Location and Mereology

First published Fri Jun 7, 2013; substantive revision Mon Mar 18, 2024

Substantivalists believe that there areregions of space orspacetime. Many substantivalists also believe that there are entities(people, tables, social groups, electrons, fields, holes, events,tropes, universals, …) that arelocated at regions.These philosophers face questions about the relationship betweenentities and the regions they are located at. Are located entitiesidentical to their locations? Are they entirely separate from theirlocations, i.e., they share no parts with them?

Without prejudging these metaphysical questions, some philosophershave formulatedlogics of location—typically groups ofaxioms governing a location relation and its interaction withmereological notions. These logics aim to capture the ways in whichthe mereological properties of and relations between located entitiesmust mirror the mereological properties of and relations between thelocations of those entities.

The recent literature focuses on four questions, each corresponding toa way in which the relevant mirroring might fail:

Say that two entitiesinterpenetrate just in casethey do not share parts but their exact locations do. Isinterpenetration possible?
Say that anextended simple is an entity that hasno proper parts but is exactly located at a region that has properparts. Are extended simples possible?
Conversely, say that anunextended complex is anentity that has proper parts but is exactly located at a region thatdoes not have proper parts. Are unextended complexes possible?
Say that an entity ismultilocated just in caseit is exactly located at more than one region. Is multilocationpossible?

The present article surveys recent work on these questions andaddresses other issues along the way. The goal of the entry is not toprovide a general account of the metaphysics of location. Rather itfocuses on the issues that are concerned with location and itsinteraction with parthood (in the spirit of, e.g., the paperscollected in Kleinschmidt (2014)).

Supplement: Systems of Location

1. Preliminaries: Spacetime and Parthood

This article focuses on the recent literature on location andmereology. On the history of these topics, see Marmodoro (2017), Harte(2002), Sorabji (1983, 1988), Pasnau (2011), and Holden (2004), aswell as the entriesancient atomism,medieval mereology,atomism from the 17th to the 20th century, andmereology.

In keeping with the recent literature, we will focus on‘entity-to-region’ location relations—i.e., thosethat paradigmatically hold between entities andregions. Wewill ignore location relations that hold between entities andnon-regions.

Since our focus is on entity-to-region location relations, we willwork under the following controversial but popular assumptions. Thereare spacetime regions that comprise a fundamental four-dimensionalarena, spacetime. All spacetime regions are equally real and there isno region which is absolutely present in any non-indexical sense. Wedo not assume that there are points; we leave open the hypothesis thatspacetime is gunky. However, we do assume that if there are points,then points count as regions—specifically, they would be simpleregions.

Throughout the entry we take parthood as primitive and take forgranted several standard mereological definitions. We usePfor parthood,PP for Proper Parthood, andO forOverlap—see the entrymereology, and Cotnoir and Varzi (2021).

We address questions framed in modal terms. Are extended simplespossible? Is it necessary that nothing is multilocated? The relevantmodality is metaphysical. In keeping with current orthodoxy, we assumethat being metaphysically necessary (a property of propositions orsentences) is not to be identified with being a logical truth, beingan analytic truth, being a conceptual truth, or being anapriori truth—see the entryvarieties of modality. Although metaphysical necessity is not identified with conceptualtruth—and, correlatively, metaphysical possibility is notidentified with conceivability—one might still think thatconceivability (or something in that vicinity) is evidence formetaphysical possibility—see the entry onthe epistemology of modality.^[1]

One last preliminary. The recent literature on location and mereologytends to bracket considerations of vagueness and indeterminacy (thoughsee Eagle 2016a, Leonard 2022) and quantum theory (though see Pashby2016, Calosi 2022a). We will do the same.

2. Location

2.1 Which Location Relation is Fundamental?

We begin by distinguishing four location relations. Often it isassumed that one of these is fundamental and involved in thedefinitions of the others—more on that shortly. For now, we giveinformal glosses of the four relations.

Exact location: \(x\) is exactly located atregion \(y\) if and only if \(x\) has (or has-at\(-y)\) exactly thesame shape and size as \(y\) and stands (or stands-at\(-y)\) in allthe same spatial or spatiotemporal relations to other entities as does \(y\).^[2] (See Casati & Varzi 1999: 119–120; Bittner, Donnelly, &Smith 2004; Gilmore 2006: 200–202; Sattig 2006: 48). In symbols:\(L(x, y)\)
Weak location: \(x\) is weakly located at region\(y\) if and only if \(y\) is ‘not completely free of’\(x\) (Parsons 2007: 203). \(\WKL(x, y)\)
Entire location: \(x\) is entirely located atregion \(y\) if and only if \(x\) ‘lies within’ \(y\)(Parsons 2007: 203; Correia 2022: 560). \(\EL(x, y)\)
Pervasive location: \(x\) is pervasively locatedat region \(y\) if and only if \(y\) is no larger than \(x\) and \(x\)‘completely fills’ \(y\) (Parsons 2008: 429; Correia 2022:560). \(\PL(x, y)\)

Figure 1 illustrates cases of these four relations.

a box diagram of regions and objects: link to extended description below

Figure 1: The dashed lines indicateregions \((r_1\)–\(r_6)\). The two shaded squares indicate twosquare objects, \(o_1\) and \(o_2\), that compose a larger rectangularobject, \(o_3\). [Anextended description of figure 1 is in the supplement.]

The table (Figure 2) indicates, incompletely, which objects bear which relations to whichregions.

	\(r_1\)	\(r_2\)	\(r_3\)	\(r_4\)	\(r_5\)	\(r_6\)
\(o_1\)	exactly weakly entirely pervasively		weakly entirely	weakly	weakly pervasively	weakly entirely
\(o_2\)		exactly weakly entirely pervasively	weakly entirely			weakly entirely
\(o_3\)	weakly pervasively	weakly pervasively	exactly weakly entirely pervasively	weakly	weakly pervasively	weakly entirely

Figure 2

Intuitively, \(o_1\) is exactly located at one and only one region,\(r_1\), which has the same size and shape, and stands in the samespatial relations to other things, as \(o_1\). However, \(o_1\) isentirely located at each region that it lies within, such as \(r_1,r_3\), and \(r_6\). It is pervasively located at each region that itcompletely fills, such as \(r_1\) and \(r_5\). It is weakly located ateach region that is not completely free from it, such as \(r_1, r_3,r_5, r_6\), as well as \(r_4\), at which it is neither entirely norpervasively located. Region \(r_2\), however, is completely free from\(o_1\), so \(o_1\) is not even weakly located at \(r_2\). Likewise,\(o_2\) is not even weakly located at \(r_1\). This should be enoughfor a pre-theoretic grasp of our four target relations.

Typically, one of the relations above is taken to be fundamental andused to define the others. This gives rise to a wide range of possibletheories, each with its own set of definitions and axioms. Some ofthese theories differ in what patterns of location they permit. Forexample, if one assumes that exact location is fundamental, then oneis free to accept the possibility of a strongly multilocated thing, athing that is exactly located at two non-overlapping regions. On theother hand, Parsons (2007) presents two theories, one that takes exactlocation as fundamental and one that takes weak location asfundamental. In the latter exact location is defined as follows:

(DS2a.1): \(x\) is exactly located at \(y\) \(=_{\df}\)\(x\) is weakly located at all and only those entities that overlap\(y\) \[L(x, y) =_{\df} \forall z[\WKL(x, z) \leftrightarrow O(y, z)]\]

According to this definition, it is analytic, hence impossible, thatnothing is strongly multilocated. To save space, we will assumehenceforth that exact location is the unique fundamental locativerelation, and that the other three relations are defined, asfollows:

(DS1.1): \(x\) is weakly located at \(y\) \(=_{\df}\)\(x\) is exactly located at something that overlaps \(y\).\[\WKL(x, y) =_{\df} \exists z[L(x, z) \amp O(z, y)]\]

(DS1.2): \(x\) is entirely located at \(y\) \(=_{\df}\)\(x\) is exactly located at some part of \(y\). \[\EL(x, y) =_{\df}\exists z[L(x, z) \amp P(z, y)]\]

(DS1.3): \(x\) is pervasively located at \(y\)\(=_{\df}\) \(x\) is exactly located at something of which \(y\) is apart. \[\PL(x, y) =_{\df} \exists z[L(x, z) \amp P(y, z)]\]

For a sketch of some theories arising from other views about whichrelations are defined, and how, see the supplementary documentSystems of Location.

2.2 The Pure Logic of Location

Most of the formal work on location has focused on how locationinteracts with parthood. But one might wonder about the logic oflocation itself. We raise two groups of questions about thislogic.

2.2.1 Logical Form

We take exact location as our unique locative primitive. We assumethat

it is a two-place relation, and
both argument places in that relation aresingular.

But both (i) and (ii) have been questioned.

For example, one might reject(i) in favor of the view that exact location is a three-place relationthat holds between a located entity, a region of space, and an instantof time (Thomson 1983; Costa 2017). This is a natural view for thosewho think of space as a three-dimensional entity that endures through,and is separate from, time. (This picture is discussed in Skow 2015and Gilmore, Costa, & Calosi 2016.) To allow for the possibilityof motion, those who endorse such a view will want to be able to say,of a given object, that it is exactly located at region \(r_1\), notat region \(r_2\), at time \(t_1\), and that the same object isexactly located at \(r_2\), not at \(r_1\), at time \(t_2\). To allowfor the possibility that time is gunky and does not contain instants,one might take exact location to be expressed by ‘\(x\) isexactly located at region \(r\) within interval \(s\)’. Adifferent option is to reject(i) in favor of the view that exact location is variably polyadic, anidea floated by Jones (2018: note 29). The thought here is that oneand the same relation is expressed both by the two-place predicate‘(…) is located at (…)’ and by (e.g.,) thethree-place predicate ‘(…) is located at (…) attime (…)’. The relation is neither two-place simpliciternor three-place simpliciter but two-place as it occurs in somepropositions and three-place as it occurs in others.

