be a material object is for the region to instantiate some property (say, having a non-zero value of mass). If it could be vague whether a region has a non-zero value of mass, then it could be vague whether that region is a material ob- ject. 81 Second, consider a sorites series of worlds, where one end (say, the actual world) contains the Eiel Tower. Throughout the series, the particles composing the Eiel Tower continuously move further and further away from one another until, at the end of the series, there is a possible world where the particles composing the Eiel Tower (in the actual world) are dispersed throughout the universe (and thus, the Eiel Tower does not exist). A number of worlds in the series will be worlds where it is vague whether the Eiel Tower is a material object (and also vague whether it is a region, for the supersubstantivalist) be- cause it will be vague whether or not there is an Eiel Tower in those worlds at all. This doesn't just show that there is reason to think that 81 Nolan (2014) takes something like this very seriously. He then argues that since being a part of something just is being a material object that is a subregion of another material object, since it can be vague whether some region is a material object, it could be vague whether that region is a part of something. 66 MATT LEONARD there are borderline material objects. If the Eiel Tower is a region, then this modal sorites shows that there can be borderline regions. All of this is compatible with both contingentism and necessitism. If con- tingentism is true, then the modal sorites above implies that there can be vague existence. 82 But if necessitism is true, no such thing is im- plied. For the necessitist, though the Eiel Tower exists in all worlds, there are a number of worlds in the series where it is vague whether the Eiel Tower is concrete. 83 So it is important not to overstate the problem for the identity the- ory: the identity theory is not inconsistent with the thesis that some material object is such that it is vague where it is located, because if there could be borderline material objects and borderline regions, then it would be vague where they are located. 84 But the identity theory is, however, inconsistent with the plausible thesis that it is vague where I 82 Vague existence is a controversial topic. See Lewis (1986: p. 213), Hawley (2002), Sider (2003), and Carmichael (2011). 83 Here is one more (albeit exotic) example. Suppose that p is a subatomic particle having no proper parts, and suppose there is an object A such that it is vague whether p is a part of A. Also suppose that it is vague whether the number seven is a part ofA, and thatA has no other denite proper parts or vague proper parts. If supersubstantivalism is true, then it seems to be vague whether A is a region. 84 Interestingly, we can show that if the simple theory is true, then even if it could be vague where some object is located, we could never know it or never assert it. Bacon (2018) argues that the following result is derivable from the modal logic T: vague identity implies vague identity at every order. As Bacon notes, if vagueness precludes knowledge and forbids assertion, if it is vague whetherx =y, then we can never know it and can never assert it. But given (Simple-Identity) and higher- order Leibniz's Law, we can substitute and show that vague location implies vague location at every order. So if there is an object such that it is vague where it is located, the simple theory implies that one can never know it and can never assert it. WHAT IT IS TO BE LOCATED 67 am located, given that I denitely have a location (and any other thing that denitely has some location). And this is sucient, I am inclined to think, to show that it is false. 10. Conclusion We have explored two important dierences between modality and vagueness. First, though distinct things are necessarily distinct, we have no reason to think that distinct things are denitely distinct. And second, though I am denitely located somewhere, it's not the case that I am necessarily located somewhere. These dierences make a dierence to the outcomes of arguments against the identity theory. As a result, the arguments I have defended have been structurally dierent kinds of arguments. I think that if I am located somewhere, it is vague whether I am located there. One cannot point to a region (even if it is in fact my location) and say that I am denitely located there. And I also think that I could have been located somewhere other than where I am located. Again, it is not metaphysically necessary that I be located where I am in fact located. But given that these plausible theses are inconsistent with the identity theory, I am inclined to think that the identity theory of location is false. 68 MATT LEONARD CHAPTER FOUR: Explaining Harmony I am interested in a fact which, at rst glance, seems to be something of a modal mystery: parts show up wherever their wholes are located. My arm is a part of my body. From this it follows that my arm's loca- tion must be a part of my body's location. 85 In other words: Necessarily, if x is located at r, y is located at s, and x is a part of y, then r is a part/subregion of s. (P-Harmony) Philosophers are great at constructing exotica, but it's hard to even imagine how this principle could be false. Only one philosopher (that I know of) has seriously considered rejecting it: Saucedo (2011) though even after developing a lengthy and fascinating combinatorial argument against the principle, he himself remains neutral about whether there could be violations of the principle (pg. 266). I happen to be inclined to reject combinatorial principles like the one Saucedo defends, though this paper is not the place where I shall explain why. I simply take (P-Harmony) to be true. And for what it's worth, my guess is that many metaphysicians do, as well. 85 By location I mean exact location (which is sometimes also called occupation). In addition to having my exact size and shape, my exact location also bears the same distance relations to things that I do. See Casati and Varzi (1999), Parsons (2007) and Gilmore (2018). WHAT IT IS TO BE LOCATED 69 But why is it true? What explanation could one possibly have for this modal mystery? The main purpose of this paper is to explore ways one might explain this principle of harmony. Let's begin by considering a pair of questions. First, what is it for a material object to be a part of another material object? And second, what is it for a material object to be located at a region of spacetime? It is natural to suppose that both being a part of and being located at are primitive relations that is, it is natural to suppose that parthood and location lack informa- tive metaphysical analyses. Let's call this supposition the primitivist package. The primitivist package has a well-known problem: it doesn't seem to be in a position to explain why (P-Harmony) is true. Of course, the primitivist is free to simply adopt an axiom which says that x is a part of y only if x's location is a part of y's location. But this doesn't answer the more interesting question of why it is true. It would be better, I would think, if one was able to give a satisfying explanation for why it is true. The way forward, I propose, is to follow a nice idea which naturally falls out of a view I am nonetheless inclined to reject: supersubstan- tivalism the view that material objects are just identical to regions of spacetime. 86 This view of material objects suggests a straightfor- ward analysis of location, and thus rejects the primitivist package: to 86 For defenses of supersubstantivalism, see Sider (2001), Skow (2005), Schaer (2009), Nolan (2014), and Eagle (2016). 