Donald Baxter's Composition as Identity Jason Turner October 2, 2012 In the late 1980s, Donald Baxter published a pair of important papers (1988a, 1988b) defending the radical view that identity could hold one-many, between - for instance - a thing and its parts. David Lewis briefly flirted with this view in Parts of Classes, (1991: §3.6) before eventually accepting a diluted cousin. An impressive list of authors have since critically discussed the view under the name Lewis gave it, 'Composition as Identity' - 'CaI', for short.1 But although citations to Baxter's papers are de rigeur, most authors focus primarily on the Lewisian version of the view. Baxterian CaI has received less attention. Lewis's dominance in contemporary analytic metaphysics partly explains this phenomenon. But it isn't the whole story. Baxter's CaI is, in his own words, 'stronger and stranger' (this volume, PP) than even the view Lewis rejected. The view's radical nature, introducing novel philosophical concepts and using familiar ones in unfamiliar ways, can give even the most liberal-minded philosophers a feeling of vertigo. My aim here is to stabize Baxter's vertiginous readers. My method is regimentation: clothing the unfamiliar theory in formal garb and writing down principles to characterize it. Why regiment? First, it aids understanding, or at least a close approximation. We can manipulate the regimented notions in accordance with the rules even if we have a tin ear for the concepts themselves. Second, it facilitates critical discussion by revealing theoretical commitments, highlighting choice points, and making definite predictions. It is much easier to tell if a theory has bad consequences if a formal system makes those consequences explicit. And finally, it can serve as an implicit definition of the novel concepts, a la Lewis 1970. We can, of course, argue about whether the regimented theory has a realization; but we cannot doubt that it is a theory. Lewisian CaI collapses the composition and identity relations: composite objects are identical to their parts taken together, but individual (proper) parts are in no sense identical to wholes. Baxterian CaI, by contrast, holds that each part is identical to an aspect of the whole - which aspect is identical to the whole 1See, e.g., Cameron 2012, Cotnoir forthcoming, Hawley forthcoming, Kleinschmidt forthcoming, McDaniel 2008, 2010a, Merricks 1999, Sider 2007, van Inwagen 1994, Wallace 2011a,b, Yi 1999, and many papers in this volume. 1 Jason Turner itself.2 Understanding how this works - and why it doesn't immediately entail that my arms are identical to each other - involves understanding both Baxter's Aspect Theory (§1) and his doctrine that existence is count-relative (§2). In §3 I discuss how these come together for a Baxterian CaI. §4 finishes with critical remarks and suggestions for further research. Though I'll draw on later work (1989, 1999, 2001) to help, Baxter 1988b is my primary source. I do not promise my discussion will be faithful to everything Baxter says later. (His 1988a, as I read it, is a less developed precursor to the 1988b, and when they seem to conflict I take the latter as canonical.) 1 Aspect Theory I am both a father and a philosopher. As a father, I spend too much time thinking about philosophy. As a philosopher, I don't. If you knew me as well as I do, you'd know these are plain, down-home truths about me. How do we analyze these truths? We might think there are two different properties - thinking about philosophy too much as a father and thinking about philosophy too much as a philosopher - and that I instantiate only one of them. We might think there is one property - thinking about philosophy too much - and instantiation is relativized to, for instance, fatherhood and philosopherhood. Or we might think there are two different objects - Jason-as-philosopher and Jasonas-father - and only one instantiates the property. Following Baxter, we'll call the kinds of objects involved in the third analysis aspects: Jason-as-philosopher is an aspect of me. (We call non-aspects 'unqualified'.) Baxter argues at length (1988a: 203–206 and 1999) for the third option, but with a twist: Jason-as-philosopher and Jason-as-father aren't different objects. They're both me. But although these aspects are both me, they needn't share exactly the same properties with me, or with each other. There are thus cases of self-differing: I differ from myself by having aspects that differ from each other. 1.1 Talking About Aspects Since Aspect Theory allows for self-differing, Leibniz' Law is going to fail, and so it's hard to predict, in advance, how this theory will play out. We have no hope of getting it under control unless we first have a systematic way to talk about aspects. 