BLOCKING THE PATH FROM VAGUENESS
TO FOUR DIMENSIONALISM*
Kristie Miller
Abstract
There is a general form of an argument which I call the ‘argument from vagueness’
which attempts to show that objects persist by perduring, via the claim that vagueness is
never ontological in nature and thus that composition is unrestricted. I argue that even if
we grant that vagueness is always the result of semantic indeterminacy rather than
ontological vagueness, and thus also grant that composition is unrestricted, it does not
follow that objects persist by perduring. Unrestricted mereological composition lacks the
power to ensure that there exist instantaneous objects that wholly overlap persisting
objects at times, and thus lacks the power to ensure that there exists anything that could
be called a temporal part. Even if we grant that such instantaneous objects exist, however,
I argue that it does not follow that objects perdure.To show this I briefly outline a
coherent version of three dimensionalism that grants just such an assumption. Thus
considerations pertaining to the nature of vagueness need not lead us inevitably to accept
perdurantism.
1 Introduction
It is now well established that there are two main views about the way objects persist through
time: four dimensionalism and three dimensionalism. Four dimensionalism is the view that
persisting objects are the mereological fusion of temporal parts: objects persist by perduring.
Three dimensionalism is the view that persisting objects have only spatial extension, and are
wholly present at each moment at which they exist: they endure. In his recent book, Theodore
Sider contends that the best argument in favour of four dimensionalism is what he calls the
argument from vagueness. This argument has two parts: the first part seeks to establish that
composition is unrestricted, that is, that any arbitrary arrangement of particulars composes some
object. The second part seeks to show that if composition is unrestricted, then it follows that
persisting objects are composed of temporal parts, and thus that four dimensionalism is true.
In general, the argument from vagueness is resisted by objecting to the first part of the
argument, usually by denying that composition can never be vague. In this paper though, I want
to concentrate on the less often considered second part of the argument. I argue that even if we
accept the first part of the argument, we are not led inevitably to accept four dimensionalism. For
it is perfectly coherent to agree both that vagueness is never ontological, and that composition is
unrestricted, and nevertheless to embrace three dimensionalism. To show this I will briefly
outline a version of three dimensionalism and show how this view avoids the conclusion of the
second part of the argument from vagueness. Consideration of this view allows us more clearly
to see the key points at issue between the three and four dimensionalist, and to clarify just what it
is for an object to be composed of temporal parts. I conclude that even if one adopts the view
that composition is unrestricted, this alone provides no reason to prefer four dimensionalism over
three dimensionalism: considerations pertaining to vagueness are strictly orthogonal to the issue
of the manner in which objects persist.
2 The Argument from Vagueness
The first part of the argument from vagueness owes its origins to David Lewis. Lewis argues that
any attempt to restrict composition in a way that is in-keeping with our intuitions about which
objects exist, must necessarily be a vague restriction. For commonplace objects all have
imprecise temporal and spatial borders and thus there is no determinate point at which, for
instance, some atom A can be said to be part of some object x, or not part of x. But if
composition itself is vague, then existence will be vague. It will be vague at exactly which
moment an object comes into existence, and vague at exactly which moment it ceases to exist: so
for every object, there will be some time at which it is indeterminate whether that object exists or
not. Since Lewis thinks that vagueness is never ontological, but rather is the result of semantic
indeterminacy, he holds that existence cannot be a matter of degree, and thus he concludes that
composition cannot be restricted. Recently, Lewis’ argument for unrestricted composition has
been further refined by Sider Expressed as a reductio, Sider’s argument is as follows.
Part I: From Vagueness to Unrestricted Composition
1. Assumption: Existence is not vague: composition either definitely occurs or definitely does
not occur.
2. Assumption: Not every arrangement of matter composes an object.
3. So there must be a continuum of cases such that at one end of the continuum composition
occurs, and at the other end composition fails to occur.
4. Each of the cases on the continuum is highly similar to the adjacent cases.
5. So there is no principled way to draw the line between a case where composition occurs and a
case where composition does not occur.
