Abstract
According to the prevalent ‘sum view’ of stuffs, each portion of stuff is a mereological sum of its subportions. The purpose of this paper is to re-examine the sum view in the light of a modal temporal mereology which distinguishes between different varieties of summation relations. While admitting David Barnett’s recent counter-example to the sum view (Barnett, Philos Rev 113:89–100, 2004), we show that there is nonetheless an important sense in which all portions of stuff are sums of their subportions. We use our summation relations to develop, as an alternative to the sum view, an analysis of stuffs that distinguishes between the ways in which different sorts of stuffs are sums of their subportions.
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Notes
Here and throughout this paper, we use ‘portion of water’ as a synonym for ‘portion of H2O’ and not, e.g., to pick out the sort of liquid mass you expect to have delivered to your table when you ask your waitperson for a glass of water (where, presumably, you do not expect to receive a glass of ice). Note, however, that our use of ‘water’ differs from that of Barnett (2000) where it is argued that ‘water’ and ‘H2O’ do not function as rigid designators for the same kind of stuff.
Zimmerman (1995, 64) makes this point.
By a ‘collection’, we mean a non-empty set of individuals. By a ‘subportion’ of the portion of stuff x, we mean a portion of stuff of the same kind as x which is part of x. For example, any portion of water that is part of Walter (e.g. the water currently in the bottom half of our glass) is a subportion of Walter. Notice, however, that very many of the portions of stuff which are part of Walter are not subportions of Walter. For example, many portions of hydrogen are parts of Walter but are not subportions of Walter since they are not portions of water.
By ‘comprise all of x’, we mean that the subportions sum to x at the time in question in the sense of the summation relation SUM introduced in our formal theory in Sect. 3. Briefly, the subportions in Sub x comprise all of x at time t if and only if every member of Sub x is part of x at t and every part of x at t overlaps a member of Sub x at t. In Walter’s case, this requirement guarantees that every part of every molecule in Walter overlaps at least one of the subportions under consideration.
It goes without saying that the survival of just any collection of Walter’s subportions does not guarantee Walter’s survival. For example, if the water in the top half of the glass is annihilated then Walter is also destroyed even though the portions of water in the bottom half of the glass survive.
In fact, Burge, Zimmerman, and Barnett all hold that a single H2O molecule is a (minimal) portion of H2O. On this view, the fact that a portion of H2O cannot gain or lose molecules is just a special case of its inability to gain or lose subportions.
Note that without condition (ii), the Constant Basis Principle would be trivially satisfied by any individual. If o is any object and {o} is the one-member collection whose only member is o, then, necessarily, whenever o is present, the member of {o} comprises all of o.
Notice that if what is described below as the ‘sum view’ of stuffs were true, it would follow that for any portion of stuff x the collection, Sub x , of x’s subportions is a basis for x.
What we are calling ‘unstructured stuffs’, Barnett calls ‘discrete stuffs’ and what we call ‘structured stuffs’, Barnett calls ‘nondiscrete stuffs’. Barnett’s terminology stems from an additional distinction he makes between the two categories of stuff—minimal portions of stuff kinds belonging to the first category are always discrete (i.e. do not share parts) while minimal portions of stuff kinds belonging to the second category can share parts. However, Barnett does not explain how the discrete/nondiscrete distinction is connected to the sum/rigid embodiment distinction. In particular, he gives no reason for thinking that every stuff kind whose portions are characterized in terms of a certain embodiment relation must have overlapping minimal portions. (We do, however, agree with Barnett that all familiar examples of structured stuffs do seem to have overlapping least portions, if they have least portions at all.) Since the discrete/nondiscrete issue is not relevant to our primary focus in this paper, we prefer to ignore it.
A quite weak (but still sufficient for our purposes) first-order axiomatization of ε is presented in Bittner and Donnelly (2007). But there is also no reason why MTM as a whole could not be conjoined with a stronger set theory as long as collection variables are restricted to non-empty sets of objects.
Simons treats his counterpart of our presence predicate as a separate primitive in CT. But since he requires that (i) x is part of y at t only if both x and y are present at t (a version of our (A1)) and (ii) x is part itself whenever it is present, it turns out that in CT, x’s being present at t is equivalent to x’s being part of itself at t. Thus, Simons might have defined his presence predicate as we do here without any change in the theorems of CT.
We do not have room in this paper to address perdurantism (i.e. the theory that ordinary objects are temporally extended and persist by having different temporal parts at different times) separately, but it is not hard to see how MTM and the summation relations introduced below can be reformulated in perdurantist terms. To do this, we need only embed MTM in a modal mereology that includes, in addition to the perdurantists’ atemporal parthood predicate, a counterpart of our present at predicate (in this context treated as a separate primitive and not as a defined predicate). A time-indexed parthood predicate corresponding to our P could then be defined along the lines suggested in Sidor (2001, 52–62) and used to formulate counterparts of MTM’s axioms and definitions.
In fact, Barnett seems to take it as a consequence of his two category theory of stuffs that some portions of stuff will spatially coincide with other portions of stuff. The idea is that at each moment of its life Crude spatially coincides with a portion of unstructured stuff which also has CrdMol as a basis. Though we agree that Barnett’s two category account of stuffs would seem to bring possibilities for coincidence along with it, we do not here make the positive assumption that distinct objects (or, in particular, distinct portions of stuff) do in fact coincide.
SUM is intended to pick out the same relation as does Sidor’s (similarly defined) ‘fusion-at-a-time’ predicate (Sidor 2001, 58).
See Simons (1987, 177–187) for other interesting possibilities.
Unless ClSub T happens to be the one member collection consisting of just Clem. But this is an uninteresting case.
In fact, though, this is not as clear as one would like, since the modal and temporal aspects of the relation between a sum and its summands are often not specified.
A somewhat stronger, but still reasonable, sum predicate would require that x is an essential bound and constant sum of a collection whose members are pair-wise discrete (i.e., do not share parts). Here, we would use a sense of summation that is more in spirit of the composition relation of van Inwagen (1990). But we do not see that the added discreteness condition helps in this context and it raises modal issues (e.g. whether distinct molecules are necessarily discrete) which we do not wish to take up here.
We are grateful for the very helpful comments of David Barnett and an anonymous reviewer. A version of the mereology MTM used in this paper is presented in greater detail in Bittner and Donnelly (2007).
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Donnelly, M., Bittner, T. Summation relations and portions of stuff. Philos Stud 143, 167–185 (2009). https://doi.org/10.1007/s11098-007-9197-6
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DOI: https://doi.org/10.1007/s11098-007-9197-6