The paper provides a new and detailed critique of Barker and Dowe’s argument against multi-location. This critique is not only novel but also less committal than previous ones in the literature in that it does not require hefty metaphysical assumptions. The paper also provides an analysis of some metaphysical relations between mereological and locational principles.
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An object is exactly located at a region iff it shares all relevant geometrical properties and relations of that region and stands in all spatiotemporal relations with other entities as does the region itself. This is how exact location is understood by many, including (Gilmore 2006, 2013; Donnelly 2010; Balashov 2010; Casati and Varzi 1999; Hawthorne 2008; Hudson 2001).
Note that RES cashes out the way we understood exact location in the first place. See footnote 1.
It may seem that this way of phrasing WLP entails what is known in the literature as the Principle of Arbitrary Partition or the Doctrine of Arbitrary Undetached Parts. If so it would conflict with the existence—or even the possibility – of so called extended simples. However things are not that easy. For nothing in the principle prevents an extended simple to be exactly located at an extended region and at a proper subregion of that region. In other words the principle does not rule out the possibility of an extended simple which is exactly located at (some) of the proper subregions of (one of) its exact locations. The principle, in this rough formulation, will however entail Arbitrary Partition if what we will label Region Dissection is in place. Nothing in the arguments we will put forward (except maybe a minor detail to be discussed later on) depends on the existence or possibility of extended simples, so we will leave it at that. Furthermore, as we pointed out, this is just a provisional formulation of the principle in question. Thanks to an anonymous referee for having pushed this point.
We focus on regions of space for ease of reading, but nothing will bear essentially on this; the same argument works for multilocation in any dimension.
Sattig (Sattig 2006) also contains a criticism of the argument. His remarks resembles our discussion of the so called Additivity principle. We will return to this point later on.
Daniels (Daniels 2013) defends both the arguments of McDaniel and of Beebe and Rush from Barker and Dowe’s responses. In the first case he argues that there is nothing paradoxical in maintaining that objects have extrinsically different shapes with respect to different regions. In the second case he argues that a multilocated entity need not have a multilocated life, which in turn, undermines Barker and Dowe’s objection against Beebe and Rush.
The reader might worry that Barker and Dowe do not take location to be our notion of exact location. In the original paper Barker and Dowe speak sometimes in terms of location or whole location. However we contend, it should be understood in terms of our notion of exact location. Here is why we believe it to be the case. Barker and Dowe’s argument against multilocation is that it entails a contradiction, namely that one and the same entity is both 3D and 4D. If location in WLP is not intended as exact location, but rather say, as Parsons’s weak location – an entity is weakly located at every region which is not completely free of it- it follows that anything can be both a 3 or 4D entity without any contradiction. Any 4D entity counts in fact as being weakly located at a 4D region (its exact location) and many 3D regions (all 3D regions that are proper subregions of its exact location). Note however that problems with this formulation do not vanish altogether as we will point out later on in the paper and in footnote 10.
Note that this “dimensionality clause” is in line with RES.
See footnote 3.
Actually it is worse than that. For the same argument applies to, say, your right arm. This formulation of WLP would then imply that you are exactly located at the exact locations of both your left foot and right arm. By RES you would have the same geometrical properties as those regions. But they are plainly contradictory. We think there is a fair amendment to the formulation of WLP that would indeed serve Barker and Dowe’s purposes and that avoids these problems, so we will not pursue this line of argument.
In what follows < stands for the primitive notion of parthood, << stands for proper parthood (defined as usual as a part of something which is distinct from it), xOy stands for x overlaps y (defined as usual as x shares a part with y), and finally x@R abbreviates x is exactly located at region R. Note that the first clause of WLP* seems necessary in order to avoid the problem we pointed out in the original formulation of Barker and Dowe, namely that it actually entails massive multilocation. It hides however a serious problem. Probably friends of multilocation want multilocated objects to be able to gain parts, or more in general to be composed of different proper parts at different regions. Now, suppose a multilocated object x is exactly located at R1 and R2 and gains a part at R2 which was not part of it at R1. In this case this first clause will turn out to be false (there will be a part of x which is exactly located at no region that is part of R1) and so will the antecedent of WLP*. We will therefore not be able to run a modus ponens argument with WLP* and its antecedent as premises to the conclusion that x is exactly located at R1. This indicates how difficult is to actually give a precise formulation of the principle in question which does not beg the question against friends of multilocation. However Barker and Dowe could still argue that in the case in which the multilocated entity does not undergo mereological change WLP* still delivers their paradox. In the following we will argue that WLP* should be abandoned anyway so we will not pursue this line of argument further. Thanks to an anonymous referee for having pointed out shortcomings of previous formal renditions of the principle. This discussion was prompted by her remarks.
In our terminology r = R1(R2), O r = x(y) and R = R.
This is the term Barker and Dowe themselves use.
The ruler argument is intended to support WLP in the following sense. If it goes through it shows that whenever we have a region that is filled up -to use Barker and Dowe’s words- there is something that is exactly located there and so there is an object with the dimensionality of the region that is filled up, as WLP requires.
As we already pointed out Sattig (Sattig 2006: 49-50) offers similar reasons.
In other words z is the sum of those entities that satisfy the formula “y is exactly located at R1 and y is exactly located at R2”.
This actually hides another assumption of the argument, namely, that x is the only material object exactly located at R1(R2). Nothing in the locative principle we discussed so far guarantees this is indeed the case, for nothing prevents the possibility of exact co-location. We could rule this out by imposing an axiom to the point that co-location is impossible: ∀x∀y(x@R∧y@R → x = y). This renders the argument less general than we originally indicated. Thanks to an anonymous referee here.
It may be that relativizing geometrical properties to regions might offer a way out. This is a delicate question which goes beyond the scope of the paper.
It actually shows something even stronger, namely that a multilocated entity is (almost) never exactly located at the sum of any two or more of its exact locations. Thanks to an anonymous referee here.
This argument assumes that the mereological fusion of an object with itself, even in cases of multilocation, is just that very object. This assumption can be questioned, in that it might be argued that, in such cases, the multilocated entity fuses with itself to constitute a further object. This is an issue that deserves an independent scrutiny so we will not pursue this line of argument here.
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We want to thank two anonymous referees of this journal for their insightful comments and suggestions on a different draft of the paper. Damiano Costa is grateful for support from the Swiss National Science Foundation (doc.mobility funding scheme).
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Calosi, C., Costa, D. Multilocation, Fusions and Confusions. Philosophia 43, 25–33 (2015). https://doi.org/10.1007/s11406-014-9566-2
- Additivity of location
- Mereological fusion