6. So there are cases where it is indeterminate whether composition occurs or not.
7. So existence is vague.
As it stands, this argument requires the rejection of either (1) or (2). In general,
arguments of this form have been resisted by disputing the truth of (1). Hence Sider has
constructed a number of ancillary arguments that aim to bolster (1) and thus lead to the rejection
of (2). In this paper, however, I want to focus on the second part of the argument from
vagueness, namely that part that moves from the falsity of (2) to the truth of four
dimensionalism. Let us turn to this second part of the argument from vagueness. Here is a
reconstruction of Sider’s argument:
Part II From Unrestricted Composition to Temporal Parts
1. Assumption: Composition is unrestricted.
2. So there is a fusion of the members of any arbitrary set S at time t, where x is a fusion of the
members of S at t iff every member of S is part of x at t, and each part of x at t overlaps at t some
member of S.
3. Objects persist through time, so we need a temporalised version of unrestricted composition.
4. An object x is a diachronic fusion of the members of sets S1, S2, and S3 at times t1, t2 and t3 if
(i) x is composed of S1 at t1, S2 at t2, and S3 at t3 and (ii) x exists only at the times t1 t2 and t3.
5. Since composition is unrestricted, any sets of objects at times has a diachronic fusion.
Following Sider, call this thesis U: Any arbitrary sets Sis and times tis has a diachronic fusion.
6. Given U, it follows that there for any y at t, there is some instantaneous object x that (i) is the
fusion of the members of set y at t, and (ii) which exists only at t.
7. y is part of x at t and every part of x at t overlaps y at t (from (i) and the definition of fusion).
8. With certain mereological principles we can then move from the claim that every part of x at t
overlaps y at t, to the claim that x is part of y at t.
9. Now consider the definition of an instantaneous temporal part: x is an instantaneous temporal
part of y at instant t=df (a) x is part of y at t (b) x exists at, but only at t and (c) x overlaps at t
everything that is part of y at t.
10. The diachronic fusion x meets the definition of an instantaneous temporal part (from 7, 8 9
and 10).
11. Since all diachronic fusions exist (see 6) it follows that every persisting object is composed
of these instantaneous objects.
13. So four dimensionalism is true.
Sider’s argument then, is that every diachronic fusion exists, and that these fusions count
as temporal parts. If this argument is successful, then it is surprising, for we might have thought
that the issue of when composition occurs, and thus which objects exist, is orthogonal to the
question of how objects persist.
3 Mereology and Composition
Let us grant part I of the argument and consider only part II. The first thing to notice is (2), the
claim that given unrestricted composition, there will exist the fusion of the members of any
arbitrary set S at time t, where x is a fusion of the members of S at t iff every member of S is part
of x at t, and each part of x at t overlaps at t some member of S. Now unrestricted mereological
composition only tells us that given that we have some particulars, we can fuse those particulars.
It does not tell us that there exist any instantaneous basic particulars that can be fused to
compose an instantaneous object. So it does not tell us that any members of S are instantaneous.
But suppose S has three members, particulars P, P1 and P2 which persist through T. We cannot
assume these particulars perdure. But if they endure, their fusion just if an enduring object that
persists through T: there exists no instantaneous object that is the fusion of P, P1 and P2 at t in T,
for there exists no object P-at-t.
Now Sider attempts to bypass this problem by talking (in 2) of the fusion of the members
of S at t.The problem is that (2) is ambiguous between the claim that x is a fusion-at-t of the
members of S, and that x is a fusion of the members of S-at-t. Construed as a fusion-at-t, (2)
involves some additional mereological axiom that allows for the fusion-at-a-time of particulars
which exist at other times, and it is difficult to see why the endurantist need accept such an
axiom. Construed as a fusion of the members of S-at-t, (2) looks like a tricky way of smuggling
in temporal parts by the back door. After all, what is S-at-t if not a temporal part?
For the sake of argument however, let us grant what the endurantist almost certainly will
not, either that basic particulars are instantaneous and thus that there exists any arbitrary fusion
of these particulars, or that there exist instantaneous fusions-at-times of particulars. Thus we
grant that there exists some arbitrary fusion of the members of S at t. Now let us consider
whether Sider’s argument establishes that such objects count as temporal parts. The crucial step
in Sider’s argument is premise 8, which relies on the mereological principle that licenses the
move from ‘every part of x at t overlaps y at t’ to ‘x is part of y at t.’