Alternatively, one might agree that exact location is a two-placerelation but reject(ii) above in favor of the view that, say, the second argument place inexact location (the ‘location’ slot) is plural. One ideais that an extended object can be exactly located at many points,collectively, without being exactly located at any one of themindividually or at the set or fusion of them. This is suggested byHudson (2005: 17); motivations are developed in Gilmore (2014b: 25). Adifferent idea is to take the first argument place (the‘occupant’ slot) to be plural, and to speak in some casesof some things collectively being exactly located at a given region.For approaches like this, but applied to a primitive relation ofpervasive location, see Loss (2023) and the supplementary documentSystems of Location.

2.2.2 Purely Locational Principles

If we assume that exact location is the one fundamental locationalrelation, that it’s two-place, and that both of its argumentplaces are singular, what should we say about its behavior? Here weconfine our attention to purely locational principles, that is,principles that can be stated in a first-order language with identitywhose only non-logical predicate is ‘\(L\)’.

Casati and Varzi (1999: 121) propose two principles:

Functionality: Nothing has more than one exactlocation. \[\forall x\forall y\forall z[(L(x, y) \amp L(x, z)) \rightarrow y = z]\]
Conditional Reflexivity: Exact locations areexactly located at themselves. \[\forall x\forall y[L(x, y) \rightarrow L(y, y)]\]

Functionality bans multilocation, which we discuss inSection 6. It tells us that nothing is exactly located at more than one region,or indeed, at more than one entity.

Conditional Reflexivity is a principle about the location of regions.It boils down—roughly—to the claim that regions arelocated at themselves. There seems to be another option for thelocation of regions, namely that they donot have anylocations, insofar as theyare locations. Varzi (2007: 1016)calls this principle Conditional Emptiness:

Conditional Emptiness: If \(x\) is exactlylocated at \(y\), \(y\) does not have an exact location\[\forall x\forall y\forall z [L (x, y) \rightarrow L (y, z)]\]

Simons (2004b: 345) endorses Conditional Emptiness, whereas Parsons(2007: 224) and Varzi (2007: 1016) both claim that the choice betweenthe two is somewhat conventional. However, as we show below,Conditional Reflexivity and Conditional Emptiness might beincompatible with different locative principles.

According to Conditional Reflexivity, exact locations are exactlylocated at themselves. (See also Donnelly (2004: 158), who presents asystem in which Conditional Reflexivity is a theorem, though shereplaces the location predicate ‘\(L\)’ with a primitivefunction symbol ‘\(r\)’ for ‘the exact locationof’.) Suppose that Obama is exactly located at region \(r\).Together with Conditional Reflexivity, this entails that \(r\) isexactly located at itself. This conflicts with a purely locationalprinciple endorsed by Simons (2004b: 345):

Asymmetry of Location:If\(x\) is exactly located at \(y\) then \(y\) is not exactly located at\(x\). \[\forall x\forall y[(L(x, y) \rightarrow \neg L(y, x)]\]

Note, however, that cases in which a region is exactly located atitself do not conflict with

Antisymmetry of Location:Notwo entities are exactly located at eachother. \[\forall x\forall y[(L(x, y) \amp L(y, x)) \rightarrow x = y]\]

Antisymmetry of Location may salvage some of the motivation forAsymmetry of Location while still harmonizing with ConditionalReflexivity. Antisymmetry of Location is a logical consequence ofFunctionality and Conditional Reflexivity (as is the view that exactlocation is transitive).

If we further assume that Obama is not identical to his exact location\(r\), we get the result that there are two different entities exactlylocated at \(r\)—namely, \(r\) and Obama. In that case, we havea counterexample to another purely locational principle that some havefound attractive:

Injectivity of Location:Notwo entities share an exact location.\[\forall x\forall y\forall z[(L(x, z) \amp L(y, z)) \rightarrow x = y]\]

Opponents of co-location may take this as areductio ofConditional Reflexivity. Others may take it as a reason to rejectInjectivity of Location in favor of a weaker variant, e.g.:

Conditional Injectivity of Location:Ifneither \(x\) nor \(y\)is identical to \(z\), then if each of them is exactly located at\(z\), then \(x\) and \(y\) are identical to each other.\[\forall x\forall y\forall z[(\neg x = z \amp \neg y = z) \rightarrow ((L(x, z) \amp L(y, z)) \rightarrow x = y)]\]

Conditional Injectivity is equivalent to the claim that whenever twodifferent entities share a given exact location, one of them isidentical to that location. This may salvage some of the motivationfor the ban on co-location, while still harmonizing with ConditionalReflexivity.

In the presence of Conditional Reflexivity the ‘regionpredicate’ can be defined as:

Regionhood: \(R(x) =_{\df} L(x, x)\)

That is, regions are the entities located at themselves. In turn thishelps formulating restricted mereological principles such as“any plurality of regions has a fusion”.

3. Interaction with Parthood

Philosophers have put forward various axiom systems to capture theinteraction between parthood and location. One idea is that themereological properties of, and relations between, located entitiesperfectly match those of their locations. This has beendubbedMereological Harmony (Schaffer 2009a; Uzquiano 2011;Leonard 2016), andMirroring in Varzi (2007).

Mereological Harmony has been captured formally in different ways byVarzi (2007), Uzquiano (2011), and Leonard (2016). Saucedo (2011:227–228) offers the following principles:

(H1): \(x\) is mereologically simple iff \(x\)’s location ismereologically simple.
(H2): \(x\) is mereologically complex iff \(x\)’s location ismereologically complex.
(H3): \(x\) has exactly \(n\) parts iff \(x\)’s location hasexactly \(n\) parts.
(H4): \(x\) is gunky iff \(x\)’s location is gunky.
(H5): \(x\) is a part of \(y\) iff \(x\)’s location is a subregionof \(y\)’s location.
(H6): \(x\) is a proper part of \(y\) iff \(x\)’s location is aproper part of \(y\)’s location.
(H7): \(x\) and \(y\) overlap iff \(x\)’s location and\(y\)’s location overlap.
(H8): The \(x\)s compose \(y\) iff the locations of the \(x\)s compose\(y\)’s location.

Some philosophers take Mereological Harmony to be a necessary truth(Schaffer 2009a: 138).^[3] The remainder of this entry considers three separate threats to theview that Mereological Harmony is necessary: interpenetration (Section 4), extended simples and unextended complexes (Section 5), and multilocation (Section 6).

There are other threats to Mereological Harmony that we will notdiscuss, e.g., threats to(H7) and(H8) that arise from ‘moderate views about receptacles’,according to which only topologically open (alternatively: onlytopologically closed) regions can be exact locations (see Cartwright1975; Hudson 2005: 47–56; and especially Uzquiano 2006), orthreats to(H4) discussed in Uzquiano (2011).)

A case of interpenetration occurs when non-overlapping entities haveoverlapping exact locations—e.g., when a ghost passes through awall. In such a case, the right-to-left direction of(H7) fails. Similar cases involve violations of the right-to-leftdirections of(H5) and(H6). An extended simple is a simple entity with a complex exact location:it violates the left-to-right direction of(H1), the right-to-left direction of the (equivalent)H2, and the left-to-right direction of the instance of(H3) that results from letting \(n = 1\). An unextended complex violates(H1) and(H2) and, depending upon cases,(H5)—seeSection 5.5. A case of multilocation occurs when a given entity hasmore than one exact location. This violates Functionality, which isleft implicit in Saucedo’s statement of MereologicalHarmony.

The four questions that we consider—Is interpenetrationpossible? Are extended simples possible? Are unextended complexespossible? Is multilocation possible?—are logically independentof one another. Thus, there is room for 32 specific packages ofviews.

Even if interpenetration, extended simples, unextended complexes, andmultilocation are all possible, some substantive principles linkingparthood and location may still survive. For example, the possibilityof interpenetration and extended simples poses no threat to:

Expansivity: Necessarily, if \(x\) is a part of\(y\), and if \(x\) is exactly located at \(z\) and \(y\) is exactlylocated at \(w\), then \(z\) is a part of \(w\): “thepart’s location is a part of the whole’s location”.^[4] \[\Box \forall x\forall y\forall z\forall w[[P(x, y) \amp L(x, z) \amp L(y, w)] \rightarrow P(z, w)]\]

Delegation: Necessarily, if \(x\) is complex andis exactly located at \(y\), then for any part \(z\) of \(y\), someproper part \(w\) of \(x\) is exactly located at some region thatoverlaps \(z\).^[5]\[\begin{align}\Box\forall z\forall x\forall y &\left[ \left[ C(x) \amp L(x,y) \amp P(z,y)\right] \right. \\& \rightarrow \exists w\exists v \left.\left[ PP(w,x) \amp O(v,z) \amp L(w,v) \right]\right]\end{align}\]

Roughly, Expansivity says that an object must extend out at least asfar as its parts: it must go where its parts go; and Delegation saysthat if an object is complex, then it must not extend out farther thanits proper parts: it must not go anywhere that its proper parts do notgo. Expansivity rules out cases like the following (Figure 3), in which the object \(a\) is a part of the object \(o\), but\(a\)’sexact location, \(r_a\), isnot a part of \(o\)’s exact location, \(r\).

a 2 by 3 box diagram: link to extended description below

Figure 3: The object \(a\) is part ofthe object \(o\), but \(a\)’s exact location \(r_a\) is not apart of \(o\)’s exact location, \(r\).Ruled out byExpansitivity. [Anextended description of figure 3 is in the supplement.]