2Baxter also endorses a non-distributive form of collective identity (1988b: 208–209), but with a distinctively Baxterian twist. See §3.3. Donald Baxter's Composition as Identity – 2 Jason Turner Our initial language will have with names, variables, predicates, and truthfunctions; we'll add quantifiers later. If α is a name in our language, call it a regular name. Here's a new bit of notation: if α is a regular name and φ(y) any formula open in y, let αy[φ(y)] an aspect name. For instance, 'Jasony[y is a father]' refers to me-as-father.3 Aspect names are, unsurprisingly, a sort of name: if α is a regular name and ψ(y) a well-formed formula, then if φ(α) is well-formed, so is φ(αy[ψ(y)]). For instance, since 'Jason spends too much time thinking about philosophy' is wellformed, 'Jasony[y is a father] spends too much time thinking about philosophy' is well-formed, too. (Choice point: Can aspect-names be used to make further aspect names? That is, do square-brackets 'stack', so that we can have 'ay[Fy]z[Gz]', for instance? The related metaphysical question asks whether aspects can themselves have aspects. For simplicity we'll assume here that aspects don't stack, though I suspect the best working-out of the theory disagrees.) Well-formedness is one thing; referring another. Even though 'Jasony[y is an airline pilot]' is well-formed, since I'm no airline pilot, it presumably names no aspect of me. Some aspect names can be empty. The natural way to say that an aspect-name is empty uses quantifiers, so we'll need to add some to our language. It turns out that sometimes we'll want to ignore aspects when quantifying, and sometimes we won't. To do this, we'll use different fonts for variables: variables in italics are restricted to ignore aspects, but variables in boldface are not. Avoid the temptation to think of this convention as 'italicized variables don't range over aspects', because these variables do range over aspects, at least insofar as they range over things that aspects are identical to. The safest way to think about the convention is in terms of what counts as a quantified formula's substitution instance. Consider: (1) Jason does too much philosophy. (2) Jasony[y is a father] does too much philosophy. Both of these count as substitution instances of (3) ∀x(x does too much philosophy). But only (1) counts as a substitution instance of 3Notice: this allows 'open' aspect names that can be quantified into. We won't be spending any time on them here, though. Donald Baxter's Composition as Identity – 3 Jason Turner (4) ∀x(x does too much philosophy). Even though (2) counts as a substitution instance of (3), it might not follow from it: If I wasn't a father, then presumably (2) would have been false even if (3) were true. Since aspect-names may be empty, we'll need a free logic, which won't license the inference from (4) to (2) without the further premise (5) ∃x(x = Jasony[y is a father]), which will be our way of saying that me-as-father exists. (We'll have a negative free logic, according to which all atomic sentences involving empty names - including identity sentences - are false.) (Another choice point: Are we going to allow empty regular names? If we don't, §2 will get more complicated than I'd like, so I'll let them in. Other than avoiding some unnecessary complications this choice will make no difference in what follows.) Under what conditions do aspects exist? Presumably, me-as-father exists but me-as-airline-pilot doesn't because I am a father but not an airline pilot. This suggests a crucial principle of Aspect Theory: Descriptive Necessity: ∀x(∃z(z = xy[φ(y)])→ φ(x)). In other words, if x-as-φ exists, then x must be φ. What about the converse - the principle that if anything is φ then it-as-φ exists? Call this Descriptive Sufficiency: ∀x(φ(x)→ ∃z(z = xy[φ(y)])). Descriptive Sufficiency has some odd consequences; it entails, for instance, that there is an aspect of you-as-self-identical-and-the-Hapsburgs-are-no-longer-onthe-throne. Even 'mere Cambridge change' will create and destroy aspects if Descriptive Sufficiency is right. Still, oddity is no bar to truth. And without either Descriptive Sufficiency or some suitably restricted variant we might worry there is no principled way to decide whether a certain aspect exists. Having flagged it as a potentially important choice point I'll take no further stand. 1.2 Aspects and Identity The above apparatus is consistent with theories of aspects which sees them as distinct from their unqualified objects. But Baxter insists that things are identical to their aspects. I am me-as-father - and me-as-philosopher, and me-as-bipedal, and so on. That is, Baxter endorses Donald Baxter's Composition as Identity – 4 Jason Turner Aspect Identity: ∀x(∃z(z = xy[φ(y)])→ x = xy[φ(y)]). If anything is x-as-φ, x is. Baxter also accepts the reflexivity, symmetry, and transitivity of identity.4 So he will accept all of: (6) Jasony[y is a father] = Jasony[y is a philosopher]. (7) Jasony[y is a father] does too much philosophy. (8) ∼(Jasony[y is a philosopher] does too much philosophy). So Leibniz' Law fails. This is no smoking gun; Baxter is upfront about it. He writes: I am urging the discernibility of identicals. To say that identicals are indiscernible is to mean that for all a and b if a and b are identical then a and b have the same