To clarify this step a little, we need to be a little clearer about the move from premise 7 to
8. Sider’s idea, I take it, is that we begin with some persisting object y. At some arbitrary time t,
y is composed of some things. Now consider the set y whose members are all of those things that
compose y at t. The argument is a little confusing because Sider uses ‘y’ to refer both the
persisting object, and to the set whose members compose y at t. I retain this terminology because
ultimately it allows us to see how the argument goes wrong. So given that composition is
unrestricted, we can fuse all of the members of that y at t, and call this fusion x. Then we
conclude that the instantaneous object x, is part of the persisting object y. We derive premise 7
from the definition of fusion, combined with the fact that x is the fusion of the members of y at t.
If x is the fusion of the members of y at t, then we know that every member of y at t is part of x
at t, and x at t overlaps at t some member of y. So 7 should more properly read:
7. Every member of y is part of x at t and every part of x at t overlaps some member of y at t.
Then in premise 8 we move from ‘every part of x at t overlaps y at t’ to ‘x is part of y at t.
This should instead read: ‘every part of x at t overlaps some member of y at t.’ And it is not
obvious that ‘x is part of y at t’ follows from this claim, even if we accept the relevant
mereological principle. For presumably the ‘y’ in ‘x is part of y at t’, refers to the persisting
object y at t: for only if x was part of this object, could it be said that we have shown four
dimensionalism to true. Moreover, since no fusion is ever part of a set, (since sets do not have
parts, they have members) it could not be that x is part of y at t, where ‘y at t’ is a set at a time. In
order to show that x is part of y at t, we need to note that every part of y at t overlaps some
member of the set y at t, and every member of set y at t is part of y at t. It then follows that x is
part of y at t.
So has Sider shown that given the truth of unrestricted composition, it follows that four
dimensionalism is true? If we grant (2), he has shown that there exists a plethora of instantaneous
objects that wholly overlap persisting objects at a time, and which are part of those objects at that
time. Isn’t that just to say that four dimensionalism is true? To answer this question we need to
consider the persisting object y. What is y? If persisting objects are diachronic fusions, then y is
a diachronic fusion. Now recall that an object is a diachronic fusion of the members of sets at
times, if it is composed of those sets at those times, and exists only at that those times. What is it
to be composed of the members of a set at a time? According to Sider, some object x is
composed of the members of set S at t, iff every member of S is part of x at t, and each part of x
at t overlaps at t, some member of S. So x is composed of the members of S at t, just if x is the
fusion of the members of S at t. Call the fusion of the members of a set at a time a synchronic
fusion. A diachronic fusion then, is the fusion of two or more synchronic fusions.
A synchronic fusion is an instantaneous object that has as spatial parts, each of the
members of the set that it fuses. A diachronic fusion is a persisting object composed of the
fusions of members of sets at times. So at each time at which it exists, a diachronic fusion has the
spatial parts of the synchronic fusion that exists at that time. But in addition to these spatial parts,
a diachronic fusion has as parts, all of the synchronic fusions of which it is composed.
Diachronic fusions not only have spatial parts, they have synchronic fusions, or instantaneous
objects, as parts. So diachronic fusions are the mereological fusion of instantaneous objects: they
are four dimensional objects.
So if y is a diachronic fusion, then it follows that y is a four dimensional object, and that
x is an instantaneous temporal part of y. And if all objects are either synchronic or diachronic
fusions, then it follows that four dimensionalism is true. But no three dimensionalist will concede
that persisting objects are diachronic fusions, nor need she understand unrestricted composition
as the claim that every synchronic and diachronic fusion exists. For the general claim that any
arrangements of matter at any different times composes some persisting object, need not be
understood as the claim that every diachronic fusion exists. For there is nothing in three
dimensionalism per se that prohibits the three dimensionalist from holding that there exists any
enduring object composed of arbitrary combinations of things at times. Even if the three
dimensionalist accepts that there exist instantaneous objects (fusions-at-times), she need not
concede that persisting objects are the fusions of these objects. She could instead hold that for
every synchronic fusion, there is some enduring object x that is constituted by those fusions at
those times. Call such an object a diachronic object.