70 MATT LEONARD be located at a region just is to be identical to that region call this the identity theory of location. The identity theory has an awesome explanation for (P-Harmony): it implies it. If x and y are identical to their locations, andx is a part ofy, then it follows by Leibniz's Law thatx's location is a part ofy's location. The identity theory has some well-known problems (it is inconsistent with contingent location, and with the possibility of co-location, for example) and thus I am inclined to reject it. However, I suggest that we follow the nice idea and develop metaphysical analyses of parthood and location which, like the identity theory of location, can explain why (P-Harmony) is true. The plan for the paper is as follows. In the rst section, I will sketch three analyses, each of which can oer a satisfying explanation for why (P-Harmony) is true each imply it. This suggests a natural ques- tion: if these three analyses imply this principle of harmony, do they imply other principles of harmony? If so, which ones? The second sec- tion of the paper explores which principles of harmony follow from the three analyses. Since each of the three analyses can explain why (P- Harmony) is true, we should take them seriously. But which of them is best? In the nal section of the paper, I oer a couple of tentative arguments for why I prefer one of the analyses to its two rivals. WHAT IT IS TO BE LOCATED 71 11. Explaining (P-Harmony): Some Initial Analyses Let's begin by stating formally our non-negotiable fact about part- hood and location (let `@' express the relation of location and `' ex- press the relation of parthood): Necessarily, if x is located at r, y is located at s, and x is a part of y, then r is a part/subregion of s. ((x@r^y@s^xy)!rs) (P-Harmony) It is necessarily the case that if something is a part of me, then it's located at a part/subregion of my location. 87 Why is (P-Harmony) true? In what follows, I introduce three analyses of parthood and location and then I will argue that each implies (P-Harmony). Each has an explanation for what initially looks to be a modal mystery, and so, all else being equal, I suggest that this is a reason to prefer them to the primitivist package. The rst is an analysis of parthood in terms of location. Let's say that the locational theory of parthood is the view that forx to be a part of y is for x to be located at a subregion of y. It is natural to think of metaphysical analyses as identications: being a part of something just 87 I say `part/subregion' because, though I take there to be one parthood relation that ranges over both material objects and regions, and thus the version of (P- Harmony) I will like will be stated purely in terms of parthood, if an account can give an explanation for a version of (P-Harmony) with `subregion', I take that to be totally acceptable. 72 MATT LEONARD is being located at a subregion of that thing's location. That is, the property being a part of something is identical to the property of being located at a subregion of that thing's location. Following Dorr (2016), let `' ' express the higher-order identication that for it to be the case that' is for it to be the case that . The rst theory says the following: (x is a part of y) (x's location is a subregion of y's location) Or, with `v' expressing the subregion relation, we have: (xy) (9r9s(x@r^y@s^rvs)) (Locational Theory of Parthood) That is, for x to be a part of y is for x and y to be located and for x's location to be a part of y's location. 88 The rough idea is rst defended (as far as I know) in Oppenheim and Putnam (1958). Markosian (2014) defends a related version, a view which he calls the subregion theory of parthood, which says that x is a part of y in virtue of the fact that x's location is a subregion of y's location. And Nolan (2014) defends a version of supersubstantivalism that ts nicely with a similar view of parthood for material objects. 89 88 I have in mind here a view of concrete parthood. That is, a relation that can hold between materials objects and a relation that can hold between regions. 89 Nolan defends a restricted version of supersubstantivalism on which every material object is a region, but not every region is a material object. Additionally, he defends WHAT IT IS TO BE LOCATED 73 Like the rst analysis, the second analysis takes location to be prim- itive. But rather than dening parthood, it denes an additional loca- tive relation in terms of the primitive location relation, together with resources from mereology. This analysis is inspired by those Williams (2008), for instance who have defended the view that there is a dis- tinction between fundamental location and derivative location. 90 On this type of view, only simples are fundamentally located; complex material objects are derivatively located, in virtue of the fundamental locations of the simples which compose them. While I will ignore issues here about fundamentality, we can follow this line of thought in giving an analysis of a locative relation that complex material objects bear to regions. Let's begin by introducing a primitive relation that ma- terial simples bear to spacetime regions (these can be points, or they can be extended regions, if there are extended simples). This relation, `p-location' (for primitive location), has no informative metaphysical analysis. We can dene a new relation, `d-location' (for derivative lo- cation), in terms of p-location as follows: (x isd-located aty) (ifx is a fusion of some simplesxx, then there are some simples yy such that y is a fusion of yy, each one ofxx isp-located at one ofyy, and each one ofyy is ap-location the view that for x to be a material part of y is for x and y to be material objects and for x to be a subregion of y. 90 Thanks to Daniel Nolan for telling me about this sort of view. 74 MATT LEONARD of one of xx) Or, with `F ' expressing the fusion relation, 91 with `' expressing the relation is one of from standard plural logic, with `S' expressing the property of being simple (that is, not having a proper part), and with `@ p ' expressing the primitive location relation, we have: (x@ d y) (8xx((xFxx^8x 0 (x 0 xx! Sx 0 ))!9yy(yFyy^ 8y 0 (y 0 yy! Sy 0 )^8x 0 (x 0 xx!9y 0 (y 0 yy^x 0 @ p y 0 ))^ 8y 0 (y 0 yy!9x 0 (x 0 xx^x 0 @ p y 0 ))))) (Derivative Theory of Location) On this picture, I am a fusion of some simple particles, each of which is p-located at some spacetime point/region. Roughly, I am d-located at the region which is the fusion of the p-locations of the simples that compose me. Unlike the rst two analyses, the third gives an analysis of location purely in terms of parthood. This view, a slightly more complex version of which is defended in Chapter 2, draws inspiration from the obser- vation that the relationship between a material object and its location is analogous in certain important ways to the relationship between the statue and the clay. The statue and the clay have dierent modal prop- erties, and so by Leibniz's Law are distinct. And yet, the statue and 91 Dened thus: yFxx = df 8z(zxx!zy)^8z(zy!9x(xxx^zx)). WHAT IT IS TO BE LOCATED 75 the clay are made up of the very same particles and thus overlap all and only the same things. According to the third analysis, the mere- ological theory of location, material objects overlap all and only the same things as their locations. According to this view, for O to be lo- cated at a regionR just is forO to be mereologically coincident withR: (x is located at y) (y is a region which is coincident x) Or, with `R' expressing the property of being a region and `' express- ing the overlap relation: (x@y) (Ry^8z(zx$zy)) (Mereological Theory of Location) On this view of location, to be located is to be coincident with a region. As far as I know, this view was rst sketched in Hawthorne (2006: pg. 118, fn 18.) and has been developed by Gilmore (2013) in an endurantist framework. Each of the above analyses have a nice virtue: they straightforwardly explain why (P-Harmony) is true. Let's start with the locational the- ory of parthood. On this view, for x to be a part of y just is for x's 76 MATT LEONARD location to be a part of y's location. This implies the following neces- sary biconditional, where x and y range over material objects: (xy $ 9r9s(x@r^y@s^rvs)) (LP) And (LP) implies (P-Harmony). Moving on, note that on the derivative theory of location, only sim- ples are located. And if some object is a simple, then its only part is itself. Assuming that simples are located, the fact that simples are parts of themselves implies the truth of (P-Harmony). 