properties. So when I say that identicals are discernible I mean there exists some a such that a has and lacks a property. For a insofar as it is one way has the property and a insofar as it is another lacks it.5 (1988b: 205–206) But while Baxter thinks there are Leibniz' Law failures, he doesn't think the law completely worthless. It holds so long as we're not talking about aspects, but the 'bare' things they're aspects of. [Leibniz' Law] is: Thing a has some property, a is numerically identical with thing b, so b has that property. I do not want to dispense with this but merely circumscribe its application. It is valid of a and b considered unqualifiedly - otherwise contradictions would be true. But it is not valid if either a or b is considered qualifiedly, that is, if either is an aspect of something. (1999: 49; cf. 1988b: 206) It is easy to formulate the version of Leibniz' Law that Baxter wants to reject: Strong Leibniz' Law: ∀x∀y(x = y→ (φ(x)↔ φ(y))) Strong Leibniz' Law is inconsistent with (6)–(8), because we can instantiate its boldface variables with 'Jasony[y is a father]' and 'Jasony[y is a philosopher]'.6 4More precisely, he accepts these principles as formulated with boldface variables and '='. Whether he accepts the transitivity of identity is something we'll come back to in §4. Note also that transitivity and symmetry usually follow from Leibniz' Law, but won't follow from any of the proposed restrictions of it discussed below. 5Baxter often uses 'insofar' in aspect names: 'Jason insofar as he is a philosoper' is long for 'Jason-as-philosopher'. 6The entailment relies on the negativity of the free background logic, which lets us infer from (6) that Jasony[y is a father] and Jasony[y is a philosopher] both exist. Donald Baxter's Composition as Identity – 5 Jason Turner It's tempting to think that Baxter wants instead: Fairly Weak Leibniz' Law: ∀x∀y(x = y→ (φ(x)↔ φ(y))) But that's not clear. Consider three arguments: Argument One: Twain = Clemens Twainy[y wrote Hucklebury Finn] is well-known. So, Clemensy[y wrote Hucklebury Finn] is well-known. Argument Two: Twain = Clemens Susany[y read all Twain's work] is well-read. So, Susany[y read all Clemens's work] is well-read. Argument Three: Twain = Clemens Twainy[y wrote Twain's work] is well-known. So, Clemensy[y wrote Clemens's work] is well-known. Fairly Weak Leibniz' Law licenses all three, but we might not want to. We can distinguish several weaker versions of Leibniz' Law. First, say that a term t occurs in name position of a description-name if it is outside of the square brackets, and say that it occurs in descriptive position if it occurs inside the square brackets. For instance, in 'Clemensy[y wrote Twain's work]', 'Clemens' occurs in name but not descriptive position, and 'Twain' occurs in descriptive but not name position. Then our three weaker laws are: Descriptively Weak Leibniz' Law: ∀x∀y(x = y → (φ(x) ↔ φ(y))), where x and y don't occur descriptive position in φ(x) and φ(y). Nominally Weak Leibniz' Law: ∀x∀y(x = y→ (φ(x)↔ φ(y))), where x and y don't occur in name position in φ(x) and φ(y). Very Weak Leibniz' Law: ∀x∀y(x = y → (φ(x) ↔ φ(y))), where x and y don't occur in descriptive or name position in φ(x) and φ(y). Descriptively Weak Leibniz' Law licenses Argument One, but not Arguments Two or Three. Nominally Weak Leibniz' Law licenses Argument Two, but not arguments One or Three. And Very Weak Leibniz' Law doesn't license any of them. Baxter clearly wants at least Very Weak Leibniz' Law. My suspicion is that an Aspect Theorist should also endorse Argument One, and thus Descriptively Donald Baxter's Composition as Identity – 6 Jason Turner Weak Leibniz' Law. After all, if 'Twain = Clemens; Twain wrote books; so Clemens wrote books' is valid - and Very Weak Leibniz' Law says it is - then the Aspect Theorist presumably thinks Leibniz' Law failures don't stem from using different names for the same unqualified object. But that suggests the Aspect Theorist should also think we pick out the same aspect whether we describe it as an aspect of Twain or as an aspect of Clemens. So, I will tentatively suppose that the Aspect Theorist endorses at least Descriptively Weak Leibniz' Law, and leave it to her to decide what else she want to endorse. 2 The Count-Relativity of Identity 2.1 Counts With Aspect Theory at the ready, we're ready to develop Baxter's CaI. It starts from the thought that existence is relative to what he calls 'counts': Consider the express check-out line in a grocery store. It says 'six items or less'. You have a six-pack of orange juice. You might well wonder if you have one item or six items. But you would never hesitate to go into the line for fear of having seven items: six cans of orange juice plus one six-pack. . . In counting, we either count the whole as one, or each part as one. If we count the whole, we do not count the parts. If we count the parts, we do not count the whole. I propose that we take the fact that there is more than one way to count, as evidence that there is more than one true number