The idea behind Delegation, in slightly different terms, is that acomplex entity cannot be weakly located at a certain region unless oneof its proper parts—a ‘delegate’—is alsoweakly located there. Regarding the formal statement of Delegation,one might wonder why it is not formulated with ‘\(\PP(u,z)\)’ in place of ‘\(O(u, z)\)’ in the consequent.The reason for this is that Delegation is meant to be friendly toextended simples. Suppose that a complex, spherical object, \(c\), isexactly located a spherical region, \(r\). Suppose that \(c\) iscomposed of two hemispherical simples, \(h\) and \(h*\), and that\(r\) is composed of continuum-many simple points, each plurality ofwhich composes a region that is a part of \(r\). Then, contrary to theproposed revision, it will not be true that for every part \(y\) of\(r\), some proper part of \(c\) is exactly located at a region thathas \(y\) as a proper part. Consider, for example, the sphericalregion \(r*\) with the same center point as, but half the volume as,\(r\) itself. \(c\) does not have a proper part that is exactlylocated at \(r*\), nor does it have a proper part whose exact locationhas \(r*\) as a proper part. But, as Delegation requires, \(c\) doeshave a proper part \((h\), for example) that has an exact locationthat overlaps \(r*\).

Delegation rules out cases like the following (Figure 4), in which \(o^*\) is a complex object that is exactly located atregion \(r^*\), but \(r^*\) has a part \(r_a\) that does not overlapan exact location of any of \(o^*\)’s proper parts:

a 2 by 3 box diagram minus box a: link to extended description below

Figure 4: The region \(r_a\) is a partof object \(o^*\)’s exact location, and object \(o^*\) iscomplex, but no proper part of \(o\) has an exact location thatoverlaps \(r_a\).Ruled out by Delegation. [Anextended description of figure 4 is in the supplement.]

Neither interpenetration nor extended simples threaten Expansivity orDelegation. One threat to Delegation comes from Pickup (2016: 260),who considers the possibility of a complex entity that is exactlylocated somewhere despite the fact that none of its proper parts isexactly (or weakly) located anywhere. One route to such entities (notPickup’s) runs as follows:

some material objects (electrons, maybe) do not have any othermaterial objects as proper parts,
any such material object is a complex entity whose only properparts are universals,
all material objects have locations, but
no universals have locations.

Bundle theorists who are platonic realists about universals, and whotake the constituents of a given bundle to be parts of that bundle,will face pressure to accept (i)–(iv) and hence to rejectDelegation. A related idea is discussed in connection with the BuryingStrategy inSection 4.1 below.

Another possible threat toDelegation comes from recentliterature on the mereological emergence of spacetime in quantumgravity. According to one account, spacetime does not exist at thefundamental level but it ismereologically composed of (more)fundamental entities that are not themselves spatiotemporal. Glossingover some details, if one holds that emergent spacetime regions areexactly located at themselves, one will then have yet anothercounterexample toDelegation. In effect, this is similar inspirit to the one we discussed already. It provides an example of acomplex entity with an exact location whose proper parts are notlocated anywhere (see, e.g., Baron 2020 and Baron & Le Bihan2022a). Naturally, one could turn the argument on its head and claimthat Delegation provides reason to think that the fundamentalentities, whatever they are, are notparts of the region.

Finally, a particular view, i.e. (unrestricted) supersubstantivalism,entails mereological harmony—seeSection 7. Therefore, any argument in favor of the former is an argument infavor of the latter.

4. Interpenetration

In this section we consider some arguments for the followingprinciple:

No InterpenetrationNecessarily, if \(x\) and \(y\) have exactlocations that overlap, then \(x\) and \(y\) themselvesoverlap. \[\Box \forall x\forall y\forall z\forall w[(L(x, z) \amp L(y, w) \amp O(z, w)) \rightarrow O(x, y)]\]

According to No Interpenetration, it is metaphysically impossible forentities of any type to ‘pass through one another’ withoutsharing parts—in the manner of a ghost passing through a solidbrick wall. There is a related principle that deserves some comment.The related principle says that, necessarily, if \(x\)’s exactlocation is a part of \(y\)’s exact location, then \(x\) is apart of \(y\). In symbols:

(1): Necessarily, if \(x\) is exactly located at a part of\(y\)’s exact location, then \(x\) is part of \(y\).\[\Box \forall x\forall y\forall z\forall w[(L(x, z) \amp L(y, w) \amp P(z, w)) \rightarrow P(x, y)]\]

This principle may seem to say basically the same thing as NoInterpenetration but to say it more simply—using the primitivepredicate ‘P’ instead of the defined predicate‘\(O\)’. Why then focus on No Interpenetration instead of(1)?

The reason for this is that some of the opposition to(1) will stem from opposition to a purely mereological principle: StrongSupplementation. It says that if every part of \(x\) overlaps \(y\),then \(x\) is a part of \(y\). Those who deny this will be very likelyto deny (1), but they might still be attracted to No Interpenetration.Consider for example the case of the statue Goliath and Lumpl, theclay it is ‘made out of’. Goliath and Lumpl have the sameexact location yet one might want to deny that Goliath is part ofLumpl (Lowe 2003). In this case they will constitute a counterexampleto (1), but insofar as they share parts, they do not constitute acounterexample to No Interpenetration.

As we noted in the introduction, in general, our task here is to setaside the purely mereological controversies (see the entry onmereology and Cotnoir & Varzi 2021) and to focus instead on the issues thatare exclusively concerned with location and its interaction withparthood. Too much of the controversy over(1) arises from controversy over ‘pure mereology’. Bycontrast, if No Interpenetration is controversial, this is onlybecause of what it says about theconnections betweenparthood and location.

4.1 For Interpenetration #1: from Universals or Tropes

Immanent realists say that a universal is in some sense ‘whollypresent’ in each thing that instantiates it (Armstrong 1978: 79;Bigelow 1988; O’Leary-Hawthorne 1995; O’Leary-Hawthorne& Cover 1998; Paul 2002, 2006, 2012; Newman 2002; Hawley &Bird 2011; Lafrance 2015; Peacock 2016). If immanent realism is true,it is plausible that disjoint universals frequentlyinterpenetrate.

Let \(e\) be an electron and suppose that it instantiates twodifferent universals: a mass universal, \(u_m\), and a chargeuniversal, \(u_c\). Suppose that \(e\) is exactly located at region\(r\). Then it will be natural for the immanent realist to saythat

\(u_m\) is exactly located at \(r\), or at some region \(r_m\)that has \(r\) as a part, and
\(u_c\) is exactly located at \(r\) or at some region \(r_c\) thathas \(r\) as a part.

If these universals are also instantiated elsewhere, then it will bedebatable whether they areexactly located at \(r\). Perhaps\(u_m\) has only one exact location, which fuses the exact locationsof its instances (Effingham 2015b). Likewise, for \(u_c\). Either way,the immanent realist will say that \(u_m\) and \(u_c\) have exactlocations that overlap by having \(r\) as a common part. Butpresumably \(u_m\) and \(u_c\) do not overlap. If these universals arenon-structural, non-conjunctive, and perfectly natural, then they areplausiblysimple, in which case they overlap only if they areidentical, which they are not. A similar point can be made in terms oftropes—particular, spatiotemporally located ‘cases’ of properties orrelations. For trope theorists who take tropes to be located atspacetime regions, it will be natural to say that mass tropes andcharges tropes, for example, frequently interpenetrate.

Three responses to this argument are worth considering.

The first response says: so much the worse for immanent universals andtropes. This response uses a mereo-locational principle,No Interpenetration, as a premise in an argument against certain metaphysical views,namely those that posit immanent universals or tropes. Is there somereason why mereo-locational principles should not be used in this way?The principles of pure mereology are often so used. For example, Lewis(1999: 108–110) rejects states of affairs and structuraluniversals on the grounds that they would violate Uniqueness ofComposition, the principle that no entities \(xx\) have more than one fusion.^[6] Why not give the same status to certain mereo-locational principles?One might, for example, say that No Interpenetration is betterjustified than is the view that universals or tropes arespatiotemporally located.

The second response says that while immanent universals or tropes arespatiotemporal entities that are ‘in their instances’,they are not exactly located anywhere. Simplified somewhat, theresponse holds that

universals are suitably related to entities that have exactlocations, and in that sense they are ‘in theirinstances’, but
universals do no themselves have exact locations and hence do nothave overlapping exact locations.

Given (ii), the universals or tropes in question no longer count asexamples of interpenetration. Call this theBurying Strategy,since it ‘buries’ universals and/or tropes in locatedentities, rather than treating them as being located—examplesare found in Armstrong (1989: 99) and Lowe (2006: 25).

The third response to the argument from universals and tropes is tosay, ‘True, universals and/or tropes can interpenetrate, butmaterial objects can’t’. This grants the argument andrejectsNo Interpenetration in favor of the weaker, restricted principle below, where \(M\)stands for the ‘material object’ predicate:

(2): Necessarily, if material objects \(x\) and \(y\) have exactlocations that overlap, then \(x\) and \(y\) themselves overlap.\[ \Box \forall x\forall y\forall z\forall w[(M(x) \amp M(y) \amp L(x, z) \amp L(y, w) \amp O(z, w)) \rightarrow O(x, y)] \]

This response also handles potential counterexamples toNo Interpenetration arising from regions, sets, events, portions of stuff, holes,spirits, and other ‘immaterial entities’.

On the location of regions, see Casati & Varzi (1999: 123), whohold that regions are located at themselves, and Simons (2004b: 345),who holds that nothing is located at itself. On the location of sets,see Maddy (1990); Lewis (1991); Effingham (2010, 2012); and Cook(2012). On the location of events, see Casati & Varzi (1999);Price (2008); Giordani & Costa (2013); Costa & Giordani(2016); and Costa (2017). On the location of portions of stuff, seeMarkosian (1998, 2004, 2015). On the location of holes and shadows,see Lewis & Lewis (1970); Casati & Varzi (1994); Wake,Spencer, & Fowler (2007); Donnelly, Bittner, & Rosse (2006);and Sorensen (2008). On the location of spirits, see Thomas (2009) andInman (2017). Sanford (1970) discusses many of these topics, andHudson (2005: 4) mentions many of them briefly.