92 This is a lit- tle unsatisfying, however, since the intuitiveness behind (P-Harmony) concerns complex objects, and not simples. We can nevertheless show that a related principle is true: Necessarily, if x is d-located at R, y is d-located at S, and x is a part of y, then R is a part of S. (P-Harmony) To see that (P-Harmony) is true, suppose that x is d-located at R, that y is d-located at S, and that x is a part of y. By the denition of d-location, we know that there are some simples xx which compose x, and that there are some simplesyy which composey. And we know that there are some simples rr which compose R and some simples 92 Throughout the paper, I am going to make the assumption that all material objects are located. For some exotic counterexamples to this claim, see Gilmore (2006), Parsons (2007), Leonard (2014; 2016), and Kleinschmidt (2016). WHAT IT IS TO BE LOCATED 77 ss which compose S, such that each of the simples composing x is p- located at one of rr (and each one of rr is a p-location of one of xx), and each of the simples composingy isp-located at one ofss (and each one ofss is ap-location of one ofyy). We need to show thatR is a part of S. To show this we will show that whatever overlaps R overlaps S. Assuming that regions obey classical mereology, 93 this suces to show that R is a part of S. So suppose that some arbitrary z overlaps R. This means that they both have a part in common, say, z 0 . Since z 0 is a part ofR and thatR is composed of simple regionsrr, we know that there exists an r 0 among rr that is a part of z 0 . 94 We also know that since x is a part of y, the simples which compose x are also parts of y. 95 Thus,r 0 a location of one ofyy which composey is a part of the d-location of y, i.e., S. So z 0 is a part of both z and S, and thus they overlap. Butz was arbitrary, and so anything that overlapsR overlaps S. Assuming that regions of spacetime obey strong supplementation, it follows that R is a part of S. Thus, (P-Harmony) is true. Last, the mereological theory of location implies(P-Harmony). As we saw in Chapter 2, suppose thatx is a part ofy, thatx is located atr and thaty is located ats. Sincex is a part ofy, we know that whatever overlapsx overlapsy. Sincer andx are mereologically coincident, and 93 Specically, that the axioms of classical mereology are true when the outer most universal quantiers in the axioms of classical mereology are restricted to regions. 94 In general, if x is composed of some simples xx, and y is a part of x, then some simple x 0 among xx is a part of y. 95 This follows by the transitivity of parthood and the denition of fusion. 78 MATT LEONARD since s and y are mereologically coincident, we know that whatever overlaps r overlaps x and that whatever overlaps y overlaps s; thus, we know that whatever overlaps r overlaps s. By classical extensional mereology for regions, it follows that r is a part of s. 96 Thus, (P- Harmony) is true. Why is (P-Harmony) true? Though the primitivist package lacks an explanation, the three analyses developed above do not: the truth of (P-Harmony) is implied by what parthood is, or by what location is. 12. What Further Principles of Harmony are True? (P-Harmony) is a principle of mereological harmony, the view that says (roughly) that the mereological (or, part-whole) structure of ma- terial objects mirrors and is mirrored by the mereological structure of those objects' locations. Consider my body and my body's location. Harmony tells us that the mereological relations which obtain between my body and its parts match the mereological relations which obtain between my body's location and its parts, and vice versa. For example, principles of harmony tell us things like the following: x is a part of y i x's location is a part of y's location, x overlaps y i x's location overlaps y's location, x is a fusion of yy i the location of x is a fusion 96 In particular, `whatever overlaps x overlaps y' implies that x is a part of y, by (Strong Supplementation): ifx is not a part of y, then there is some part of x which does not overlap y. WHAT IT IS TO BE LOCATED 79 of the locations of yy, x is gunky i x's location is gunky, and so on. 97 Interestingly, as Uzquiano (2011) and (Saucedo (2011) have pointed out, many of these harmony principles are logically independent of one another. And many have been rejected. 98 Besides(P-Harmony), which of these harmony principles are true? This is a dicult question, and not one I will attempt to answer here. However, in what follows I will explore what other principles follow from our three analyses. This, I think, counts as substantial progress. After all, we have three analyses that can explain(P-Harmony). This explanatory power is a virtue. If an analysis explains (P-Harmony), we should take seriously what else it implies. What other constraints on parthood and location do our three analyses imply? Not many, according to the rst two analyses. And quite a few, according to the other. 12.1. The Locational View of Parthood. Not many, according to our rst analysis. Earlier we saw that the locational view of parthood implies the following necessary biconditional: (xy $ 9r9s(x@r^y@s^rvs)) (LP) 97 See Uzquiano (2006; 2011), Varzi (2006), Saucedo (2011), Leonard (2016), and Gilmore (2018). 98 See Gilmore (2018) for a survey. 80 MATT LEONARD However, as noted by Uzquiano (2011), principles like (LP), which re- quire there to be a mereological match in parthood, do not imply many other similar looking principles requiring there to be analogous matches concerning overlap, proper parthood, and fusion (and their subregion- dened counterparts). And as noted by Saucedo (2011), principles like (LP) do not imply many other similar looking principles requiring there to be analogous matches in gunkiness, simplicity, number of parts, and so on. The locational view of parthood places very little constraint on the mereological relationship between material objects and their loca- tions. 12.2. The Derivative Theory of Location. Like the rst analysis, the derivative theory of location also places very little constraint on the mereological relationship between material objects and their loca- tions. 99 Consider, for instance: Necessarily, if x is d-located at R, y is d-located at S, then x overlaps y i R overlaps S.. ((x@ d R^y@ d S)! (xy$RS)) (Overlap) Notice that a material simple can bed-located at a point. If two simples a andb are co-d-located at a pointp, then whilep overlaps itself,a and b do not overlap. And so (Overlap) is false. 