of things that exist. (1988b: 200) This might sound initially sound like an observation about human activity: when we count, we tend to count by sortals ('bottle', 'six-pack', etc.), ignoring what doesn't fall under them. But that wouldn't mean that 'there is more than one true number of things that exist'. After all, why couldn't we just count all the self-identical things, and let that be the 'one true number'? (Cf. Rayo 2003: 105–106) Baxter is after something deeper, though; his remarks here, coupled with his insistence that 'there is a count which includes the several parts and a count which includes the whole, but no count which includes both' (1988b: 201), hearken back to Ryle's 'category mistake' (1949: 22): you can buy a left and a right glove, or a pair of gloves, but you're confused if you think you're buying three things. Peter van Inwagen (1998: 236–237) sees in Ryle the view that 'existence' Donald Baxter's Composition as Identity – 7 Jason Turner is equivocal: there is no sense of 'exist' that covers the several parts and the whole. I interpret Baxter likewise: there is no one true number of things that exist because there is no one true sense of 'exist' we could use in answering this question. More precisely, there is no one true answer to the question because the question presupposes that material objects exist or not simpliciter, but they don't: 'what exists is relative to a count' (1988b: 201), and 'exists' is equivocal insofar as it leaves open which count it is relativized to. If the thought seems unfamiliar, perhaps a deflationary variant can help fix ideas. Some philosophers think the world ultimately consists of 'stuff' spread throughout regions, with no privileged division of this stuff into distinct, countable 'things' (Jubien 1993, 2009; Einheuser 2011). Nonetheless, we can in thought divide the stuff into things. Perhaps we think of the stuff over there as three separate things, with thus-and-so boundaries, and the stuff over here as a single thing, with such-and-such boundaries.7 On these views, we choose how to carve stuff into things. Each choice imposes an ontology on the world, and we can represent these different choices of ontology with different quantifiers. If '∃c' corresponds to one choice and '∃d' to another, then '∃cxFx' will be true if the c-choice carves a boundary around some F stuff, and '∼∃dxFx' will be true if the d-one doesn't. Baxter calls these choices 'counts', both because they determine how many things exist (according to the choice), and because they determine which parcels of reality 'count' as single things. Baxter's account differs from the above only by rejecting its deflationism: counts aren't things we project onto the world, but are part of its fundamental metaphysical structure. Count-relativity is thus a form of 'ontological pluralism' (McDaniel 2009, 2010b, Turner 2010, 2012), using multiple quantifiers to capture the different, relativized senses of 'exist'. Formally, our quantifiers will be tied to different counts by way of subscripted indices - I'll use 'c' and 'd', with or without subscripts. '∃xFx' and similar are longer well-formed, but the corresponding '∃cxFx' and similar are. Informally, I'll use 'c' and 'd' to talk about the counts associated with respective quantifiers. If we have multiple inequivalent quantifiers we must be careful about their inference rules. In particular, we cannot deduce 'Fa' from '∀cxFx' without the premise '∃cx(x = a)', and similarly for existential generalization. (Cf. Turner 2010: §5) 7Einheuser (2011) argues that the deflationist should think we must do more in thought than simply chop stuff into chunks. I agree, but the subtleties won't matter here. Donald Baxter's Composition as Identity – 8 Jason Turner 2.2 Count Relativity and Aspects The principles of §1 were written with unindexed quantifiers; we'll need to rewrite them now. Most of these principles involve multiple quantifiers. Are we allowed to 'mix' indices in these principles, or must all the quantifiers in any given principle be tied to the same count? This technical question relates to a metaphysical one: If an object doesn't exist in one count, might it still have an aspect that exists in that count? I'm not aware of Baxter anywhere explicitly addressing this question, but it's clear from his 1988b that he thinks it can't. As a result, and with one exception, we can endorse any of the principles from §1 even when their quantifiers are tied to different counts. For instance, Aspect Identity becomes Rewritten Aspect Identity: ∀cx(∃dy(y = xz[φ(z)])→ x = xz[φ(z)]), (where 'c' and 'd' can be replaced for any count-terms we like). We can say that aspects follow objects in counts with: Count Coordination: ∀cx(∃dy(y = xz[φ(z)])→ ∃dy(y = x)). If we add an inference rule that lets us move from ∃cy(y = a) to ∃cy(y = a), this follows from Rewritten Aspect Identity.8 Descriptive Sufficiency, if we endorse it, forms the one exception to our policy. If we allowed its quantifiers to be tied to different counts, we could use an object in any