The next two pro-interpenetration arguments count equally againstNo Interpenetration and(2), but we will continue to focus on No Interpenetration forsimplicity.

4.2 For Interpenetration #2: from Conceivability

Some think that it is possible for two disjointmaterialobjects to have overlapping exact locations. Perhaps there are noactual cases of the relevant sort. Such cases may even be nomicallyimpossible—ruled out by the laws of nature (though see the nextsection). But one might still think that these cases aremetaphysically possible.

After all, what is it that keeps material objects frominterpenetrating in the actual world? Repulsive forces, presumably.But a standard view is that the laws governing such forces are notmetaphysically necessary.^[7] And on that assumption it is natural to conclude that there aremetaphysically possible worlds in which any repulsive forces thatexist can be overridden in such a way as to allow material objects tointerpenetrate. (For more on this, see Zimmerman 1996a and Sider2000.)

A similar line of thought is sometimes framed as a conceivabilityargument. One might take cases of interpenetration to be conceivableor intuitively possible, and one might take this to be some evidencefor their possibility. InNew Essays the Human Understanding(II.xxvii.1), Leibniz writes that

we find that two shadows or two rays of light interpenetrate, and wecould devise an imaginary world where bodies did the same. (1704[1996]).

Sanford (1967: 37) describes a similar scenario in more detail.

4.3 For Interpenetration #3: from Bosons

Does contemporary physics provide us with examples of disjointfundamental particles that have the same, or overlapping, exactlocations? Hawthorne and Uzquiano apparently claim that the answer is‘Yes’. They write that

particles having integral spin—otherwise known asbosons—in modern particle physics (…) are generallythought to be point-sized. Moreover (…) bosons are perfectlywell able to cohabit a single spacetime point. (2011: 3–4)

Schaffer (2009a) suggests that in the case at hand, we are not forcedto consider the conceived scenario as one in which there are twoco-located yet disjoint bosons. Rather,

[a] more sophisticated treatment of these cases involves field theory.Instead of there being two bosons co-located at regionr,there is a bosonic field with doubled intensity atr. (2009a:140).

Whereas Hawthorne and Uzquiano apparently take bosons to provideactual examples of interpenetration, McDaniel (2007a: 240)suggests that they at least reinforce the conceivability of suchcounterexamples and therefore their possibility should not bediscardeda priori.

If one’s goal, in constructing a theory of location, is toarticulate the necessary anda priori truths governinglocation and its interaction with parthood, then even McDaniel’smodest point still counts against includingNo Interpenetration in one’s theory. For if McDaniel is right, then that principleis not ana priori truth, though perhaps it is still anecessary truth. (See Simons 1994 & 2004a for further discussionof bosons and for related considerations in support ofinterpenetration. For further discussion of Hawthorne and Uzquiano,see Cotnoir 2016.)

4.4 For Interpenetration #4: from Recombination

Sider (2000: 585–6), McDaniel (2007a), and Saucedo (2011) haveall objected toNo Interpenetration on the grounds that it conflicts with plausible broadly Humean‘principles of recombination’. The following is areconstruction of the argument in McDaniel’s (2007a: 241).

Let \(o_1\) and \(o_2\) be two different objects, let \(r\) be aregion, and consider the following states of affairs:

\((s_1)\): \(o_1\)’s being simple and exactly located at \(r\)

\((s_2)\): \(o_2\)’s being simple and exactly located at \(r\)

Then we can reconstruct the argument as follows:

(P1): \(s_1\) is a contingent state of affairs.
(P2): \(s_2\) is a contingent state of affairs.
(P3): \(s_1\) is distinct from\(s_2\).
(P4): For any \(x\) and any \(y\), if \(x\) and \(y\) are eachcontingent states of affairs, and if they are distinct from eachother, then possibly, both \(x\) and \(y\) obtain.

Therefore

(C): Possibly, both\(s_1\) and\(s_2\) obtain.

If it’s possible for both\(s_1\) and\(s_2\) to obtain, then it’s possible for a given region to be theexact location of two different simples. And since no two simples canoverlap, this would mean that it’s possible for disjoint things(the simples) to have identical (hence overlapping) exactlocations.

Is the argument successful? As Sider and McDaniel are well aware, thenotion of distinctness in the formulation of Humean recombinationprinciples needs to be handled with care if P4 is to get off theground. As a way of illustration, it cannot be simplenumericaldistinctness. If it were, the state of affairs thatpand the state of affairs thatnotp would berecombinable to yield a genuine metaphysical possibility. For anotherexample, the state of affairs thatx is green and the stateof affairs thatx is scarlet could be recombinable to yieldyet another genuine metaphysical possibility.^[8] But it is no easy matter to give ‘distinct from’ ameaning that makesP3 andP4 simultaneously plausible. If it means ‘shares no parts orconstituents with’, then P4 avoids the counterexample givenabove, but P3 ceases to be plausible, since\(s_1\) and\(s_2\) do plausibly share a constituent, namelyr. If ‘\(s\)is distinct from \(s^*\)’ is defined as

possibly, \(s\) obtains and \(s^*\) does not,
possibly, \(s\) does not obtain and \(s\) does,
possibly, neither \(s\) nor \(s^*\) obtains, and
possibly, both \(s\) and \(s^*\) obtain’,

thenP4 is trivially true, butP3 begs the question—see also Lo and Lin (2023).

5. Extended Simples and Unextended Complexes

A simple is an entity that has no proper parts. Are there any simples?Within the realm of spatiotemporal entities, some natural candidatesare: spacetime points, fundamental particles such as electrons (orinstantaneous temporal parts of them), and perhaps certain universals,certain tropes, or certain sets. On the other hand, it would seem tobe an empirically open possibility that all spatiotemporal entitiesare gunky.

Say that an entity isextended just in case it is aspatiotemporal entity and does not have the shape and size of a point.In this sense of ‘extended’, a solid cube would count asextended, but, given natural assumptions, so would a fusion of twopoint-particles that are one foot apart. Although such a fusion isnaturally taken to have zero length, it would be a scattered objectand so would not have the shape of a point.

Are there any extended simples? Could there be? Those who answer‘No’ to both questions will be inclined to accept

No Extended Simples (NXS)Necessarily,if \(x\) is exactly located at \(y\)and \(y\) is complex, then \(x\) is complex. \[\Box \forall x\forall y[[L(x, y) \amp C(y)] \rightarrow C(x)]\]

Strictly speaking,NXS does not say that extended simples are impossible; rather, it saysthat simples with complex exact locations are impossible. It leavesopen the possibility that there are extended simple regions andextended simple entities that are exactly located at them. (For moreon extended simple regions and discrete space or spacetime, seeForrest 1995; Tognazzini 2006; Braddon-Mitchell & Miller 2006;McDaniel 2007b, 2007c; Dainton 2010: 294–301; Spencer 2010,2014; Hagar 2014; Jaeger 2014; Kleinschmidt 2016; Goodsell et al.2020; and Baron & Le Bihan 2022b.) And NXS rules out thepossibility that there is a point-sized material simple that isexactly located at a point-sized but mereologically complex region(e.g., a region that is the fusion of several point-sized tropes eachof which is at zero distance from each of the others).

For the most part, however, it will do no harm to treat the debateover extended simples as a debate over NXS. We can do so if we assumethat, necessarily, a region is extended if and only if it is complex.So, in what follows, we will operate under that assumption unless weexplicitly note otherwise.

Unextended complexes are objects that are mereologically complex andexactly located at regions that are simple and so, we assume,pointlike. Are there unextended complexes? Could there be? Those whoanswer ‘No’ to both questions will be inclined toaccept:

No Unextended Complexes (NUC)Necessarily,if \(x\) is exactly located at \(y\)and \(y\) is simple, then \(x\) is simple. \[\Box \forall x\forall y[[L(x, y) \amp \neg C(y)] \rightarrow \neg C(x)]\]

Strictly speaking,NUC says that complexes with simple exact locations are impossible, butfor the most part, it will do no harm to treat the debate overunextended complexes as a debate over NUC.

5.1 For Extended Simples #1: from Conceivability

An initial argument appeals to the claim that extended simples areconceivable and takes that to be some evidence in favor of theirpossibility. To conceive of an extended simple, think of anextended—say, cubical—object that has no proper parts. Theidea is not, or not merely, that the cubecannot be physicallysplit or cut up. Whether or not it can be split is a separatequestion.

Debates about extended simples typically focus on the question ofwhether extended simple material objects are possible. But entities inother ontological categories (tropes, universals, sets, regions) aresometimes thought to be located. So it is worth keeping in mind that,whatever one thinks about material objects, one might hold thatextended simples in other categories are possible. With that said, wewill focus on material objects for the remainder of this section.

5.2 For Extended Simples #2: from String Theory

As McDaniel (2007a: 235–6) notes, some physicists interpretstring theory as positing extended simples. McDaniel quotes a passagefrom Brian Greene:

What are strings made of? There are two possible answers to thisquestion. First, strings are truly fundamental—they are“atoms,”uncuttable constituents…. Fromthis perspective, even though strings have spatial extent, thequestion of their composition is without any content. (1999: 141)

Can strings be treated as being identical to the spacetime regions atwhich they are exactly located? Greene does not explicitly addressthis question. If the answer is ‘Yes’, however, and ifstrings are exactly located only at complex regions, then stringtheory would not be committed to extended simples after all. For anargument that string theory does not posit extended simples, see Baker(2016). For a discussion of different arguments for and againstextended simples in quantum gravity see Baron and LeBihan (2022b).