99 Thanks to John Hawthorne here for discussion. WHAT IT IS TO BE LOCATED 81 Consider further a principle like the following: (Necessarily, if x is d-located, then x is gunky i x's location is gunky.) (x@ d R! (8y(y x!9z(z < y))$8y(y R!9z(z < y)))) (Gunkiness) Consider an extended simple s that is d-located at an extended gunky region r. This violates (Gunkiness) . It might, however depending on one's background mereology imply the other direction of (P-Harmony) : Necessarily, if x is d-located at R, y is d-located at S, and R is a part of S, then x is a part of y. ((x@ d R^y@ d S^RS)!xy) (Parts(()) Assuming that x is d-located at R, that y is d-located at S, and that R is a part of S, we can show that by the derivative theory, what- ever overlaps x overlaps y, by paralleling the argument we gave for (P-Harmony) . However, in our nal step to show that R is a part of S, we relied on (Strong Supplementation). While it is uncon- troversial that regions obey (Strong Supplementation), it's more 82 MATT LEONARD controversial whether material objects obey it many argue that they don't. In the presence of(StrongSupplementation), the derivative view implies (Parts(()) , but without it, it doesn't. 12.3. The Mereological Theory of Location. As we saw in Chapter 2, there are two types of coincidentalists corresponding to two princi- ples. 100 The S-Coincidentalist accepts (Strong Supplementation) and the negation of (Uniqueness). If the statue and the clay overlap the same things, it follows from (Strong Supplementation) that the statue and the clay are parts of each other; that is, the statue and the clay are mutual parts. 101 The A-Coincidentalist, on the other hand, accepts the conjunction of (Anti-Symmetry) and the negation of (Uniqueness). From A-Coincidentalism it follows that the statue and the clay are not mutual parts. Again, A-Coincidentalists typically make an additional mereological claim: the clay is a part of the statue (and thus not vice versa). 102 100 Since (Strong Supplementation) and (Anti-Symmetry) jointly imply (Uniqueness), coincidentalists must reject at least one of them: If x is not a part of y, then there is some part of x which does not overlap y. (Strong Supplementation) If x and y are parts of each other, then x =y. (Anti-Symmetry) 101 This view has been developed and/or endorsed in Thomson (1983), (1998), Cot- noir (2010), (2013), (2016), Cotnoir and Bacon (2012), Hovda (2013), and Gilmore (forthcoming). 102 This view is defended or discussed in Goodman (ms), Walters (forthcoming), and Gilmore (forthcoming). WHAT IT IS TO BE LOCATED 83 In the presence of S-coincidentalism the mereological theory implies a particularly strong list of constraints: (Necessarily, x is a part of y i x's location is a part of y's loca- tion.) (x@R^y@S! (xy$RS)) (Parts) (Necessarily, x is a proper part of y i x's location is a proper part of y's location.) (x@R^y@S! (x<y$R<S)) (Proper Parts) (Necessarily, x overlaps y i x's location overlaps y's location.) (x@R^y@S! (xy$RS)) (Overlap) (Necessarily, x is simple i x's location is simple.) (x@R! (:9y(y<x)$:9y(y<R))) (Simplicity) (Necessarily, x is gunky i x's location is gunky.) (x@R! (8y(yx!9z(z <y))$8y(yR!9z(z <y)))) (Gunkiness) (Necessarily,x has a proper expansion ix's location has a proper expansion.) 84 MATT LEONARD (x@R! (9y(x<y)$9y(R<y)))) (Expansions) And as we also saw, in the presence of A-coincidentalism, the mereo- logical theory implies that a number of the related principles above are not true. The following are not true (counterexamples are sketched in Chapter 2): (Necessarily, if x's location is a part of y's location, then x is a part of y.) ((x@R^y@S^RS)!xy) (Parts(()) (Necessarily, ifx is a proper part ofy, thenx's location is a proper part of y's location.) ((x@R^y@S^x<y)!R<S) (Proper Parts())) (Necessarily, ifx is a proper part ofy, thenx's location is a proper part of y's location.) (x@R! (:9y(y<R)!:9y(y<x))) (Simplicity(()) This, I think, is progress. Whereas before we merely had a list of harmony principles from which to pick and choose, we now have some non-arbitrary options on the table, option which corresponds to our three analyses of parthood and location. WHAT IT IS TO BE LOCATED 85 13. A Tentative Defense of the Mereological Theory of Location In this nal section, I will give some rough reasons why I am inclined to tentatively prefer the mereological theory of location (in particular, the S-Coincidentalist version) over its two rivals. I will rst argue that the derivative theory of location has a straightforward limitation. And then I will argue that the locational view of parthood fails to have a satisfying explanation for a number of further theses about parthood and location. 13.1. A Problem for the Derivative Theory of Location. The central limitation of the derivative theory of location is simple: it doesn't t well with the possibility of gunky matter. (P-Harmony) is true, regardless of whether or not material objects are gunky. If the world is gunky, the derivative theory of location no longer seems to have a straightforward explanation for why a principle like (P-Harmony) is true. 13.2. A Problem for the Locational Theory of Parthood. We've spent some time exploring what principles of harmony follow from our three analyses. But let's set harmony principles to the side. Are there any other plausible principles governing parthood and location? One, I think, is a principle which places a natural constraint on a complex material object (an object with proper parts) and its location: 86 MATT LEONARD (Necessarily, some disjoint material objects have a fusion if and only if some material object is located at a region which fuses their locations.) ((8x(x xx! Mx)^8w8z((w xx^z xx^w6= z)! woz))! (9y(yFxx)$9z9r9ss(rFss^Mz^^z@r^8x(x xx!9s(sss^x@s))^8s(sss!9x(xxx^x@s))))) (Regionalism) This principle was rst proposed in Markosian (2014), who defends it as an answer to the special composition question, the question which asks: Under what circumstances do several material objects compose another object? 103 (Regionalism) is compatible with a number of more famil- iar answers to the special composition question (for example: moderate views, universalism, and nihilism) because it is consistent with a wide range of views about which regions have complex material objects lo- cated at them. Even though(Regionalism) doesn't settle the dispute between, say, the universalist and the nihilist, it is a plausible thesis in its own right. Both directions are very tempting. If I am a fusion of my right half R and left half L, then how could it fail to be the case that some object is located at the fusion ofR andL's locations? After all, I am located there! Moreover, if I am located at the region which is the fusion ofR andL's locations, then how can it fail to be the case thatR 103 Well, Markosian's formulation is ambiguous between a few readings. See Gilmore and Leonard (forthcoming) for two disambiguations. My formulation here is slightly dierent. WHAT IT IS TO BE LOCATED 87 and L fuse something? 104 I am inclined to think that (Regionalism) is true. But why is it true? Markosian's central argument for the thesis is that it follows from (LP), which again, is: (xy $ 9r9s(x@r^y@s^rvs)) (LP) However, Gilmore and Leonard (forthcoming) show that this is not true. Neither (LP) nor the locational theory of parthood implies (Region- alism). Here is a model where (LP) is true and yet (Regionalism) is false. In this model, the complex material objecto is composed of the simple material objectso 1 ;o 2 ; ando 3 . Ando 1 ando 2 do not compose anything. 104 Maybe a ghost can be located at the region which is the fusion of the locations of a bunch of particles, without the particles composing something. I do not take these sorts of exotic cases very seriously. 