one count to form an aspect in any other, which by Count Coordination would put every object in every count. We can, however, endorse a variant of Descriptive Sufficiency where both quantifiers are tied to the same count. One upshot of Count Coordination is that if a = b, then a and b exist in the same count. As a result, the identity relation '=' can be thought of as a kind of intra-count identity - and indeed, Baxter often calls it that. 3 Composition as Identity Let's start with an example. Here is Baxter's: 8Suppose ∃cx(x = a) and ∃dy(y = az[φ(z)]). We need to show ∃dy(y = a). By Rewritten Aspect Identity, a = az[φ(z)]. But since ∃dy(y = az[φ(z)]), by (plural) existential generalization, ∃dy(y = a). So by our new inference rule, ∃dy(y = a). Donald Baxter's Composition as Identity – 9 Jason Turner You are showing a child an orange then its parts. First you say, 'Here is an orange. It's juicy inside but not outside.' After peeling you say, 'Here is the inside of the orange - it's juicy. And here's the outside - it's not juicy.' In the before case, you are talking about one thing, the orange. The orange inside is one way and the orange outside is another. . . In the after case, the inside of the orange is one thing that is one way, and the outside of the orange is a second thing that is another way. (Baxter 1988b: 206) Let's call the orange 'Otto,' the peel 'Peely,' and the juicy inside 'Innie'. (See figure 1.) Peely and Innie are clearly parts of Otto. In the before case, we're Otto{ }} { Peely Innie Figure 1: Otto and parts talking about Otto. According to Baxter, wholes and parts never exist in the same count. So in the before case, we must be talking in a count that includes Otto but excludes Peely and Innie. If it excludes Peely and Innie, what are we doing when in the before case we talk about 'the orange outside'? Baxter says this is short for 'the orange insofar as it occupies the location of the peel,' (Baxter 1988b: 206). So we're talking about aspects of the Orange - aspects tied to location. Let P be Peely's location and I Innie's; then in the before case, we're talking about Otto-as-occupying-P and Otto-as-occupying-I. Using '@' as a shorthand for 'occupies', we're talking about Ottoy[y@P] and Ottoy[y@I]. The move to CaI consists in identifying Ottoy[y@P] with Peely. But this identity can't be the intra-count identity '=' discussed above, because Peely and Ottoy[y@P] can't be in the same count. Indeed it isn't: Baxter calls it cross-count identity. Donald Baxter's Composition as Identity – 10 Jason Turner Why think cross-count identity deserves to be called identity? The idea is perhaps clearest from the deflationary perspective described in §2.1. Suppose cO is a count where Otto exists, and we go searching for Peely there. Well, Peely doesn't exist in count cO, not properly, not as such, so we won't find it. But we might think there's a way in which Peely has snuck in: Even though Peely doesn't exist in cO, Peely's stuff does. It's hiding inside Otto. In cO we cannot find anything that exactly occupies P.9 But we can find something that covers P - something that occupies P and more besides. It's Otto. Otto contains all Peely's stuff, and more. If we had a way, while staying in cO, to focus just on the stuff inside P, that would be a way to indirectly think about Peely within in cO. But we do: Ottoy[y@P] lets us focus on exactly this stuff. Since 'Ottoy[y@P]' and 'Peely' seem to be ways of getting at the same stuff from the perspective of different counts, it's not too much of a stretch to say they are identical. That, at any rate, is the main idea. Let's sharpen it up a bit. 3.1 Regions Baxterian CaI relies heavily on occupation relations. We should make its occupational commitments explicit. Capital letters will be terms (names or variables) for regions. Regions' existence won't be count relative, so we won't quantify over them with count-relative quantifiers. We might as well quantify over them with unsubscripted '∃' and '∀', since those aren't being used for anything else right now. We'll also need a subregion predicate '⊆' along with the 'occupies' predicate '@' from above. And for simplicity we assume that regions don't have aspects: 'Ry[φ(y)]' isn't well-formed. Our earlier use of '@' treats it as what Josh Parsons (2007: 203) calls pervading: if I pervade a region, then I fill all the region and perhaps more besides. We can also define a notion of exact occupation: Exact Occupation: x@!R =df. x@R ∧ ∀S(x@S→ S ⊆ R). In other words, x exactly occupies R iff it occupies (pervades) R and no other region other than R's subregions. Following Parsons (2007: 205), we'll endorse two principles about location: Exactness: ∀cx∃R(x@!R)10 9At least, no unqualified thing; see §4.1. 