A diachronic object is an enduring object: it is wholly present whenever it exists. So too a
synchronic fusion is wholly present when it exists. At any time at which any diachronic object
exists, there will be some synchronic fusion that wholly overlaps that object at that time. At that
time, these two objects are related by the constitution relation. The constitution relation is the
relation that holds at a time, between any two objects that are materially coincident at that time.
So if x and y are related by constitution at t, then x is an improper part of y at t, and y is an
improper part of x at t.
Now let us suppose that y is a diachronic object, and consider again the question of
whether or not this object is composed of instantaneous objects. Well if what it is to be composed
of certain objects is to be the fusion of those objects, then no diachronic object is composed of
instantaneous objects, for no diachronic object is the mereological fusion of instantaneous
objects. Rather, there exist instantaneous objects, and enduring objects (diachronic objects) both
of which are wholly present whenever they exist, and where these diachronic objects are at each
time at which they exist, constituted by some instantaneous object. Moreover, since every
diachronic object exists, each instantaneous object will constitute more than one diachronic
object at each time at which it exists: for there will exist diachronic objects that wholly overlap
for some period of time. Just as an instantaneous temporal part will, if unrestricted composition
is true, be a part of more than one four dimensional object, so too for the three dimensionalist,
these instantaneous objects will constitute more than one enduring object at the time at which
they exist.
4 Diachronic Fusions and Diachronic Objects
But is there any real difference between the view I am describing, and four dimensionalism?
Well the debate between the three and four dimensionalist is not a debate about whether or not
objects can have other objects as parts at times: for no one need deny that. Nor is it a debate
about which objects exist. For there is nothing about four dimensionalism that prescribes that one
accept unrestricted composition. Crucially, four dimensionalism is the view that persisting
objects are temporally extended: that is, that at every time at which they exist, some of their parts
at not present at those times. An object does not have a temporal part in virtue of having some
improper spatial part for a period of time. An object O has a temporal part P if it is true that ‘O
has P simpliciter.’
For the four dimensionalist, ‘P is part of O’ is true at every time at which O exists, even if
P is not present at the time of utterance. So a four dimensional object that is the mereological
fusion of instantaneous objects is an object that has each of these instantaneous objects as parts
simpliciter. For the three dimensionalist though, these instantaneous objects are not parts
simpliciter of the diachronic object that they constitute at a time. The three dimensionalist will
say that the instantaneous object x that exists at t, is an improper part of the diachronic object O
at t, but is not part of O simpliciter. This version of three dimensionalism, therefore, is not
simply a way of accepting that there exist ersatz temporal parts. For it involves rejecting the idea
that objects are temporally extended, and thus rejecting the idea that there is anything answering
to the description of ‘temporal’ part.
We can further clarify this distinction if we consider how Sider’s argument would
proceed with respect to extended temporal parts. Consider daisy*. Daisy* is a diachronic fusion
of the fusions of the members of sets C at t, C1 at t1, C2 at t2 and C3 at t3. Let’s say that at t,
every member of C is part of some other object Daisy, and every part of Daisy at t overlaps some
member of C at t. So too with C1 at t1 and so forth. Daisy is a cat, and therefore exists at many
times at which daisy* does not. So is daisy* an extended temporal part of Daisy? To answer this,
consider the following definition, due to Sider, of an extended temporal part.
ETP: x is an extended temporal part of y during T iff (1) x exists at, but only at,
times in T (2) x is part of y at every time during T, and (3) at every moment in T x
overlaps everything that is part of y at that moment.
Daisy* exists through times t to t3. Call this duration T. So daisy* exists at and only at
times in T. Daisy* is part of Daisy at every time during T, and at every moment in T, daisy*
overlaps everything that is part of Daisy at that moment. So we can conclude that daisy* is a
temporal part of Daisy. All well and good since the three dimensionalist denies that daisy*
exists.
But now let us consider the diachronic object snowy*. Snowy* is an object that is
constituted by the fusions of the members of sets D at t, D1 at t1, D2 at t2 and D3 at t3. At each of
those times, the fusions of those sets also constitute Snowy the dog, who is five years old. Both
snowy* and Snowy are enduring objects that are wholly present whenever they exist: snowy* is
not, for the three dimensionalist, a temporal part of Snowy. When we look to Sider’s definition
of an extended temporal part, however, we run into difficulties. For snowy* exists only during T.
Snowy* overlaps at every moment in T, everything that is part of Snowy at that moment, and
snowy* is part of Snowy at every time in T. So by Sider’s definition of an extended temporal
part, snowy* is a temporal part of Snowy.
What is snowy*? Snowy* is simply a persisting object that exists between and only
between certain times, and which happens to overlap another object, Snowy, at the times at
which it exists. None of this precludes snowy* (and Snowy) from being wholly present at every
time at which each exists. For consider a more familiar example: the statue and the lump of clay.
A statue and the lump of clay that composes the statue are materially coincident at certain times.
But suppose that the statue is squashed, and thus ceases to exist at some time, while the lump
persists for some time longer. It surely does not follow merely from the fact that the statue and
the lump wholly overlap for some period of time, and that the statue exists during and only
during that period of time, that the statue is a temporal part of the lump. The same is true for the
relation between snowy* and Snowy.
The problem lies in Sider’s definition of an extended temporal part. For on this definition,
something is an extended temporal part of y if it completely overlaps y for some period of time,
and exists only during that period of time. Once we accept, however, that three dimensionalists
can coherently accept a version of unrestricted composition, then this definition of an extended
temporal part is seen to be lacking. For given unrestricted composition, a three dimensional
diachronic object can overlap another such object during and only during some period of time T.
But nothing about this suggests that one of those diachronic objects is a part simpliciter of the
other. Specifically, nothing about snowy* shows that it is part of Snowy simpliciter. At each time
at which snowy* exists, snowy* is an improper part of Snowy at that time. But there is no
timeless sense in which snowy* is part of Snowy, because neither Snowy nor snowy* are
temporally extended.
The problem with Sider’s ETP definition of an extended temporal part is that clause (2) is
couched in terms of parthood at times, rather than atemporal parthood. This was an attempt to
accommodate the three dimensionalist, who rejects the idea of atemporal parthood. The difficulty
is that clause (2) does not distinguish between a case where we have a diachronic object such as
snowy*, where snowy* is an improper part of Snowy at each time t in T, and the case where we
have a diachronic object daisy* that is part of Daisy simpliciter. For if daisy* is part of Daisy
simpliciter then it is also true that daisy* is part of Daisy at each time in T, though the reverse is
not the case. But only if daisy* is part of Daisy simpliciter, does it follow that Daisy is a
temporally extended object, of which daisy* is an extended temporal part. Sider’s atemporal
version of ETP achieves just this:
AETP: x is an extended temporal part of y during T iff (1) x exists at, but only at,
times in T (2) x is part of y and (3) at every moment in T, x overlaps every part of
y that exists at that moment.
Considering again clause (2), we can see that in the case of daisy* and Daisy, it is true
that daisy* is a part of Daisy, and thus that daisy* is an extended temporal part of Daisy. In the
case of snowy*, however, since both snowy* and Snowy are wholly present at each time at
which they exist, there is no atemporal sense in which snowy* is part of Snowy. While there are
some times at which snowy* is part of Snowy, there are other times at which snowy* is not part
of Snowy. So (2) is not true of snowy* and Snowy, and thus snowy* is not an extended temporal
part of Snowy.
This brings us back nicely to the question of why the second part of the argument from
vagueness does not show that the fusion of the members of y at t, namely x, is part of the
persisting object y. Recall that Sider employed a mereological principle according to which we
can move from the claim that every part of x at t overlaps y at t, to the claim that x is part of y at
t. I earlier conceded that Sider had shown that x at is part of y at t, where y is a persisting object.