88 MATT LEONARD Further, o is located at r, o 1 is located at r 1 , o 2 is located at r 2 , and o 3 is located at r 3 . As the reader can check, (LP) is satised in the model. For any case in which x is located at s 1 and y is located at s 2 , x is a part of y i s 1 is a part of s 2 . But while r is a fusion of the locations of o 1 and o 2 , and there is a material object located at r in the model, o 1 and o 2 do not compose anything, so (Regionalism) is not satised. Thus, the locational theory of parthood does not imply (Regionalism). Interestingly,(Regionalism) suggests a series of similarly plausible principles which have not been discussed in the literature: (Necessarily, a material objectx is a proper part of some material object if and only if x's location is a proper part of the location of some material object.) (Mx! (9y(My^x<y))$9r9s9z(Mz^x@r^z@s^r<s))) (9-Proper Parthood) If my arm is a proper part of my body, how could my arm's location fail to be a proper part of some material object's location? Or if my arm's location was a proper part of the location of some material object, how could my arm fail to be a proper part of some material object? Relat- edly: WHAT IT IS TO BE LOCATED 89 (Necessarily, a material object x overlaps some material object if and only if x's location overlaps the location of some material object.) (Mx! (9y(My^xy))$9r9s9z(Mz^x@r^z@s^rs))) (9-Overlap) And similarly: (Necessarily, a material object x is simple if and only if there is not a material object located at a proper part of x's location.) (Mx! (:9y(My^y<x)$:9r9s9z(Mz^x@r^z@s^s< r))) (9-Simplicity) How could something be simple, say, an electron, if a material object is located at a subregion of the electron's location? Or how could there be a material object located at a subregion of something's location if that thing is a material simple? One speculative idea is that perhaps there are ghosts, or particles smaller than electrons which can interpenetrate electrons pass through them without overlapping them. Perhaps this is possible. I am inclined to set these sorts of speculative cases aside, however. It's not obvious to me, for what it's worth, that if this were 90 MATT LEONARD to happen, that the tiny particle (or the tiny ghost) would not be a part of the electron when passing through the electron. These interesting principles are structurally dierent from harmony principles. I happen to nd them plausible. They seem to me to be true. But why are they true? Not only does the locational theory of parthood fail to imply(Regionalism), but it fails to straightforwardly imply these additional principles. The S-Coincidentalist version of the mereological theory of location, however, in the presence of some natural assumptions, does straightfor- wardly imply(Regionalism), (9-ProperParthood), (9-Overlap), and (9-Simplicity). For the sake of space, I will provide the argu- ment for (Regionalism). The argument relies on two further prin- ciples about parthood and location, each of which as we saw above are also implied by the mereological view of location (given the S- Coincidentalist spin, according to which a material object and its loca- tion are mutual parts): Necessarily, ifx is located atR,y is located atS, thenx is a part of y if and only if R is a part/subregion of S. (Parts) Necessarily, ifx is located atR,y is located atS, thenx overlaps y if and only if R overlaps S. (Overlap) WHAT IT IS TO BE LOCATED 91 (Parts) follows from the fact that material objects and their loca- tions are parts of each other, and (Overlap) follows from the fact that material objects and their locations overlap all and only the same things. The argument that the mereological theory of location implies (Re- gionalism) is as follows. Consider arbitrary disjoint material objects o 1 ;:::;o n . Let's start with the left-to-right direction of (Regional- ism). Suppose thato 1 ;:::o n have a fusion, say, F . 105 And suppose that o 1 ;:::o n and F are all located. 106 We need to show that some mate- rial object is located at the region which is the fusion of the locations' of o 1 ;:::o n . By classical mereology for regions, we know that each the locations of o 1 ;:::o n have a fusion, say, R. We know that F has some location, say,L F . We'll show thatL F =R. By(Parts), we know that since eacho i is a part ofF , the location of eacho i is a part ofL F . But 105 As Gabriel Uzquiano has pointed out to me, it is worth noting that there are two ways of thinking about fusion here. On the one hand, letting range over both material objects and regions, we might dene fusion as follows: yFxx = df 8z(z xx!zy)^8(y!9x(xxx^x)). This says that if I am a fusion of, say, my left half (l) and my right half (r), thenl andr are parts of me, and for any (whether it be a material object or a region), if is a part of me, then it overlaps either l or r. On another notion of fusion, however, we restrict to only range over material objects. Using `m' here instead of, and to only range over material objects, we have: yFxx = df 8z(zxx!zy)^8m(my!9x(xxx^mx)). This says that if I am a fusion of, say, my left half (l) and my right half (r), then l and r are parts of me, and for any material object m, if m is a part of me, then it overlaps either l or r. It is worth noting that I am relying on the rst notion of fusion (the notion) throughout the argument for regionalism, and that the argument will break down if we were to use the second notion. 106 Things get tricky if we start considering exotic cases where material objects which lack exact locations. As I mentioned earlier, I will ignore these cases in this paper. 92 MATT LEONARD if R is the fusion of the locations of each o i , then the location of each o i is a part of both L F and R. And if every part of R is a part of L F , then it follows that R is a part of L F . We will now show that L F is a part of R. Once we have shown this, we will have shown that R and L F are mutual parts. Given Anti-Symmetry for regions, it will follow thatR andL F are identical. Thus, we will have shown that something is located at the fusion of the locations of o 1 ;:::;o n . What follows is a bit tedious but it is worth working through. To show that L F is a part of R, we will show that whatever overlaps L F , overlaps R. Given Strong Supplementation for regions, this suces to show that L F is a part of R. So suppose something z overlaps L F . We will show that z also overlaps R. Since z overlaps L F , there is something z 0 that is a part of both L F and z. Since L F is a part of F , it follows that z 0 is a part of F . Since z 0 is a part of F and since F is a fusion of o 1 ;:::;o n , we know that z 0 overlaps some o i among o 1 ;:::;o n , by the denition of fusion. So there is some z 00 which is a part of o i . By the mereological theory of location, we know that o i is coincident with its location, call it Lo i . Since z 00 overlaps o i , we know that z 00 also overlaps Lo i . There- fore, there is somez 000 that is a part of bothz 00 andLo i . Soz 000 is a part of both z 0 and R. Thus, R and z 0 overlap. But z 0 was arbitrary. So whatever overlaps L F overlaps R. By (Strong Supplementation) for regions, it follows that L F is a part of R. But since R is also a part of L F , it follows by (Anti-Symmetry) for regions that L F =R. WHAT IT IS TO BE LOCATED 93 Therefore, we have shown that something is located at the fusion R of the locations of o 1 ;:::;o n , namely, F . Now for the right-to-left direction of (Regionalism). Suppose that a material objectO is located at the fusion (call itR) of the locations of some disjoint material objectso 1 ;:::;o n . We need to show thato 1 ;:::;o n have a fusion. That is, given the standard denition of fusion in the literature, we need to show that there is some F such that (i) each o i is a part of F and that (ii) every part of F overlaps at least one of o 1 ;:::;o n . We'll do this by showing that O = F . Condition (i) follows from (Parts), the fact that O is located at R, and the fact that the location of each o i is a part of R. Now for condition (ii). Suppose some object o 0 is a part of O. We need to show that it overlaps the location of some o i . Since o 0 is a part of O, we know by (Parts) that the location of o 0 is a part ofR. ButR is the fusion of the locations of o 1 ;:::;o n , and since the location of o 0 is a part of R, it follows from the denition of fusion that the location of o 0 overlaps