5.3 For Extended Simples #3: from Recombination

As with interpenetration, one might offer a recombination argument forthe possibility of extended simples (Sider 2007; McDaniel 2007b;Saucedo 2011). One could claim that being simple and being a simpleregion are accidental properties that can be recombined to yield astate of affairs in which a simple is exactly located at acomplex—and therefore, we take it, extended—region. Sincethis argument does not appear to raise any issues that are specific toextended simples, we will move on.

5.4 Against Extended Simples #1: from Qualitative Variation

One might argue that if extended simples were possible, then theycould vary qualitatively across space or spacetime.^[9] An ordinary hammer can vary qualitatively over space by having awhite handle and a non-white (say, gray) head. Likewise, one mightthink that if extended simples were possible then there could be anextended, hammer-shaped simple that varies in color across space inthe manner of an ordinary hammer with a white handle and a non-whitehead. It is tempting to say that, if there were such a simple, thenone part of it would be white and one part would be non-white. Butsince the simple has only one part, itself, this would entail that thesimple itself is both white and non-white. This being impossible, onemight conclude that extended simples quite generally areimpossible.

One might resist the argument by insisting that extended simples arepossible only if qualitatively homogeneous across spacetime (seeSpencer 2010, Jaeger 2014, and Spencer 2014 for discussion). But mostfriends of extended simples try to resist the argument in otherways.

In this connection, it is useful to see that the problem ofqualitative variation perfectly mirrors the infamous problem of change(a.k.a., temporary intrinsics), which deals with the case of apersisting entity exhibiting qualitative variation across time.Consequently, several solutions developed for the problem of changeapply,mutatis mutandis, to the case of extended simples. Forexample, a friend of extended simples might adopt regionalizedproperties or regionalized instantiation (the terminology is due toSchaffer 2010). In the first case, a seemingly monadic property suchas being white is really taken to be a relation to a region indisguise, such as beingwhite at. In the second case, oneregionalizes instantiation rather than the property by claiming, forexample, that the extended simple instantiates-here whiteness. Thesetwo strategies parallel the classic relativization strategies of,e.g., Mellor (1981) and adverbialist strategy of, e.g., Johnston(1987) and Haslanger (1989).

Yet another strategy is worth mentioning here, because it wasdeveloped originally to deal with qualitative variation in extendedsimples. This is Parsons’ (2000) solution involvingdistributional properties. Parsons proposes that if a simple is whitein one region and gray in another, then it has a fundamental,intrinsic, distributional property. Some distributional properties,such as being black all over, are uniform. Others, such as beingpolka-dotted, are non-uniform. When a simple has a non-uniformdistributional property, this fact is not grounded in it having properparts, configured in a certain way, that each have simpler, uniformproperties. Nor is it grounded in the simple’s standing indifferent relations (being white at and being gray at) to differentspacetime regions. Rather, it is an ungrounded fact about the simple.This apparently avoids the worries faced by previous approaches (onwhich see Haslanger 2003). As McDaniel (2009) notes, however,Parsons’s solution faces several difficulties. For example, itseems unable to provide an account of what is it for \(x\) to be \(F\)at \(r\). What is it, for example, for something to be gray at aregion \(r\)? It can’t simply be for it to have a givendistributional property \(D\), such as being gray all over. And thisfor at least two reasons. First, something could be gray at \(r\) invirtue of having other distributional properties, such as being halfgray and half white. Second, something could have the relevantdistributional property without being gray at \(r\), for examplebecause it is not located at \(r\). The problem is not solved if wefurther require the thing to be located at \(r\). Indeed, two circlesthat are co-located at \(r\) and have both the distributional propertyof being half gray and half white might be such that one is gray atthe top part of their exact location while the other is gray at thebottom part.

As we point out in the supplementary documentSystems of Location some theories of location rule out extended simples bydefinition.

5.5 For Unextended Complexes

What about unextended complexes? McDaniel (2007b), Pickup (2016), andCalosi (2023) all discuss their possibility (but see also Leonard2016, which labels them “crowded simples”).

A first argument, due to McDaniel, goes as follows:

point-like entities are possible;
co-located point-like entities are possible;
fusions of point-like co-located entities are possible.

Fusions of co-located point-like entities qualify as unextendedcomplexes. Pickup suggests that there is another way a complex entitymight be exactly located at a single point: the parts of the pointycomplex do not have exact locations, but the pointy complex has one,namely the relevant point. (We touched upon this when discussingpossible violations of Delegation.) For the purpose of this entry, itis interesting to note that the two cases discussed above violate verydifferent principles about the interaction between parthood andlocation. In the first case both Injectivity and ConditionalInjectivity of Location inSection 3 are violated. Therefore, any argument against interpenetration willcount against this particular kind of unextended complex.

In the second case, the following principle will be violated:

Expansivity*:Necessarily,if \(x\) is part of \(y\) and \(y\) isexactly located at \(w\), then there is a subregion \(z\) of \(w\)such that \(x\) is exactly located at \(w\). \[\Box \forall x\forall y \forall w [ P(x, y) \amp L(y, w) \rightarrow \exists z(P(z, w) \amp L(x, w))] \]

We should note that Expansivity* is similar (in spirit) to ExpansivityinSection 3, but is slightly stronger. Depending on whether one takes the parts ofthe pointy complex to have at least weak locations—Pickup beingsilent on that—one would also have a violation of

Exactness +:Necessarily,if a thing is weakly located somewhere, then it is exactly locatedsomewhere. \[\Box \forall x [\exists y \WKL (x, y) \rightarrow \exists y L(x, y)]\]

Pickup offers yet another argument in favor of the possibility ofunextended complexes. The argument has it that unless a reason isgiven for the difference between the case of extended simples and thecase of unextended complexes one should treat their possibilitiesequally. That is, if one finds extended simples possible, then oneshould find unextended complexes possible as well. A possible reply isthat, as we saw, extended simples and unextended complexes violatevery different principles of location. One could have differentattitudes towards those principles which would then warrant differentattitudes towards the metaphysical possibility of the (allegedly)problematic entities—see for example, Calosi (2023).

6. Multilocation

To say that an object is multilocated is to say that it has more thanone exact location: ‘\(x\) is multilocated’ means

\[\exists y_1\exists y_2 [L(x, y_1) \amp L(x, y_2) \amp y_1\ne y_2].\]

(For an attempt to motivate a slightly different definition ofmultilocation, designed to allow for cases of multilocation in absenceof exact location, see Calosi 2022a, Correia 2022.) We consider aseries of putativeexamples of multi-location inSection 6.3.

The debate over multilocation concerns

Functionality+Necessarily,nothing has more than one exactlocation. \[\Box \forall x\forall y_1\forall y_2 [[L(x, y_1) \amp L(x, y_2)] \rightarrow y_1 =y_2]\]

Opponents of multilocation accept Functionality+. Friends ofmultilocation typically want to affirm something stronger than thenegation of Functionality+. They typically accept the possibility ofan entity that is exactly located at each of two regions that do noteven overlap.

Earlier we glossed ‘\(x\) is exactly located at \(y\)’ as‘\(x\) has (or has-at\(-y)\) the same size and shape as \(y\),and stands (or stands-at\(-y)\) in all the same spatiotemporalrelations to things as does \(y\)’. Thus, spheres are exactlylocated only at spherical regions, cubes only at cubical regions, andso on. When an entity is said to be multilocated, then, it is said tostand in this relation toeach of several regions: informallyput, it has the same size, shape, and position as region \(r_1\); ithas the same size, shape, and position as region \(r_2\); and so on.No claim is made to the effect that the object is exactly located atthefusion of \(r_1, r_2,\ldots\), or at anyproperparts of any of these regions.

To clarify the idea of multilocation in an informal way, it may beuseful to considerFigure 5, inspired by Hudson (2005: 105) and Kleinschmidt (2011: 256).

Figure 5a diagram: link to extended description below

(a) A scattered, singly located object

Figure 5b diagram: link to extended description below

(b) A non-scattered, multilocated object

Figure 5: [Anextended description of figures 5a and 5b are in the supplement.]

The object \(o_1\) is scattered: its shape is that of the sum of twonon-overlapping circles. It is not multilocated. Rather, it has justone exact location: the scattered region \(r_3\). It is not exactlylocated at any proper part of that region, such as \(r_1\) or\(r_2\).

The object \(o_2\) is multilocated. It has two (and only two) exactlocations. It is exactly located at the circular region \(r_3\); andit is exactly located at the circular region \(r_4\), which does notoverlap \(r_3\). It is not exactly located at their fusion, and it isnot located at any of their proper parts. Since \(o_2\) is exactlylocated at \(r_3\), which is circular, \(o_2\) is circular, at leastat \(r_3\). For parallel reasons, \(o_2\) is circular at\(r_4\). By contrast, \(o_1\) is not circularsimpliciter,nor is it circular at any region.

Everything we have said so far is neutral with respect to whethereither of the material objects is simple. It may be that both objectsare simple, or that both are complex, or that \(o_1\) is simple and\(o_2\) is complex, orvice versa. This is worth emphasizing,since questions about the possibility of extended simples andquestions about the possibility of multilocation are sometimes runtogether.

It is natural to think that if these two objects were visible, theywould be visually indistinguishable. Indeed, it is tempting to thinkthat there would be no empirical difference between \(o_1\) and\(o_2\). For those with verificationist leanings, this may lead to thebelief that there is no difference at all between \(o_1\) and \(o_2\)and hence that there must be something defective about the initialset-up of the case.

6.1 For Multilocation #1: from Conceivability

As with interpenetration and extended simples, one might offer aconceivability argument for the possibility of multilocation. Onecould claim that multilocation is conceivable and take this to beevidence that multilocation is possible. Since this argument does notappear to raise any issues that are specific to multilocation, we willmove on.