We can now see why conceding this was not conceding that x is a temporal part of y. For
consider again snowy* and Snowy. Snowy* at t overlaps Snowy at t. So by the mereological
principle, we can conclude that snowy* at t is part of Snowy at t. Even having shown this,
however, we have not shown that snowy* is a temporal part of Snowy. To show that, it needs to
be the case that we can move from the claim that ‘snowy* at t overlaps Snowy at t’ to the claim
‘snowy* is part of Snowy.’ For the three dimensionalist does not deny that at t, snowy* is part of
Snowy: snowy* is an improper part of Snowy at t. Rather, she denies that snowy* is part of
Snowy simpliciter. So while the mereological principle is sound, applying this principle to two
wholly overlapping persisting objects that exist for different durations, tells us only that at each
time at which both exist, each is an improper part of the other. There is a further question as to
whether one is a part of the other simpliciter. Only if this latter is the case, can we conclude that
one is a temporal part of the other. So all Sider’s earlier argument shows is that x at t is part of y
at t. It does not show that x is part of y simpliciter, and thus does not rule out the possibility that
y is a diachronic object, and not a diachronic fusion.
So it is hardly surprising that it is the atemporal, and only the atemporal version of the
definition of an extended temporal part that permits us to draw the distinction between
diachronic objects and diachronic fusions. For showing that some object is a temporal part of
another object involves more than just showing that the former is at some times an improper part
of the latter, it involves showing that the former is part of the latter simpliciter, and the temporal
definition of an extended temporal part is unable to accomplish this. Thus what it ends up
defining is not a temporal part at all.
5 Unrestricted Composition and Fusions-at-Times
But why would a three dimensionalist accept that there exist any such instantaneous objects at
all? Perhaps most wouldn’t. This would not preclude the endurantist from accepting unrestricted
composition and its attendant advantages. As we noted previously, that composition is
unrestricted does not entail that there exist instantaneous objects that could count as
instantaneous temporal parts. The fusion of materially coincident enduring particulars is an
enduring object. But suppose there exist two basic particulars P1 and P2. P1 endures through T,
and P2 endures through T*, where T and T* are non-contiguous intervals. Given unrestricted
mereological composition, there exists a fusion of P1 and P2, call it P, and P appears to be a four
dimensional object in that it has P1 and P2 as parts, albeit enduring parts. So while P1 is wholly
present through T, P is only partly present during T.
Although this would not be disastrous for the endurantist (it would certainly not be
typical perdurantism) the endurantist might want to understand unrestricted composition not in
mereological terms, but rather, in the same non-mereological terms as does the endurantist who
accepts the existence of instantaneous objects–as constitution at a time.Thus she might contend
that any arbitrary combination of enduring particulars constitutes some persisting object, and
thus that P is constituted by P1 during T, and constituted by P2 during T*. Hence P is wholly
present whenever it exists. Thus unrestricted composition would be understood as the claim that
for any arbitrary arrangement of particulars, there is some enduring object that is constituted by
those particulars at times.
Still, if empirical discovery revealed that basic particulars are instantaneous, I do not see
that this would entail the truth of four dimensionalism, rather, it would entail something like the
three dimensionalist view I have been considering according to which enduring objects are
constituted by instantaneous objects at times. So too there are certain advantages for the three
dimensionalist in accepting Sider’s idea of a fusion-at-a-time. For instance, if there exist
instaneous objects that constitute enduring objects at times, then the endurantist can make sense
of the idea that in some sense enduring objects can have properties simpliciter: an enduring
object O is red at t just if O is constituted at t by some object that is red simpliciter. The
instantaneous objects that constitute enduring objects can play much the same explanatory role
as temporal parts do for the four dimensionalist, without the need to claim that objects are only
partly present whenever they exist.
6 Conclusion
What all this shows is that even if one endorses part I of the argument from vagueness, this does
not compel one to accept four dimensionalism. The endurantist can still reject the idea that basic
particulars are instantaneous, or that there exist fusions-at-times, and can thus reject the idea that
enduring objects are diachronic objects constituted by shorter lived objects at times. On the other
hand, even if one grants Sider’s contention that such instantaneous objects exist, it does not
follow that persisting objects perdure, for it does not follow that persisting objects are the fusions
of these instantaneous objects. Thus the argument from vagueness provides no reason to prefer
four dimensionalism to three dimensionalism.
University of Queensland
Brisbane, Queensland
Australia 4072
email: kristie_miller@yahoo.com
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