the location of one ofo 1 ;:::;o n , say, the location of o i . But then by (Overlap), it follows that o 0 overlaps o i , and so condition (ii) is true. This completes the argument for the right-to-left direction of (Regionalism). Though the locational view of parthood does not imply (Region- alism), Gilmore and Leonard (forthcoming) have shown that it does, when supplemented with a number of additional premises, one of which 94 MATT LEONARD is (Overlap) and one of which is: (Necessarily, if x is complex and is located at R, then for any yy of whichx is a fusion, each partr 0 ofR overlaps a location of one of yy). ((9y(y < x)^x@r)!8yy(xFyy!8r 0 (r 0 r!9y 0 (y 0 yy^y 0 @s^r 0 s)))) (Strong Delegation) This principle says that if a complex material object x is located at a region R, then every part of R must overlap the location of one of x's parts. (Strong Delegation) is compelling. In fact, it lls a previously unnoticed gap in Parsons' (2007) theory of location. The theory there is satised by the following model: In this model, a complex material objecto is located at a regionr, and a region which is a part ofr does not overlap the location of any of the WHAT IT IS TO BE LOCATED 95 parts which composeo. I am inclined to think that this is not possible. (Strong Delegation) rules out such models. Even if an advocate of the locational theory of parthood introduces (Strong Delegation) as an axiom to explain(Regionalism), and to rule out models like the above, we are left wondering why it is that (Strong Delegation) itself is true. Once again, (Strong Delegation) is implied by the mereological view of location. Here is a quick argument. Suppose a complex material object o is located at r. Consider arbitrary yy of which o is a fusion. We need to show that any part ofr overlaps a location of oney among the yy. Let r 0 be an arbitrary part of r. Since r 0 is a part of r, it overlaps r. Since o and r are coincident, r 0 overlaps o. Since yy fuse o, anything that overlaps o overlaps at least one y among yy. And so r 0 overlaps at least one y among yy. And since y is coincident with its location, r 0 overlaps y's location. And so (Strong Delegation) is true. 13.3. An Extension of the Locational Theory of Parthood. Be- fore concluding, it's worth brie y considering one extension of the loca- tional view of parthood, an extension which would straightforwardly imply all of the principles that the locational view itself does not. Markosian (2014), one of the prominent defenders of a view in the vicinity of the locational view of parthood (again, recall that his for- mulation uses the `in virtue of' ideology), writes that one reason he 96 MATT LEONARD likes (Regionalism) is that: \...it is a part of a general approach to mereology that includes STP [that is, (LP)], and that reduces the mereology of physical objects to the mereology of the spatial regions that they occupy. This is what I call the ... Spatial Approach to Mereology (SAM): The mereological properties and relations of physical objects are determined by the mereological properties and relations of the spatial regions those objects occupy" (Markosian 2014, pg. 84). This suggests an extension of the locational view of parthood. In addi- tion to the thesis that being a part of something y just is having your location be a subregion of y's location; perhaps, for any mereological relation ', for some material objects to be '-related just is for their locations to be ' -related, where a formula ' (x 1 ;:::;x n ) is just like '(x 1 ;:::;x n ), except that any mereological relation R in ' is replaced by its subregion counterpart in ' . That is: (x 1 @y 1 ^:::^x n @y n )! (('x 1 ;:::;x n ) (' y 1 ;:::;y n )) This view not only implies (LP), but it also implies all of the harmony principles implied by the mereological theory of location (in the pres- ence of S-coincidentalism). However, I happen to think it overshoots. WHAT IT IS TO BE LOCATED 97 One instance of it is: (x 1 @y 1 ^x 2 @y 2 )! ((x 1 =x n ) (y 1 =y 2 )) Since a higher-order identication implies a necessary biconditional, the above implies: (x 1 @y 1 ^x 2 @y 2 )!((x 1 =x n )$ (y 1 =y 2 )) And this is inconsistent with co-located distinct objects, which I happen to think is possible (and actual, in fact, when it comes to things like statues and the clay from which they are made). So while it's got a lot of things going for it, those of us who take seriously co-location have reason to prefer the mereological theory of location. 14. Conclusion We began with a modal mystery: (P-Harmony). Why is it true? The central thought of this paper has been to analyze at least one of either parthood or location. Three such analyses straightforwardly im- ply (P-Harmony), and thus explain why it is true. All else being equal, this is a reason to prefer these theories to the primitivist pack- age. Though all three theories are preferable to the primitivist package, one in particular has a lot going for it the mereological theory of lo- cation. It straightforwardly implies a number of further principles that 98 MATT LEONARD plausibly govern the relationship of parthood and location. Thus, of our three analyses, I am inclined to go for the mereological theory of location. WHAT IT IS TO BE LOCATED 99 CHAPTER FIVE: Concluding Remarks Once again, I think that a good theory of location T ought to have the following features: 1. T is consistent with contingent location. 2. T is consistent with two material objects having the same location. 3. T implies that necessarily, a material object has the same size and shape as its location. 4. T implies that necessarily, if x is located at R, y is located at S, and x is a part of y, then R is a part of S. We've now seen that the MG view has features (1)-(4). The identity theory fails to have features (1) and (2). It is also incompatible with the plausible thesis that it is vague where I am located: no region is such that I am denitely located at it. And we've seen that the primitivist package fails to have features (3) and (4). Near the end of our discussion, we've seen that the M theory of location, given some natural background assumptions, can oer an ex- planation for some further plausible principles governing parthood and location: (Regionalism) and (Strong Delegation). Since being 100 MATT LEONARD coincident with a region is part of what it is to being MG equiva- lent with a region, the MG view can help itself to these explanations. Though lots of metaphysical analyses of parthood and location have lots of things going for them, I think that the MG view emerges as the best. At least given our brief exploration of what it is to be located. WHAT IT IS TO BE LOCATED 101 References [1] David K. Albert. Elementary quantum metaphysics. In Sheldon Goldstein J. T. Cushing, Arthur Fine, editor, Bohmian Mechanics and Quantum the- ory: An Appraisal, pages 277{284. Kluwer Academic Publishers, 1996. [2] Frank Arntzenius. Space, Time, and Stu. Oxford: Oxford University Press, 2012. [3] Andrew Bacon. Vagueness at every order. manuscript a. [4] Andrew Bacon. Vagueness and Thought. Oxford University Press, 2018. [5] Andrew Bacon. The broadest necessity. Journal of Philosophical Logic, Forth- coming. [6] Andrew Bacon. Relative locations. In Dean Zimmerman, editor, Oxford Studies in Metaphysics. Oxford, Forthcoming a. [7] Andrew Bacon. The broadest necessity. Journal of Philosophical Logic, Forth- coming b. [8] Elizabeth Barnes and R.G. Williams. Vague parts and vague identity. Pacic Philosophical Quarterly, 90(2):176{187, 2009. [9] Chad Carmichael. Vague composition without vague existence. Nous, 45(2):315{327, 2011. [10] Roberto Casati and Achille Varzi. Parts and Places: The Structures of Spatial Representation. MIT Press, 1999. [11] Aaron Cotnoir. Anti-symmetry and non-extensional mereology. Philosophical Quarterly, 60:396{405, 2010. [12] Aaron Cotnoir. Strange parts: The metaphysics of non-classical mereology. Philosophy Compass, 8:834{845, 2013. [13] Aaron Cotnoir. Does universalism entail extensionalism? No^ us, 50:121{132, 2016. 