paradoxes of multi-location

106 stephen barker & phil dowe

Paradoxes of multi-location

Stephen Barker & Phil Dowe

Call theses of multi-location philosophical doctrines that hold that one and the same entity can be wholly present and located at distinct space-time regions. Several substantial metaphysical theses are theses of multi-location. One is endurantism, the doctrine that in persisting, an object Ois wholly temporally located at each time of its persistence.1 The secondthesis of multi-location is the immanence conception of universals accord-ing to which they do not transcend space-time. On this view, universals arespatio-temporally located where they are instantiated (either by space-timeregions or by particulars located at space-time regions).2

In this paper we argue first, assuming 4-Dimensionalism, or eternalism,about time, that multi-location doctrines, given reasonable assumptionsabout mereology and location, entail contradictions: that one and the sameentity is both 3- and 4-dimensional – paradox 1 below. We then show that,given even leaner assumptions about location, we get another version of

Analysis 63.2, April 2003, pp. 106–114. © Stephen Barker and Phil Dowe

1 See Armstrong 1997, for the view that simples endure; van Inwagen 1990 and Mellor1998, for the view that temporal slices endure; and Erhing 1997 and Bordes 1998,for the view that tropes can endure. A third minor thesis of multi-location followsfrom endurantism and the possibility of time travel. If you travel back in time andmeet your earlier self in a room, you are spatially bi-located. As this is a consequenceof endurantism, we do not consider it separately here.

2 Armstrong (1997) defends this kind of view.

paradoxes of multi-location 107

the paradox – paradox 2. Finally, we show that a tensed version of paradox2 arises if we assume 3-Dimensionalism or presentism about time.3

1. The mereological paradox

Take a multi-located entity O, be it enduring entity or universal. Say thatO is multi-located throughout a 4D space-time region R. Thus there is adivision of R into sub-regions r, such that O is wholly located at each r. IfO is an enduring entity, the rs will be temporal slices of R. If O is a uni-versal, the rs will either be temporal slices or spatio-temporal slices of R,say points. Consider then the following paradox:

Paradox 1:At each r that is a sub-region of R, there is an entity – a universal, or endur-ing entity – of a certain kind. Call it Or. Take the fusion, or mereologicalsum, of all such Ors. Call the fusion F(Or):

(i) Each such Or is a 3D entity, since it is located at a 3D sub-regionr. Or is an entity with non-zero spatial extent and zero temporalextent. Each Or is identical to every other. So each Or is identicalwith F(Or). So, F(Or) is a 3D entity.

(ii) F(Or) has parts at every sub-region of R. So it has non-zero spatialand temporal extent. F(Or) is a 4D entity.

Conclusion: F(Or) is both 3D and 4D, but that is a contradiction sincebeing 3D means having no temporal extent, and being 4D meanshaving temporal extent.4

Van Inwagen (1990) in his discussion of endurantism derives somethinglike this conclusion in a slightly different argument, but does not recognizethe contradiction. He fallaciously infers that this just means that persistingentities lack determinate temporal extent. But the issue is not one of anenduring entity lacking determinate temporal extent. It is rather that thereis an apparently sound argument that F(Or) is 3D and one that it is 4D.There is no indeterminacy, just contradiction.

This problem is not to be confused with the problem of temporary intrin-sics, viz. how an object a can be bent at one time and not bent at another

3 Some endurantists accept eternalism, e.g. Mellor (1998), whereas some do not, e.g.Merricks (1995).

4 Some endurantists may reject the idea that the enduring entity O is a 3D object ofzero temporal extent, since there are atoms of time of non-zero extent n. O is of tem-poral extent n. In this case the paradox has to be rephrased as follows. The first stepof the argument concludes that F(Or) is of n-temporal extent and no greater, becauseO’s extent is that of an atom of time. The second step of the argument concludes thatF(Or) has a temporal extent greater than n. F(Or)’s temporal extent is and is notgreater than n.


if the bent a is literally identical to the not-bent a. For a start, paradox 1concerns spatio-temporal predicates, so the standard ‘time-indexing’ solu-tion looks problematic: if the concern were how a could be at t1 and alsoat t2, then one can hardly say by being ‘at t1 at t1’ and ‘at t2 at t2’.5 Sec-ondly, our problematic predicates are attributed respectively to a wholeand a part, so since these occur at overlapping times the time-relativizingsolution is not available.

2. Mereology

Paradox 1 depends upon mereological assumptions. One response for themulti-locationist is to deny that either enduring entities or universals canmereologically fuse. Take the case of enduring objects first. It might beobjected that fusions of enduring entities are simply not entities that we rec-ognize in common sense ontology. However, this misses the point; to makethe paradox work we do not have to assume that such fusions are mater-ial objects or things denoted by ordinary language singular terms.

Moreover, our argument doesn’t require commitment to a totally per-missive mereology. For example, we could accept the view that an unbri-dled mereology that entails that for every set of entities there is anotherentity having those entities as parts should be rejected, because entitiesmust bear certain relations to each other before they can be parts of genuinewholes. But what are these relations? We think space-time contiguity is sufficient. If one accepts that space-time is a genuine whole, then it seemsone must accept that a mereological fusion of sub-regions of space-time,bearing relations of temporal or spatial contiguity to each other, is agenuine whole.6 If so, it cannot be denied that enduring entities and instan-tiations of universals throughout a space-time region correspond to 4Dentities, since parts, in these cases, bear relations of temporal and spatialcontiguity to each other.

But what about mereology and universals? Our reply here is that we arenot concerned with the applicability of mereology to universals in general,but to universals understood on the immanence conception, that is, aslocatable entities. (By immanence we don’t mean that universals are locatedjust in the sense that they are instantiated by located entities, or that theirinstantiations are located, but that they themselves are spatio-temporally

5 See also Ehring 2002. Ehring points out that a universal multi-locationist is commit-ted to the same universal being, say, 1 metre and 2 metres away from the same entity.This looks like a contradiction. This supports our case, although we think it’s not adeep paradox, since the multi-locationist could deny that possessing the first rela-tional property entails not possessing the second.

6 We are currently assuming 4-Dimensionalism about time, but dispense with thatassumption later to deal with endurantists who are presentists.


located.) It is our intuition that mereology applies to located entities, thingsin space and time. Thus it applies to immanent universals. Secondly, on theimmanence view, universals are parts of states of affairs, parts of concreteentities. So universals can meaningfully be thought of as potential parts.Hence, that they are locatable and are potential parts of concrete entitiesare good reasons to treat them as mereologically fusible.7

3. Part location

Paradox 1 introduces the idea of a fusion of located entities, Ors at r, andconcludes – in step (ii) – that the fusion is a 4D entity at R. Why not denythis conclusion by denying the principle that underpins it, namely, thepart/whole location principle, WLP below:

WLP: If an entity W and a space-time region R are such that for somedivision of R into sub-regions r, W has a part p located at each sub-region r, then W is located at R and is a 3 or 4D entity according tothe dimension of R itself.

The response of the multi-location theorist is that WLP is only correctassuming the located parts are non-identical with each other. In the presentcase they are not. If an object O has parts wholly located at distinct regionsr of some region R, and these parts are identical with each other, they donot fuse to form an entity F(Or) located at R. Rather, we simply find themulti-location of O throughout R.

We think this response is untenable. By our lights, there is an entity Orlocated at each r in R, and so consequently, R is, as a region, filled up. Onemight cash out that metaphor in this way. The space-time region in whichO is multi-located is the region R. Take the space-time region adjacent to R, R*, which shares its times but not its spaces. Assume that there areentities E located at each 3D slice of R*. Suppose now that none of theseentities is identical with the other. Let F(E) be the fusion of all these non-identical entities. F(E) is a 4D object. Take any part of F(E), and there isan Or that is located at the same time. There is a one-to-one correspon-dence between parts of F(E) and parts of F(Or). We might think of F(E) asa temporal ruler for F(Or). F(Or) has the same temporal extent as F(E). AsF(E) is a 4D object, so is F(Or).

Take a spatial analogy. Say a person time travels 10 times, and 10 timetravelling individuals appear in the same room. Say they squash togetheron a bench. Then the bench is a ruler. Its parts are in one-to-one corre-spondence, in the relevant sense, to human parts. The fusion of the humanparts is an object. That 3D object is one-bench length. But we are told thatthe ten humans are identical. Their fusion then is simply a human. But a

7 Lewis (1990) is an example of someone who applies mereology to abstract entities.


human is not one bench in width. Ergo, there can be no spatially multi-located individual. It seems to us nonsensical to attempt to rebut this argu-ment by claiming that the fusion of individuals is not a bench-long thing.We think the response to the 4D case is in exactly the same boat. There isno relevant difference between the spatial case and the 4D case. No crassspatialization of time is afoot here. We are simply assuming that time likespace is ordered with respect to relations such as next-to or before. DenyingWLP is no way out of the paradoxes.

4. Paradoxical lives

Even if our defence of the mereological and part-location assumptionsgiven in the previous two sections fails, there is a related paradox of multi-location that does not require commitment to either of these assumptions.Take again a multi-located entity O, be it enduring entity or universal. Saythat O is multi-located throughout a 4D space-time region R. Intimatelyconnected with O and R, there is, we submit, a 4D entity which we call thelife of O, or L(O). If O is a persisting thing, a chair, L(O) is the persistenceof the chair through the temporal region of R. If O is a universal, say theuniversal Blue, L(O) is the qualitative persistence of blue throughout R. Lives are part of common sense ontology; we speak of entities – be theypeople, animate entities or inanimate – having long, interesting, varied,good, etc. lives. Lives are 4D things; they have beginnings, middles, andends. L(O) is just like an event occurring at a region R; it is located at Rwith proper parts located at each sub-region r in R.8

Assuming doctrines of multi-location, lives are paradoxical entities.Where there is a persisting thing, material object or universal, there is a life,L(O). There is a necessary connection between a persisting thing and itslife. We should accept the principle, call it Independence,9 that there are nologically or metaphysically necessary connections between distinct exis-tences. Thus if there is a necessary connection between O’s persisting andL(O)’s existing, O and L(O) can’t be distinct things. That means, we think,that O is part of L(O); L(O) is somehow constituted out of O and pos-sibly other entities. But if O is a part of L(O) then there is the followingcontradictory reasoning. Take an object O that endures by multi-locationthrough the temporal region of R. There is a 4D object L(O) with proper

8 In making these claims about L(O), we are not disputing the strong distinction thatsome endurantists want to make between objects, e.g. people and material things, andevents. Mellor (1998: 85) insists on this. We are simply arguing that where there ispersistence of an object, there is an event-like entity, which is 4D, located where theobject persists. We are not assuming that this entity is identical with the persistingthing.

9 This is Armstrong’s (1997) name for the principle, which comes from Hume.


parts located at each sub-region r of R. Split L(O) into two proper parts.Call them L(O)1 and L(O)2, which are located at distinct regions R1 andR2. In which case:

Paradox 2:

(i) L(O)1 and L(O)2 are entities bounded by distinct regions R1 andR2. Qua event-like entities they occur or are located at R1 and R2respectively, and nowhere else.

(ii) L(O)1 and L(O)2 have as parts the entity O. O is located at rs inR2, so, as O is part of L(O)1, L(O)1, is not confined to R1; it ispartly located in R2. O is located at rs in R1, so L(O)2 is not confined to R2; it is partly located in R1.

Conclusion: L(O)1 and L(O)2 are and are not bounded by R1 and R2.

Step (ii) in this reasoning invokes the principle that a thing is locatedwhere its parts are and in particular, a thing with a multi-located part isitself multi-located. Suppose an object [p, O] is comprised by p, a singlylocated thing, located at r, and O a multi-located thing, located at k1, k2,etc. Where is [p, O] located? [p, O] is located where its parts are. It is multi-located: located at (r + k1), (r + k2), etc., where these are mereological sumsof the space-time regions. So L(O)1 and L(O)2 are, by step (i), singlylocated, but by step (ii), multi-located.

One obvious response to Paradox 2 is to deny Independence. We think that is highly implausible, but won’t argue the point here because itbrings in general metaphysical issues. Another response is to deny thatthere are lives. We translate claims like, O had a long life, as, O persisteda long time. But we doubt that such translations can always be carried out.Moreover, lives are instances of a broader ontology of events, whose exis-tence cannot be denied since they are the relata of causation. The sameparadox can be put forward with respect to any event, say, O’s movingfrom space s1 to s2. This requires O’s persisting, from which it can beargued, by Independence, that O is a part of the event of movement. Thisevent has parts singly located at certain spatio-temporal regions. But givenO is a proper part of these event-parts, they must be multi-located.Paradox 2 emerges again.10

Another response is to deny step (i) in Paradox 2, and affirm that, con-trary to appearances, lives and events are not singly located entities. So ifan event of movement occurs at r, it also occurs elsewhere, at (r + k), forall k where O is located. But this looks highly implausible since the wholepoint of event talk depends upon events being singly located.

10 For Lewis (1986) events are intrinsic properties of space-time regions. Events do nothave whole objects as parts, for then, according to Lewis, who is a perdurantist, theywould not be local entities. Instead, events have temporal stages of objects as parts.


It might be thought by immanent universalists that the way out of theparadox is to admit that there are lives, and events, but construe them asfusions of states of affairs, the latter being the instantiations of universalsby bare particulars. If L(O) is a fusion of states of affairs, each distinct fromthe other and located at a sub-region r of R, it might seem there is noparadox with respect to the location of L(O) and its proper parts. But thereis a paradox. To explain the necessary connection between persisting uni-versals and lives we must still admit that universals are parts of lives andthus parts of the states of affairs that constitute them. We can then applyour paradox of location to these entities. States of affairs are meant to besingly located entities, but, given that they contain universals as parts, theymust be multi-located. Contradiction again.

Similarly, it might be proposed that the life corresponding to the persist-ing universal is a fusion of tropes – understood classically as singly locatedentities11 – located at each sub-region r of R. In conceiving of lives asfusions of tropes, there is no multi-location paradox, it seems. But theproblem here is to explain the necessary connection between the existenceof the persisting universal and the life, which is to say, the connectionbetween the persisting universal and the fusion of tropes. If we retain Inde-pendence, then there must be a part-overlap between the universal and eachtrope. If the universal is part of each trope, then we have the multi-locationparadox, since each trope ceases to be singly located – it is partly locatedelsewhere. If each trope is part of the universal, we fail to get multi-location of the same universal, since the universals at distinct sub-regionsare distinct entities.

5. Paradoxes and presentism

We think that there are no other ways out of the paradoxes, so concludethat theses of multi-location, be they theses of endurance or immanent uni-versals, are genuinely paradoxical. It might be objected that in the case ofendurantism we have simply shown that endurantists should not accept 4-Dimensionalism about time, or eternalism. We think we have shown morethan this; even a presentist about time – holding that only the present is real– should feel the bite of the paradoxes.12 It is done by putting the para-doxical reasoning into tensed language and noting that endurantists whoare presentists still need to speak of the persistence of things over time andalso countenance events. Superficially, events look like 4D entities; they

11 See Bennett 1988 for a theory of classical tropes.12 Merricks (1995) argues that endurantists can’t accept eternalism about time, but must

be presentists. We think he is wrong to the extent that we hold that endurantism isequally problematic whether one is an eternalist or a presentist.


have beginnings, middles and ends; they are not wholly present at any par-ticular time. That might look as if presentists can’t make any sense of eventtalk. Fortunately, that is not the case. When we say that an event is occur-ring now, what we mean, assuming presentism, is that it has a part E2, itsmiddle, existing in the present. It had a part, E1, in the past and will haveone, E3, in the future. So, in speaking of WWII, we speak of somethingwhich had parts, in 1939, in 1940, and so on.13 In short, events are enti-ties that are not wholly present at any time, and have parts whose relationto the whole event is characterized through tensed language, as indicated.We can now state a paradox of multi-location for the endurantist, assum-ing presentism. It is a tensed version of paradox 2. Say that O has been andis persisting, and will continue to do so. Then:

Paradox 3:There is an event-like entity, L(O), the life of O:

(i) Its beginning, L(O)B, occurred in the past, and is not now partlyin the present. It is wholly completed.

(ii) But L(O)B has the enduring entity O as a part. O is multi-located:O was located at past times, is located in the present. But then asO is a part of L(O)B, and O is now in the present, L(O)B is notwholly in the past.

Contradiction: L(O)’s beginning, L(O)B, is and is not wholly completed.

Again, for those who want to reject lives, there is a version of the paradoxthat uses any event in which O is a participant, and appeals to Indepen-dence to argue that O is a part of that event. To conclude: endurantistscannot invoke presentism to escape paradoxes of multi-location.

University of NottinghamNottingham NG7 2RD, UK

[email protected]

University of TasmaniaSandy Bay, Hobart 7001, Australia

[email protected]

13 The name ‘WWII’, just as with the name ‘Napoleon’, does not denote a concreteentity. Rather it must be taken to denote a property, possibly a complex conjunctiveone, such that we can talk of this property, or its conjunct parts, being instantiated atparticular times. To say that Napoleon existed is to say that a certain property wasinstantiated. To say that WWII occurred is likewise to say that a certain property wasinstantiated.

114 storrs mccall & e. j. lowe

References

Armstrong, D. 1997. A World of States of Affairs. Cambridge: Cambridge UniversityPress.

Bennett, J. 1988. Events and Their Names. Indianapolis: Hackett Publishing Company.Bordes, M. 1998. Abstract particulars in a four-dimensional ontological frame.

Dialectia 52: 3–12.Ehring, D. 1997. Causation and Persistence: A Theory of Causation. New York:

Oxford.Ehring, D. 2002. Spatial relations and universals. Australasian Journal of Philosophy

80: 17–23.Lewis, D. 1990. Parts of Classes. Oxford: Blackwell.Lewis, D. 1986. Events (plus postscripts). In his Philosophical Papers, Volume 2.

Oxford: Oxford University Press.Mellor, H. 1998. Real Time II. London: Routledge.Merricks, T. 1995. On the incompatibility of enduring and persisting entities. Mind 104:

523–31.van Inwagen, P. 1990. Four-dimensional objects. Noûs 24: 245–55.

paradoxes of multi-location

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