6.2 For Multilocation #2: from Recombination

As with interpenetration and extended simples, one might offer arecombination argument for the possibility of multilocation. One couldclaim that exact location is fundamental and accidental and take thisto be evidence that multilocation is possible. Since this argument,too, appears not to raise any issues that are specific tomultilocation, we will move on.

6.3 For Multilocation #3: from Examples

Arguments in favor of multilocation may simply come from concreteexamples of multilocated entities. These include: immanent universals,enduring material objects, enduring tropes—Ehring (1997a,b,2011), four-dimensional perduring objects—Hudson (2001),backward time travelers—(MacBride 1998, Keller & Nelson2001; Gilmore 2003, 2006, 2007; Miller 2006; Carroll 2011;Kleinschmidt 2011; Effingham 2011), fission products—Dainton(2008: 364–408), transworld individuals—McDaniel (2004),works of music—Tillman (2011), and an omnipresentGod—(Hudson 2009; Inman 2017).^[10] We will focus on the first two examples here for they are arguablythe more widely discussed.

6.3.1 Immanent universals

As we have noted, immanent realists say that universals arespatiotemporal entities that are in some sense ‘whollypresent’ in the things that instantiate them. One natural way totranslate immanent realism into the terminology of exact location isvia the following principle:

(3): Necessarily, for any \(x\), any \(y\), and any \(z\), if \(x\) isexactly located at \(y\) and \(x\) instantiates \(z\), then \(z\) isexactly located at \(y\).

To see how this leads to multilocation, suppose that some monadicuniversal \(u\) is instantiated by an entity \(e_1\) that is exactlylocated at region \(r_1\) and by a different entity, \(e_2\), that isexactly located at region \(r_2\), disjoint from \(r_1\). Then, given(3), \(u\) itself is exactly located both at \(r_1\) and at \(r_2\)(Paul 2006; Lafrance 2015).

(3) is not inevitable, even for immanent realists. Some of them mightprefer to say that a monadic universal is exactly located only at thefusion of the exact locations of its instances (Bigelow 1988:18–27, can in places be read as embracing this, and Effingham2015b argues that this is what immanent realists should say). On thisview, a simple monadic universal might be scattered but would not bemultilocated. Others (Armstrong 1989: 99) prefer to say thatuniversals do not have exact locations at all, though they are partsor constituents of things that have exact locations or of spacetimeitself. This was dubbed the ‘Burying Strategy’ inSection 4.1.^[11]

6.3.2 Enduring material objects

The debate over persistence ofmaterial objects through timecenters around two rival views, endurantism and perdurantism.^[12] Endurantists often say that a persisting material object istemporally unextended and in some sense ‘wholly present’at each instant of its career. Perdurantists often say that apersisting material object is a temporally extended entity that has adifferent temporal part at each different instant of its career and isat most partially present at any one instant (Informally, aninstantaneous temporal part of Obama is an object that is a part ofObama, is made of the exactly same matter as Obama is whenever itexists, and has exactly the same spatial location as Obama doeswhenever it exists, but exists at only a single instant.)^[13]

Some philosophers have suggested that the traditional endurantismversus perdurantism dispute runs together a pair of independentdisputes about persistence: a mereological dispute concerning theexistence of temporal parts, and a locational dispute concerning exactlocations (Gilmore 2006, 2008; Hawthorne 2006; Sattig 2006; Donnelly2010, 2011b; Eddon 2010; Rychter 2011; Calosi & Fano 2015). Statedloosely, the mereological dispute is between the following views:

Mereological perdurance: there are persistingmaterial objects, and each such object has a different temporal partat each different instant at which the object exists.
Mereological endurance: there are persistingmaterial objects, but none of them has a different temporal part ateach different instant of its career. (Perhaps none of them have anyinstantaneous temporal parts—or any temporal parts aside fromthemselves—at all.)

To frame the locational dispute, it will be useful to have one furtherpiece of terminology. Say that \(y\) is apath of \(x\) ifand only if \(y\) is a fusion of the exact locations of \(x\) (Gilmore2006: 204). Informally, a path of an object is a region at which theobject’s complete career is exactly located.

We can then state the locational dispute as follows:

Locational perdurance: there are persistingmaterial objects, and each of them has exactly one exactlocation—its path.
Locational endurance: there are persistingmaterial objects, and each of them has many different exact locations,each such location being instantaneous or ‘spacelike’.Typically, each of these exact locations will count as aninstantaneous temporal part of the object’s path.

Philosophers on both sides of this dispute can agree about whichspacetime regions are the paths of which materialobjects—provided they agree that the relevant persisting objectsexist. They will disagree about which spacetime regions are the exactlocations of which objects. The locational perdurantist will say thatmaterial objects are exactly located only at their paths. Thelocational endurantist will say that a persisting material object isexactly located at many regions, each of them a slice of its path. Theinteraction between the two disputes about persistence is summarizedinFigure 6 (from Gilmore 2008: 1230).

Four complex diagrams in a 2 by 2 layout: link to extended description below

Figure 6: Persistence, the locationaland mereological disputes. [Anextended description of figure 6 is in the supplement.]

Locational endurance entails multilocation: it says that some materialobjects are exactly located at many different regions (for alocational characterization of endurantism that does not entailmultilocation see Garcia forthcoming). Mereological endurance, whichmerely rejects temporal parts, does not entail multilocation. Thus,one might reject temporal parts while retaining Functionality. This isthe position of Parsons (2000, 2007). It corresponds to the lowerleft-hand box inFigure 6.^[14]

6.4 Against Multilocation #1: from Definition

As we noted inSection 2.1, Parsons (2007) develops a theory of location on which weak locationis primitive and exact location is defined, via definition (DS2a.1).According to that definition, ‘\(x\) is exactly located at\(y\)’ means the same as ‘\(x\) is weakly located at alland only those entities that overlap \(y\)’. Those who endorsethis definition may deny the possibility of multilocation, on thebasis of the following argument:

(4): Necessarily, for any \(x\) and any \(y, x\) is exactly located at\(y\) if and only if for any \(y^*\), \(x\) is weakly located at\(y^*\) if and only \(y\) overlaps \(y^*\) (Definition of‘\(L\)’).
(5): So, necessarily, for any \(x\), any \(y_1\), and any \(y_2\), if\(x\) is exactly located at \(y_1\) and \(x\) is exactly located at\(y_2\), then \(y_1\) overlaps exactly the same things as \(y_2\)(from(4)).
(6): Necessarily, for any \(y_1\) and any \(y_2\), if something isexactly located at \(y_1\) and something is exactly located at \(y_2\)and \(y_1\) overlaps exactly the same things as \(y_2\), then \(y_1=y_2\).

Therefore

(7): Necessarily, for any \(x\), and \(y_1\) and any \(y_2\), if \(x\)is exactly located at \(y_1\) and \(x\) is exactly located at \(y_2\),then \(y_1 =y_2\) (from(5) and(6).)

To see that the inference from(4) to(5) is valid, suppose that object \(o\) is exactly located at regions\(r_a\) and \(r_b\). Since \(o\) is exactly located at \(r_a, o\) is(by (4)) weakly located at all and only the entities that overlap\(r_a\). Likewise, since \(o\) is exactly located at \(r_b, o\) isweakly located at all and only the entities that overlap \(r_b\). So\(r_a\) overlaps a given entity if and only if \(o\) is weakly locatedat that entity; and \(r_b\) overlaps a given entity if and only if\(o\) is weakly located at that entity. Hence \(r_a\) and \(r_b\)overlap exactly the same entities. The rest of the argument isself-explanatory.

The argument may persuade some. However, those who are initiallyinclined to take the possibility of multilocation seriously may seethis argument as a reason to doubt the first premise and theassociated definition (Gilmore 2006: 203; Effingham 2015b).

Interestingly, in the supplementary documentSystems of Location, we present three systems—namely systems 3, 4 and 5—thatallow for multilocation but rule out specific kinds of multilocation,in particularnestedmultilocation, in whichsomething is exactly located at a region \(r\) and at one or more of\(r\)’s proper subregions. Kleinschmidt (2011) argues thatcertain types of nested multilocation entail a violation of thepartial ordering axioms of parthood—seeSection 6.6.

6.5 Against Multilocation #2: from Qualitative Variation

Extended simples face a problem arising from qualitative variation.Multilocated entities face a similar problem, insofar as amultilocated entity might instantiate incompatible properties atdifferent locations. When such locations are temporally separated,such cases are in fact cases of change.

Some friends of multilocation might insist that multilocation ispossible, but only for entities, such as universals or tropes, that donot vary qualitatively between locations. However, friends ofmultilocation usually defend the claim that multilocation is possibleeven for entities that do vary between locations and try to resist theargument by adopting other strategies. Such strategies mirror thoseapplied to the case of the problem of change and that of qualitativevariation in extended simples, and they appear to have the samevirtues and vices here as in those contexts.

6.6 Against Multilocation #3: from the Mereological Structure of Occupants

There are a few arguments against multilocation that share a commonstructure. These arguments have it that multilocation is inconsistentwith particular mereological structures of occupants. If one holdsthat occupants have at least the relevant mereological structure, onehas an argument against multilocation. Following Varzi (2003) [2019])we stipulate:

Ground Mereology: The mereological theory thatonly comprises the partial ordering axioms for parthood.
Minimal Mereology: Ground Mereology plus WeakSupplementation.
Classical Extensional Mereology: GroundMereology, plus Strong Supplementation and UnrestrictedComposition.

Given these stipulations, the different arguments take a more specific shape:^[15]

Ground Mereology Argument: Multilocation is inconsistentwith Ground Mereology (Kleinschmidt 2011).
Minimal Mereology Argument: Multilocation is inconsistentwith Minimal Mereology (Effingham & Robson 2007).
Classical Mereology Argument: Multilocation isinconsistent with Classical Extensional Mereology (Calosi 2014).

The Classical Mereology Argument depends crucially on other admittedlycontroversial principles of location we did not mention. We willtherefore not discuss the argument (see Smid 2023a for a discussionand response).

6.6.1 Ground Mereology and Multilocation

Kleinschmidt (2011) argues that multilocation is inconsistent withGround Mereology for occupants.^[16] More precisely, what is inconsistent with Ground Mereology foroccupants is a particular kind of multilocation, nested multilocation.In Kleinschmidt’s own words:

Claim 1: It is possible that thereexists some objects, \(x\) and \(y\), and regions \(r_1, r_2\), and\(r_3\), such that \(x\) is located at \(r_1, y\) is located at \(r_2,x\) is located at \(r_3\), and \(x\) is (at \(r_1)\) a proper part of\(y\) (at \(r_2)\) which is a proper part of \(x\) at \((r_3)\)(Kleinschmidt 2011: 256)

Consider the following scenario.Clifford is a statue of adog that is made of smaller statues. One such smaller statue isKibble, a statue of a biscuit.Kibble itself is madeof smaller statues, in particular a small statue of a dog,Odie. Kleinschmidt maintains we should agree to thefollowing:

(8): Kibble is a proper part ofClifford; \(\PP(k, c)\)
(9): Odie is a proper part ofKibble; \(\PP(o, k)\)

But it turns out thatOdie is a time travelingClifford that shrank a little. Thus,

(10): Clifford is numerically identical withOdie; \(c = o\)

SettingClifford \(=\)Odie \(= x\) andKibble \(= y\) one gets an example of the locational patterninClaim 1. Indeed,Clifford \((=\)Odie) is multilocated attwo regions which are a proper part and a proper extension of thelocation ofKibble. It is easy to see that(8)–(10) violate the conjunction of Transitivity and Asymmetry of properparthood, which are theorems of Ground Mereology. Hence theconclusion: Ground Mereology is inconsistent with multilocation.

Let us consider some possible replies. A first one consists in notingthat Kleinschmidt’s case rests on the possibility of a veryparticular kind of multilocation, ‘nested multilocation’.One might simply deny the possibility of such particular kind. Indeed,this is exactly the case according to some systems of location wediscuss in the supplementary documentSystems of Location.

Another response has it that, once we are told thatClifford=Odie (i.e.,(10) above) we should simply deny thatOdie is a proper part ofKibble (i.e.,(9) above). Kleinschmidt anticipates something similar and replies:

When we started describing the case, we noted thatOdie was aproper part ofKibble, which was a proper part ofClifford. Finding out thatOdie is actually atime-traveler shouldn’t change the parthood relations we say hestands in at that time. (2011: 257)

This, one might contend, can be resisted. Finding out that somethingis a time-traveler ought to change our beliefs in, for example,numerical claims about what exists at a certain time. If you are infront of what seem to be three dogs at disjoint locations, and you aretold that ‘one of them’ is a time traveler, present infront of you at least twice over, then you ought to revisit yourbelief about there being three dogs. Indeed, banning perfectco-location—which ought to have caused you to revisit the beliefthat there are three dogs in the first place—the scenario isactually inconsistent with there being three dogs: either there aretwo dogs one of which is multilocated at two disjoint regions, or onedog which is multilocated at three disjoint regions. And, so theargument continues, what goes for numerical claims goes formereological claims. Note that, if one believes that the locationalpattern inClaim 1 is possible, one will then not have any reason to read off themereological structure of occupants from the mereological structure oftheir exact locations.

6.6.2 Minimal Mereology and Multilocation

Effingham and Robson (2007) argue that multilocation is inconsistentwith Minimal Mereology for occupants. To be more precise, it isinconsistent with the conjunction of the following metaphysicaltheses: endurantism, the possibility of time travel, and WeakSupplementation.

Effingham and Robson consider a case in which a certain enduringbrick, \(\textit{Brick}_1\), travels backward in time repeatedly, sothat it exists at a certain time, \(t_{100}\), ‘many timesover’. At that time there exist what appear to be one hundredbricks, \(\textit{Brick}_1 \ldots \textit{Brick}_{100}\), though infact each of them is identical to \(\textit{Brick}_1\) (on one oranother of its journeys to the time \(t_{100})\), and a bricklayerarranges ‘them’ into what appears to be a brick wall,Wall.

Given the scenario just described, Effingham and Robson maintain thatwe should agree on:

(11): \(\textit{Brick}_1\) is numerically identical with\(\textit{Brick}_{2 (3, …, 100)}\); \(b_1 = b_2 = \ldots = b_{100}\)
(12): \(\textit{Brick}_{1 (2, 3, …, 100)}\) is a proper part ofWall; \(\PP(b_1, w), \PP(b_2, w),\ldots, \PP(b_{100}, w)\)

It is easily seen that(11) and(12) violate Weak Supplementation in that there is no part ofWall which is disjoint from \(\textit{Brick}_{1(2, 3, …,100)}\).

Indeed, the scenario envisaged by Effingham and Robson violates almostevery decomposition principle discussed in mereology, includingprinciples that are strictly weaker than Weak Supplementation, such asCompany, Strong Company, and Quasi Supplementation, the last one underthe assumption thatBrick is atomic—see the entry onmereology. Be that as it may, the conclusion remains that, given thepossibility of endurantist time travel, multilocation is inconsistentwith Minimal Mereology.

One possible reaction to this argument is to simply take it as anargument against endurantism rather than againstmultilocation—as Effingham and Robson themselves do. See Daniels(2014) for a reply.

6.6.3 General Replies

So far, we have discussed some strategies to resist the argumentsindividually. Other things being equal, one should prefer amore systematic reply that applies to all such cases independently of(some of) their respective details. We will consider two such generalstrategies. First, Smid (2023b) argues that at leastsomerelevant premises inall the arguments derive theirplausibilitysolely from controversial principles linkingparthood and location such as:

Strong Partition:If\(x\) is exactly located at a subregion of the exact location of\(w\), it is part of \(w\) \[\forall x\forall y \forall w\forall z [ L(x, y) \land L(w, z) \land P(y, z) \rightarrow P(x, w)]\]
Strong Proper Partition:If\(x\) is exactly located at a proper subregion ofthe exact location of \(w\), it is a proper part of \(w\)^[17] \[\forall x\forall y \forall w\forall z [L(x, y) \land L(w, z) \land \PP(y, z) \rightarrow \PP(x, w)]\]

If he is right, then one can reject these principles and undermine thearguments against multilocation. Second, one could relativizemereological claims of parthood. This raises two relatedquestions:

If we relativize mereological claims what adicity shouldthe parthood relation have? Arguably, the leading contenders are thatparthood is a three-place relation, and that parthood is a four-placerelation.
What goes in the third and fourth argument slots if wetake parthood to be three or four-place respectively?

Suppose one answers(i) by claiming that parthood should be three-place. How should we answer(ii)? ‘Natural’ candidates include external time, personaltime, the exact location of the part, and the exact location of thewhole. Kleinschmidt (2011) argues that none would work. For the sakeof brevity, we will focus on the case in which one takes parthood tobe a four-place relation (thus answering (i) above) where the twoadditional slots are filled by the exact location of the part and theexact location of whole respectively, thus answering (ii). (This isthe “Location Principle” below.) This is suggestedindependently by both Gilmore (2009) and Kleinschmidt (2011). Gilmore(2009) provides a more detailed proposal so we will stick to that.Indeed Gilmore (2009) argues that friends of multilocation haveindependent reasons—reasons having nothing to do with timetravel—to treat the fundamental parthood relation as afour-place relation. Let \(P^4(x, y, z, w)\) stand for “\(x\) at\(y\) is part of \(z\) at \(w\)”. Then, according to Gilmore,four-place parthood obeys the following principles:

Location Principle:If\(x\) at \(y\) is a part of \(z\) at \(w\), then: \(x\) is exactlylocated at \(y\) and \(z\) is exactly located at \(w\).\[\forall x\forall y\forall z\forall w[P^4 (x, y, z, w)\rightarrow[L(x, y) \amp L(z,w)]]\]
Reflexivity_4P:If\(x\) is exactly located at \(y\), then \(x\) at\(y\) is a part of \(x\) at \(y.\) \[\forall x\forall y[L(x, y)\rightarrow P^4 (x, y, x, y)]\]
Transitivity_4P:If\(x_1\) at \(x_2\) is a part of \(y_1\) at\(y_2\) and \(y_1\) at \(y_2\) is a part of \(z_1\) at \(z_2\), then\(x_1\) at \(x_2\) is a part of \(z_1\) at \(z_2\).\[ \begin{align}&\forall x_1\forall x_2\forall y_1\forall y_2\forall z_1\forall z_2 \\ &\qquad[[P^4 (x_1, x_2, y_1, y_2) \amp P^4 (y_1, y_2, z_1, z_2)] \\&\qquad\qquad\rightarrow P^4 (x_1, x_2, z_1, z_2)] \end{align} \]
Weak Supplementation_4P:If\(x_1\) at \(x_2\) is a part of \(y_1\) at\(y_2\) and either \(x_1\) is not identical to \(y_1\) or \(x_2\) isnot identical to \(y_2\), then for some \(z_1\) and some \(z_2\):\(z_1\) at \(z_2\) is a part of \(y_1\) at \(y_2\) and \(z_1\) at\(z_2\) does not overlap \(x_1\) at \(x_2\), \[ \begin{multline}\forall x_1\forall x_2\forall y_1\forall y_2 [[P^4 (x_1, x_2, y_1, y_2) \amp [x_1\ne y_1 \vee x_2\ne y_2]] \\ \rightarrow \exists z_1\exists z_2 [P^4 (z_1, z_2, y_1, y_2) \amp \neg \exists w_1\exists w_2 [O^4 (z_1, z_2, x_1, x_2)]] \end{multline} \]

where four-place overlapping is defined via:

Overlapping_4P:‘\(x_1\)at \(x_2\) overlaps \(y_1\) at\(y_2\)’ means ‘some \(z_1\), at some \(z_2\), is a partboth of \(x_1\) at \(x_2\) and of \(y_1\) at \(y_2\)’\[\kern-3pt O^4 (x_1, x_2, y_1, y_2) =_{\df} \exists z_1\exists z_2 [P^4 (z_1, z_2, x_1, x_2) \amp P^4 (z_1, z_2, y_1,y_2)] \]

It is easy to see how this handles the Minimal Mereology argument. Ineffect, Effingham and Robson’s scenario simply respects WeakSupplementation_4P. Consider the following simplifiedrepresentation of the case:

Figure 7 [Anextended description of figure 7 is in the supplement.]

Here,Brick at \(r_1\) is a part ofWall at \(r_w\).Moreover,Brick at \(r_1\) is, in the relevant sense, a‘proper part’ ofWall at \(r_w\), since either\(\textit{Brick}_1 \ne \textit{Wall}\) or \(r_1 \ne r_w\)—infact, both disjuncts hold. So, we have a case in whichWeakSupplementation_4P applies: itsantecedent is satisfied. Accordingly, that principle tells us thatthere must be an \(\langle x, r\rangle\) pair such that \(x\) at \(r\)is a partWall at \(r_w\) but does not overlap\(\textit{Brick}_1\) at \(r_1\). One such pair is\(\langle\textit{Brick}_1, r_3\rangle\): \(\textit{Brick}_1\) at\(r_3\) is a part ofWall at \(r_w\), but\(\textit{Brick}_1\) at \(r_3\) does not overlap \(\textit{Brick}_1\)at \(r_1\). There is no \(\langle x, r\rangle\) pair such that \(x\)at \(r\) is a partboth of \(\textit{Brick}_1\) at \(r_1\)and of \(\textit{Brick}_1\) at \(r_3\). Hence the consequentis satisfied as well.

What about the Ground Mereology argument? Gilmore (2009) does notdiscuss this case. However, the four-place notion of parthood might behelpful here as well, even if things are a little lessstraightforward. Once proper parthood is defined (and a lot might hangon this definition), plausibly the four-place counterparts ofTransitivity and Asymmetry of Proper Parthood are given by:

Proper Parthood Transitivity_4P:If \(x_1\) at \(x_2\) is a proper part of y\(_1\)at y\(_2\) and y\(_1\) at y\(_2\) is a proper part of \(z_1\) at\(z_2\), then \(x_1\) at \(x_2\) is a proper part of \(z_1\) at\(z_2\). \[ \begin{multline}\forall x_1\forall x_2\forall y_1\forall y_2\forall z_1\forall z_2 [[\PP^4 (x_1, x_2, y_1, y_2) \\ \amp \PP^4 (y_1, y_2, z_1, z_2)] \rightarrow \PP^4 (x_1, x_2, z_1, z_2)] \end{multline} \]
Proper Parthood Asymmetry_4P:If \(x_1\) at \(x_2\) is a proper part of \(y_1\)at \(y_2\), then \(y_1\) at \(y_2\) is not a proper part of \(x_1\) at\(x_2\). \[(\forall x_1\forall x_2\forall y_1\forall y_2 [\PP^4 (x_1, x_2, y_1, y_2) \rightarrow \neg(\PP^4 (y_1, y_2, x_1, x_2)]\]

Now, go back to Kleinschmidt (2011) case, and toClaim 1 inSection 6.6.1. Clearly \(x_1 = z_1 =\)Clifford =Odie, \(x_2 =r_3, y_1 =\)Kibble, \(y_2 = r_2\), and, finally, \(z_2 =r_1\). Consider Asymmetry first. There we have that

Kibble at \(r_2\) is a proper part ofCliffordat \(r_3\), and
Odie at \(r_1\) is a proper part ofKibble at\(r_2\).

But, plausibly, we have that neither

Clifford at \(r_3\) is a proper part ofKibbleat \(r_2\), nor that
Kibble at \(r_2\) is a proper part ofOdie at\(r_1\).

At first sight the notion of four-place parthood can handle theviolation of Asymmetry in the Kleinschmidt’s case.

What about transitivity? In that case we have that

Odie at \(r_1\) is a proper part ofKibble at\(r_2\), and
Kibble at \(r_2\) is a proper part ofCliffordat \(r_3\).

Transitivity_4P yields that

Odie at \(r_1\) is a proper part ofClifford at\(r_3\).

Note that this does not violate the 4-place counterpart ofIrreflexivity of Proper Parthood, which is, arguably:

Proper Parthood Irreflexivity_4P:If\(x\) is exactly located at \(y\), then \(x\) at\(y\) is not a proper part of \(x\) at \(y\). \[\forall x\forall y[L(x, y)\rightarrow \neg \PP^4 (x, y, x, y)]\]

Thus, one may argue that at first sight the notion of four-placeparthood can handle the violation of Transitivity as well. It shouldbe noted however that the success or failure of the arguments abovecrucially depend on the interaction of four-place parthood withidentity. For example, the Asymmetry argument depends upon whether onecan plausibly deny thatClifford at \(r_3\) is identical toOdie at \(r_1\). And the Transitivity argument depends uponwhether one can plausibly deny the following: if \(x\) at \(r_1\) is aproper part of \(x\) at \(r_2\) (with \(r_{1} \neq r_{2}\) ), then \(x\neq x\).

7. Supersubstantivalism and Harmony

As we noted inSection 3, a particular metaphysical thesis,supersubstantivalism,roughly the view that material objects areidentical to theirexact locations, entails full blown mereological harmony.

It is both interesting and important to distinguish two versions ofSupersubstantivalism.RestrictedSupersubstantivalism only subscribes toSup-Sub 1 below,whereasUnrestricted Supersubstantivalism maintains bothSup-Sub 1 andSup-Sub 2—the terminology is dueto Schaffer (2009).

Sup-Sub 1: Necessarily, for every material object \(x,x\) is exactly located at \(r\) iff \(x = r\).
Sup-Sub 2: Necessarily, for every region \(r\), there isa material object \(o\) such that \(o\) is exactly located at \(r\)iff \(o = r\).

The first version is calledRestricted Supersubstantivalismbecause it is compatible with there being arestriction onwhich regions can be identified with material objects. For instance,one can maintain that empty regions should not be identified withmaterial objects, or regions with a given dimensionality should not beidentified with material objects (e.g., regions that arefour-dimensional cannot be the exact locations of objects, say becauseone endorses some variant of endurantism—see, e.g., Nolan2014).

(Unrestricted)Supersubstantivalism entails:
Perfect Harmony: For any mereological predicate \(P, x\) is\(P\) iff\(x\)’s exact location is\(P\).

One obtains H1–H8 inSection 3, by substituting the relevant predicates for \(P\) inPerfectHarmony. Let us see the arguments for the four cases wediscussed.

Interpenetration.SupersubstantivalismentailsNo Interpenetration. Assume the antecedent, i.e., suppose \(L(x, z), L(y, w)\), and \(O(z,w)\). BySup-Sub 1, \(x = z\), and \(y = w\). Therefore\(O(x, y)\), which is the consequent.
Extended Simples.Supersubstantivalismentails No Extended Simples. Assume the antecedent, i.e., suppose\(L(x, y)\), and \(y\) is complex, \(C(y)\). BySup-Sub 1,\(x = y\), and therefore \(C(x)\), which is the consequent. Theargument forNo Unextended Complexes is exactlyparallel.
Multilocation.Supersubstantivalismentails there cannot be “object multilocation”. Forreductio, suppose an object \(x\) is multilocated, that is, exactlylocated at least at twodistinct regions \(y\) and \(w\).Then bySup-Sub 1, \(x = y\) and \(x = w\). By symmetry andtransitivity of identity, \(y = w\). Contradiction.

8. Further Issues

We conclude by listing some important issues about which we have sofar said little. These include—but are not limited to:

the interaction of parthood and location with other notions suchas
- topological connection (Cartwright 1975; Hudson 2005;Bays 2003; Uzquiano 2006; S. Smith 2007; Wilson 2008; Zimmerman 1996a,1996b; Casati & Varzi 1999; Donnelly 2004; Hudson 2005; Varzi2007),
- dependence and grounding (Brzozowski 2008; Schaffer2009b; Markosian 2014), and
- vagueness and indeterminacy (McKinnon 2003; Hawley 2004;N. Smith 2005; Donnelly 2009; Barnes & Williams 2011; Carmichael2011; Eagle 2016a);
questions about
- locational pluralism (Fine 2006; Leonard 2014;Kleinschmidt 2016) and
- topic neutrality of location (Simons 2004a,b; Cowling2014b; Gilmore 2014a);
applications to particular domains such as
- social (Effingham 2010; Hindriks 2013) and
- personal ontology (Lowe 1996, 2000, 2001; Olson1998);
the impact of
- relativistic (Balashov 1999, 2000, 2008, 2010,2014a,b;Gibson & Pooley 2006; Gilmore 2006, 2008; Sattig 2006, 2015;Calosi & Fano 2015; Davidson 2014; Calosi 2015) and
- quantum physics (Pashby 2013, 2016; Calosi 2022a).

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Acknowledgments

We want to thank the editors of theStanford Encyclopedia ofPhilosophy and especially the subject editor, Daniel Nolan, forcomments that improved the entry substantially. We also thank FabriceCorreia, Antony Eagle, Matt Leonard, Achille Varzi, and the eidosgroup in Geneva for useful feedback. Claudio Calosi acknowledgessupport from the Swiss National Science Foundation, SNSF EccellenzaProject "The Metaphysics of Quantum Objects PCEFP1_181088. DamianoCosta acknowledges support from the Swiss National Science Foundation,SNSF Starting Grant Project "Temporal existence", Project NumberTMSGI1_211294.

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