102 MATT LEONARD [14] Aaron Cotnoir and Andrew Bacon. Non-wellfounded mereology. Review of Symbolic Logic, 5:187{204, 2012. [15] Cian Dorr. To be f is to be g. Philosophical Perspectives, 30(1):39{134, 2016. [16] Antony Eagle. Location and perdurance. In Dean Zimmerman, editor, Oxford Studies in Metaphysics, volume 2010, pages 53{94. Oxford, 2010. [17] Antony Eagle. Multiple location defended. Philosophical Studies, 173(8):2215{ 2231, 2016a. [18] Antony Eagle. Persistence, vagueness, and location. Journal of Philosophy, 113(10):507{532, 2016b. [19] Gareth Evans. Can there be vague objects? Analysis, 38(4):208, 1978. [20] James Garson. Modal logic. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/archives/spr2016/entries/logic-modal edition, 2014. [21] Cody Gilmore. Where in the relativistic world are we? Philosophical Perspec- tives Philosophical Perspectives Philosophical Perspectives, 20:199{236, 2006. [22] Cody Gilmore. Building enduring objects out of spacetime. In Claudio Calosi and Pierluigi Graziani, editors, Mereology and the Sciences, pages 5{34. Springer, 2014. [23] Cody Gilmore. Location and mereology. In The Stanford Encyclopedia of Phi- losophy (Spring 2018 edition). Edward N. Zalta, 2018. [24] Cody Gilmore. Quasi-supplementation, plenitudinous coincidentalism, and gunk. In Substance: New Essays. Philosophia Verlag, Forthcoming. [25] Cody Gilmore and Matt Leonard. Composition and the logic of location: An argument for regionalism. Mind, Forthcoming. [26] Jeremy Goodman. Matter and mereology. manuscript. WHAT IT IS TO BE LOCATED 103 [27] Jeremy Goodman. An argument for necessitism. Philosophical Perspectives, 30(1):160{182, 2016. [28] Katherine Hawley. Vagueness and existence. Proceedings of the Aristotelian Society, 102(2):125{140, 2002. [29] John Hawthorne. Metaphysical Essays. Oxford: Oxford University Press, 2006. [30] Richard Heck. That there might be vague objects (so far as concerns logic). The Monist, 81(1):277{299, 1998. [31] Paul Hovda. What is classical mereology? Journal of Philosophical Logic, 38(1):55{82, 2009. [32] Paul Hovda. Tensed mereology. Journal of Philosophical Logic, 42:241{283, 2013. [33] Hud Hudson. A Materialist Metaphysics of the Human Person. Oxford: Oxford University Press, 2001. [34] Shieva Kleinschmidt. Shaping up location: Against the humean argument for the extrinsicality of shape. Philosophical Studies, 172(8):1973{1983, 2015. [35] Shieva Kleinschmidt. Placement permissivism and logics of location. Journal of Philosophy, 113(3):117{136, 2016. [36] Saul Kripke. Identity and necessity. In Milton K. Munitz, editor, Identity and Individuation, pages 135{164. New York University Press, 1971. [37] Dennis Lehmkuhl. The metaphysics of super-substantivalism. Nous, 50(4), 2016. [38] Matt Leonard. On the contingency and vagueness of where i am. Unpublished manuscript b. [39] Matt Leonard. What is it to be located? Unpublished manuscript a. [40] Matt Leonard. Locating gunky water and wine. Ratio, 27(3):306{315, 2014. [41] Matt Leonard. What is mereological harmony? Synthese, 193(6):1949{1965, 2016. 104 MATT LEONARD [42] Matt Leonard. Enduring through gunk. Erkenntnis, 83(4):753{771, 2018. [43] David Lewis. Extrinsic properties. Philosophical Studies, 44:197{200, 1983. [44] David Lewis. On the Plurality of Worlds. Wiley-Blackwell, 1986. [45] David Lewis. Vague identity: Evans misunderstood. Analysis, 48(3):128{130, 1988. [46] E.J. Lowe. Substantial change and spatiotemporal coincidence. Ratio, 16:140{ 160, 2003. [47] Ruth Barcan Marcus. Identity of individuals in a strict functional calculus of second order. Journal of Symbolic Logic, 12(1):12{15, 1947. [48] Ned Markosian. A spatial approach to mereology. In Shieva Kleinschmidt, ed- itor, Mereology and Location, pages 69{90. Oxford: Oxford University Press, 2014. [49] Kris McDaniel. Gunky objects in a simple world. Philo, 9(1):39{46, 2006. [50] Kris McDaniel. Extended simples. Philosophical Studies, 133(1):131{141, 2007. [51] Vann McGee. Kilimanjaro. Canadian Journal of Philosophy, 27(sup1):141{163, 1997. [52] Richard Montague. Syntactical treatments of modality, with corollaries on re exion principles and nite axiomatizability. Acta Philosophica Fennica, 16:153{167, 1963. [53] Daniel Nolan. Balls and all. In Shieva Kleinschmidt, editor, Mereology and Location, pages 91{116. Oxford University Press, 2014. [54] Josh Parsons. Theories of location. In Dean Zimmerman, editor, Oxford Studies in Metaphysics, volume 3. Oxford University Press, 2007. [55] W.V.O. Quine. Reference and modality. In From a Logical Pointy of View. Cambridge, MA: Harvard University Press, 1953. [56] W.V.O. Quine. Things and their places in theories. In Theories and Things, pages 1{23. Cambridge: Harvard University Press, 1981. WHAT IT IS TO BE LOCATED 105 [57] Aust n Rayo. The Construction of Logical Space. Oxford: Oxford University Press, 2015. [58] Jerey Sanford Russell. Vague parts. unpublished book manuscript. [59] Nathan Salmon. Reference and Essence. Oxford: Basil Blackwell, 1982. [60] Thomas Sattig. The Language and Reality of Time. Clarendon Press, 2006. [61] Raul Saucedo. Parthood and location. In Karen Bennett and Dean Zimmer- man, editors, Oxford Studies in Metaphysics, volume 6, pages 223{284. Oxford University Press, 2011. [62] Jonathan Schaer. Spacetime the one substance. Philosophical Studies, 145:131{148, 2009. [63] Theodore Sider. Four-Dimensionalism: An ontology of persistence and time. Oxford University Press, 2001. [64] Theodore Sider. Against vague existence. Philosophical Studies, 114:135{146, 2003. [65] Theodore Sider. Parthood. Philosophical Review, 116(1):51{91, 2007. [66] Jonathan Simon. Fragmenting the wave function. In Dean Zimmerman and Karen Bennett, editors, Oxford Studies in Metaphysics, volume 12. Oxford University Press, Forthcoming. [67] Peter Simons. Parts: A Study in Ontology. Clarendon Press, 2000. [68] Peter Simons. Location. Dialectica, 58(3):341{347, 2004. [69] Lawrence Sklar. Space, Time, and Spacetime. Berkeley: University of California Press, 1974. [70] Bradford Skow. Once Upon a Spacetime. PhD thesis, NYU, 2005. [71] Bradford Skow. Are shapes intrinsic? Philosophical Studies, 133(1):111{130, 2007. [72] Richmond H. Thomason. Identity and vagueness. Philosophical Studies, 42:329{332, 1982. 106 MATT LEONARD [73] Judith Jarvis Thomson. Parthood and identity across time. Journal of Philos- ophy, 80:201{220, 1983. [74] Judith Jarvis Thomson. The statue and the clay. No^ us, 32:149{173, 1998. [75] Gabriel Uzquiano. Receptacles. Philosophical Perspectives, 20(1):427{451, 2006. [76] Gabriel Uzquiano. Mereological harmony. In Dean Zimmerman, editor, Oxford Studies in Metaphysics, volume 6, pages 199{224. Oxford, 2011. [77] Achille Varzi. Spatial reasoning and ontology: Parts, wholes, and locations. In Marco Aiello Pratt-Hartmann, Ian E and Johan F.A.K. van Benthem, editors, Handbook of Spatial Logics, pages 945{1038. Springer, 2007. [78] Achille Varzi. Mereology. Stanford Encyclopedia of Philosophy, 2016. [79] Lee Walters. Are the statue and the clay mutual parts? Nous, Forthcoming. [80] Ryan Wasserman. Material constitution. In The Stanford Encyclopedia of Phi- losophy. Edward N. Zalta (Ed.), 2017. [81] Brian Weatherson. Many many problems. Philosophical Quarterly, 53(213):481{501, 2003. [82] Timothy Williamson. Vagueness. Routledge, 1994. [83] Timothy Williamson. Modal Logic as Metaphysics. Oxford: Oxford University Press, 2013. [84] Jessica Wilson. Are there indeterminate states of aairs? yes. In Elizabeth Barnes, editor, Current Controversies in Metaphysics. Taylor and Francis, 2017.