10My statement of exactness differs from Parsons, in that mine implies that everything (ranged Donald Baxter's Composition as Identity – 11 Jason Turner Functionality: ∀cx∀R∀S((x@!R ∧ x@!S)→ R = S) Thus each material object (and aspect) exactly occupies just one region. I will call this region its location, and say it is located at this region. We also assume that everything occupies all subregions of its location. Call this principle Inheritance.11 3.2 Cross-Count Identity With these resources sorted, we can get to the main business. Baxter takes crosscount identity as a primitive notion, but wants it to behave in a certain way. It ought to be reflexive, symmetric, and transitive; let's take these as read. More importantly, though, parts in one count are meant to be identical to wholes-aslocated-at-the-part's-region in another. Peely is meant to be cross-count identical to Ottoy[y@P] precisely because Peely is located at P. Using '≈' for cross-count identity, the principle we want reads: Cross-Count: ∀cx∀dy∀R((x@!R ∧ y@R)→ x ≈ yz[z@R]) (Notice: This doesn't say that x is cross-count identical to y-as-exactly-occupyingR - for y may not exactly occupy R, and if it doesn't, then by Descriptive Necessity, there is no such aspect of y.) We might think that, if ≈ is really an identity relation, it should obey some version of Leibniz' Law. Baxter agrees; he writes: Let me [stipulate] that in my mapping, [aspect of the whole] must exactly resemble [aspect-free part], in every way that does not entail that [whole] and [part] exist in the same count, or are identical with the same things. (Baxter 1988b: 208) Call φ(x) purely qualitative iff it is (i) open only in x, (ii) contains neither '=' nor '≈', (iii) contains no count-relativized quantifiers, and (iii) is with x both nominally and descriptively bare in it. Then Baxter's stipulation corresponds to Cross-Count Leibniz' Law: ∀cx∀dy(x ≈ y→ (φ(x)↔ φ(y))) for purely qualitative φ(x). over by the boldface variables) is located, and his does not. Since the boldface variables are only meant to range over material objects and their aspects, this difference won't be important below. 11In the interests of space, this section rides roughshod over a host of complex issues; cf. Hudson 2005: ch. 4 and McDaniel 2007. Donald Baxter's Composition as Identity – 12 Jason Turner 3.3 Parthood and Many-One Identity Neither parthood nor any other mereological notion is primitive for Baxter. But we can define parthood using crossand intra-count identity. Baxter writes: The identity of part with whole is really the cross-count identity of part with whole as in sub-location, and then intra-count identity of that with whole. So the identity between part and whole seems to be between two things considered unqualifiedly, but it is so only by the mediation of identity with something considered unqualifiedly - i.e. the whole as in sub-location. (Baxter 1988b: 214) This means we can define a parthood predicate by: Parthood: x is a (proper or improper) part of y =df. ∃R(x ≈ yz[z@R]∧ yz[z@R] = y). Notice that, in this definition, parthood only holds between 'bare' things: aspects neither are nor have parts. We can now see why the view doesn't make my arms identical to each other. Each of my arms is cross-count identical to an aspect of me, and those aspects are intra-count identical. But although cross-count and intra-count identity are each transitive, their mixture isn't, so there's nothing forcing us to cut out the middlemen and make my arms identical. If we allow plural quantification, we can also define a notion of many-one identity. Like the singular, plural quantification over material objects will need to be indexed to counts.12 That's enough to get us a sort of 'automatic', distributive kind of many-one identity: the Xs are (distributively) many-one identical to y if and only if each of the Xs is identical to an aspect of y. But this kind of cross-count identity is weaker than we want. Consider a count with Peely, and then two (exhaustive) parts of Innie: Lefty and Righty. Each of Peely, Lefty, and Righty is cross-count identical with (an aspect of) Otto. So Peely and Lefty are each cross-count identical with (an aspect of) Otto, making the collection with just Peely and Lefty many-one identical, in this sense, with Otto. This doesn't quite capture the idea behind many-one identity though, for part of Otto (namely, Righty) is missing. 12I have ignored plural quantification thus far. It introduces further complexities (do pluralities have aspects?) that I can't address here. But we'll consider it long enough to discuss the sense in which, for Baxter, identity is 'many-one'. Donald Baxter's Composition as Identity – 13 Jason Turner We can define a stronger version of many-one identity that doesn't leave bits out. If we use '≈' ambiguously for both the singular/singular and singular/plural relations, we define the singular/plural one by: Many-One Identity: ∀cX∀dy(X ≈ y =df. ∀cx(x is one of X ↔ ∃R(x ≈ yz[z@R]). In other words, the Xs are strongly many-one identical to y only if they're wealky many-one identical and don't leave out anything (in their count) that (an aspect of) y is also cross-count identical to. 4 Critical Discussion 4.1 Saying Enough In Baxterian CaI, parthood relations are supposed to track subregion relations. Cross-Count gives us a sufficient condition: it guarantees that, if x's location is a subregion of y's, then x is a part of y. But we want the converse, too: if x is a part of y, then x's location should be a subregion of y's. We haven't quite said enough to guarantee this. Suppose Freddy is a ghost located at R f and living in one count, Daphne is a ghost located at a nonoverlapping Rd and living a different count, and that there are no other counts or unqualified objects. If R−d is a subregion of Rd and R − f a subregion of R f , nothing blocks Freddy being cross-count identical to Daphney[y@R−d ] and Daphne being cross-count identical to Freddyy[y@R−f ]. (See figure 2.) One Count Another Count Freddyy[y@R−f ] Daphney[y@R − d ] ≈ ≈ Freddy Daphne Figure 2: Mismatched ≈ The gap is easy to close. Thanks to Cross-Count Leibniz' Law, a part's location is the same as the location of the aspect it's cross-count identical to. So in Donald Baxter's Composition as Identity – 14 Jason Turner this case, Daphney[y@R−d ] would be located at R f and Freddyy[y@R − f ] would be located at Rd. This suggest we can block the counterexample by insisting that regional aspects are located at their respective regions: Aspect Location: ∀cx∀R(∃dz(z = xy[y@R])→ xy[y@R]@!R) And this does the trick: given Aspect Location, parts must be located at subregions of wholes.13 4.2 Mereological Principles How, in Baxter's system, does the defined notion of parthood behave? Not very well, unfortunately; it isn't even guaranteed to obeys the axioms of 'minimal mereology' (Varzi 2009: §2.2). It is transitive and reflexive,14 but it need not be anti-symmetric. Suppose there's one count with one thing located at R, and another count with a different thing located at R. Then these two things will count as parts of each other, but are distinct. It also needn't obey most other mereological principles. For instance, supplementation principles can fail: if we made Daphne's location is a subregion of Fred's but left everything else about the above case alone, Daphne would be Fred's only proper part. We could strengthen the mereology with additional principles, although this may be trickier than it looks. Lacking space to say more, I leave it for future work. 4.3 But is it Identity? Here's a straightforward objection to Baxterian CaI: since '=' and '≈' don't obey (Strong) Leibniz' Law, they can't be identity! (Cf. Wallace 2011a: 809) Baxter has worked hard to answer this charge (see especially his 1999), and I'm not going to assess his efforts here. Note only that, whether we accept it or not, Baxter's rejection of Strong Leibniz' Law is independently motivated by his Aspect Theory, not an ad hoc fix for Baxterian CaI. 13Suppose x is a part of z; then x ≈ zy[y@R]. So z@R, which means z is located at a superregion of x. But by Aspect Location, zy[y@R]@!R, and so by Cross-Count Leibniz' Law, x@!R. 14Reflexivity: if x is located at R, then x occupies R, so xy[y@R] exists, is intra-count identical to x, and by Cross-Count is cross-count identical to x. Transitivity: if x is a part of y which is a part of z, then there are regions Rx ⊆ Ry ⊆ Rz where x is located at Rx, y at Ry, and z at Rz. But by Inheritance z occupies Rx, and so x will be a part of it. Donald Baxter's Composition as Identity – 15 Jason Turner A related objection complains that Baxterian CaI violates the univocality of identity. The theory posits two different identity relations: '≈' and '='. But (goes the objection), identity is just identity! There can't be two!15 A Baxterian can deny the univocality of identity, insisting there can be two kinds of identity. She must then say why both relations count as identity. Or she can accept the univocality claim, insisting that = and ≈ are restrictions on a single underlying identity relation, ∼=. The ∼= relation can hold crossor intracount; 'cross-count identity' is just ∼= holding between things in different counts, and 'intra-count identity' is ∼= holding within a count. I'm not terribly optimistic about the second response. First, ∼= had better not be transitive, or else my right and left arms will be (in some sense) identical after all. Neither this identity nor the denial of transitivity for ∼= - our only options - are very attractive. Furthermore, it's not clear = and ≈ can coherently thought of as restrictions on ∼=. Ideally, we would want a system of axioms that, coupled with definitions for = and ≈, yield the various principles endorsed above. But = and ≈ code up information that we can't recover in any straightforward way from a single ∼= relation. For instance, if we want to say that something x exists both in counts c and d, we do it with (9) ∃cy∃dz(x = y∧ x = z). But (10) ∃cy∃dz(x ∼= y∧ x ∼= z) can't say that x exists in both c and d; for it would be true if x were cross-count identical to both y and z. (Perhaps x is a part of y, which is in turn a part of z.) It's unclear what we could add to (10) to give it the force of (9). The missing information is that x exists in both counts; but that was what we were trying to express in the first place. I do not say a full formalization of the view in terms of ∼= can't be given. But I do not see how to give it, so I'll set that strategy aside. The other strategy denies univocality: = and ≈ are simply two different kinds of identity. One reason to think they are both kinds of identity is that they both share certain structural features, such as reflexivity, transitivity, and obedience to a restricted version of Leibniz' Law. Another reason can come from 15Note that various 'collapse' arguments for this conclusion, such as in Harris 1982 and Williamson 1988, won't apply here. They rely on an unrestricted version of Leibniz' Law that neither '=' nor '≈' obey. Donald Baxter's Composition as Identity – 16 Jason Turner our native grasp of the relations. This is perhaps clearest on the deflationary version of Baxterian CaI. Consider again Otto, Ottoy[y@P], and Peely. When we carve the world into an Otto-shaped chunk, we see that Otto and Ottoy[y@P] are the same thing. After all, Ottoy[y@P], if it exists, just is whatever object we've carved that occupies P - and on this carving Otto is the only candidate. So of course the relation between Otto and Ottoy[y@P] is a kind of identity. We might gloss it as 'carving-identity'. On the other hand, when we look across the carvings, we see that the stuff in Ottoy[y@P] - a sub-portion of the stuff in Otto - is the very same stuff in Peely. But since it's precisely the same stuff, we should think that Ottoy[y@P] just is Peely. So ≈ is also a kind of identity. We might gloss it as 'stuff-identity'. This line of thought suggests a further principle. Suppose that x and y both exist in a single count, and that x ≈ y. Then x and y are made up of exactly the same stuff. But since they exist in a single count, and all a count does is divide stuff into object-sized bites, whatever object-sized bite we carve x into had better be the same one we carve y into. That is: Intra-Count Collapse: ∀cx∀cy(x ≈ y→ x = y). Notice that the two quantifiers in Intra-Count Collapse are linked to the same count; when x and y are in different counts (as Peely and Peelyy[y@P] are), this needn't hold. Likewise, Intra-Count Collapse's converse has no claim on us; although Otto = Ottoy[y@P], the aspect ignores some of Otto's stuff, so we're under no pressure to think of them as stuff-identical. Intra-Count Collapse can motivate Baxter's insistence that parts aren't in the same count as wholes. If there were a count where Peely and Otto both existed, Intra-Count Collapse would tell us Peely = Otto.16 But since Peely is located at P and Otto isn't, if they were numerically identical we'd contradict even Very Weak Leibniz' Law. Intra-Count Collapse has another, more disturbing upshot: no two things in a count can spatially overlap. Consider siamese twins, Jay and Sonny. They share an arm, 'Arm', located at a region A. The following claims both seem natural: (11) Jay and Sonny each occupy a superregion of A. (12) Some count c contains both Jay and Sonny. 16By Count Coordination, Ottoy[y@P] exists in every Otto-containing count. So if Peely were in a count with Otto, it would be in a count with Ottoy[y@P]. But since Peely ≈ Ottoy[y@P], if they shared a count, then by Intra-Count Collapse Peely = Ottoy[y@P], in which case by Aspect Identity and transitivity, Peely = Otto. Donald Baxter's Composition as Identity – 17 Jason Turner But we can't have both. Suppose (11); then Arm ≈ Jayy[y@A] and Arm ≈ Sonnyy[y@A], and so by transitivity Jayy[y@A] ≈ Sonnyy[y@A]. But if we have (12), then by Count Coordination these aspects share a count. And if they're in the same count, they'll be intra-count identical by Intra-Count Collapse. So by Aspect-Identity and Transitivity, Jay = Sonny. Disaster; one of (11) and (12) have to go. This is an unfortunate result; we naturally suppose that many things, including siamese twins, can share parts. We might try to get around it in various ways. Option One: Neither Jay nor Sonny has that arm; it's an entity in its own right, although under their control. Option Two: one count has Jay (who partly occupies A) and Sonny-minus (who doesn't), whereas another has Sonny (who partly occupies A) and Jay-minus (who doesn't). Either seems to me a serious cost. Perhaps the cost of Option One can be mitigated by stressing the non-mereological, functional sense in which the twins share an arm. And perhaps the cost of Option Two can be mitigated by raising doubts about the naïve thought that the twins have overlapping bodies. Just how costly these strategies are - and whether their costs are worth paying - isn't something I have tried to decide here. 5 Conclusion Used to the univocality of existence and the ubiquity of a strong Leibniz' Law, many will find Baxterian CaI unfamiliar. My regimentation will hopefully make the view's structure clear enough that those who find its concepts unfamiliar can nonetheless reason about it. As should be clear by now, the view is rich and complex. It raises a host of interesting questions: Do aspects stack? 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