Abstract (if available)

Abstract Consider two boring facts about myself. First, I have a certain size and shape. And second, I am composed of a number of elementary particles. Upon reflection, however, these facts become a bit more interesting. Though there are infinitely many regions of spacetime, one of them is a lot like me—my location. It, too, has my exact size and shape. And for each particle p that is a part of me, p’s location is a part of my location. In fact, for any of my parts whatsoever, its location is a part of my location. Moreover, these facts are necessarily true. But this is a bit puzzling. Why are they true? This puzzle leads to the central question of this dissertation: what it is for a material object to be located at a particular region of spacetime? In my second chapter, “What is it to be Located?”, I begin by sketching the two main theories of location suggested by the literature. I then propose a new theory of location—the mereogeometrical theory (or the ‘MG theory’ for short)—where to be located at a region just is to be geometrically congruent with the region (to have the same size and shape) and mereologically coincident with the region (to overlap all and only the same things). In my third chapter, “On the Contingency and Vagueness of Where I Am”, I develop two arguments against the identity theory of location (one of the main rivals in the literature). In my fourth chapter, “Explaining Harmony”, I explore in more detail the interaction of parthood and location.

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Asset Metadata

Creator Leonard, Matt (author)

Core Title What it is to be located

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Philosophy

Publication Date 10/16/2019

Defense Date 10/16/2019

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag location,mereology,metaphysics,OAI-PMH Harvest,parthood,spacetime

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Hawthorne, John (committee chair), Uzquiano, Gabriel (committee chair), Kleinschmidt, Shieva (committee member), Russell, Jeffrey Sanford (committee member), Schein, Barry (committee member)

Creator Email matt523@usc.edu,mleonard@calbaptist.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-224792

Unique identifier UC11674981

Identifier etd-LeonardMat-7861.pdf (filename),usctheses-c89-224792 (legacy record id)

Legacy Identifier etd-LeonardMat-7861.pdf

Dmrecord 224792

Document Type Dissertation

Rights Leonard, Matt

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA