Change: A Story of Paradox
It seems uncontroversial that objects can
change by gaining or losing parts.
An old argument suggests that this is
impossible.
Paradox of Increase:
Before:
A
B
After:
A
B
Object A hasn’t changed by gaining B as a
part.
 What A was before the addition exists
unchanged after the addition.
 We have a new object composed of A
and B (or, we have changed a
‘scattered’ object into a ‘connected’ one,
but this object hasn’t gained any parts).
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The reverse argument
The paradox of shrinking:
Before:
Z
Y
=X
After:
Z
X?
Y
X?
 What was X before the amputation
seems no longer to exist (neither Z nor
Y is identical to X.)
o Or, X has become a scattered object
but hasn’t lost any parts.
So, the old argument concludes, it is
impossible for any object to change by
gaining or losing parts.
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Strange conclusion
Mereological constancy: if x is a part of y at
any time, then it is a part of y at any time at
which y exists.
Put another way, if something gains or loses
any parts, it ceases to exist.
This is a startling conclusion. Since, for
example, you gain and lose atoms all the
time, one of the following must be true:
1. The person now reading this sentence is
a different being from the one who read
the words at the top of this page.
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Second option
2. You exist so long as the atoms that
compose you exist: so, you
existed…1000 years ago, today, and in
1000 years… (in all cases but one,
scattered).
This case, however, is just as bad:
The atoms that composed the human sitting
in your seat a second ago are different from
those that compose you right now.
 So, one second from now you will
continue to exist (as a scattered object)
but there will be a different human being
in your seat.
 So, as in #1 above, you can never
shake hands with the same person
twice.
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Consequences
Think of the implications for morality, the
law, planning for the future. E.g.:
 The person who stole from you
yesterday no longer exists, so there is
nobody to hold responsible…
 The person who will benefit from your
retirement savings is not you…
 The object that was stolen from you no
longer exists, so there is nothing to
return to you…
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Re-picturing the argument
Maybe we have the wrong picture. Not:
Before (T1):
After (T2):
A
B
But, rather:
Before (T1):
A
A
B
After (T2):
B
C
B
A
In other words, A changes by coming to be
made of B plus something else.
Problem: what is C?
It appears to be exactly what A was before
B became attached to it.
So A (=C) hasn’t changed.
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The argument spelled out
1. A acquires B as a part (assumption)
2. If A acquires B as a part, A comes to be made of
B and ‘the rest of A’ = C (from 1)
3. C does not acquire B as a part (from 2 and
definition of ‘compose’)
4. C exists before B is attached (assumption)
5. C coincides mereologically with A before B is
attached (from 2, 3, & 4)
6. If X and Y coincide mereologically, then X = Y
(assumption)
7. Therefore, C = A (from 5 & 6)
8. Therefore, A does not acquire B as a part (from 3
& 7)
So, (1) leads to a contradiction, and must be
denied.
The argument appears valid. Is it sound?
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Five responses
1. Relative identity: A and C can be identical
at T1 but distinct at T2. [Denies 6]
So, even if C doesn’t gain a part, it doesn’t
follow that A doesn’t, since A and C might
be distinct when A gains a part.
Objection: When A and C are identical, A is
about to grow and C isn’t. So, they have
different properties. Therefore, they can’t
be the same.
Leibniz’s Law: if x = y, then every property
of x is a property of y and vice versa.
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2. Sparse Ontology: There is no such object
as ‘the rest of A’ (i.e. C). [Denies 2]
There are many smaller things (D, E, F)
which once were an object (A), but now
aren’t:
 T1: A = D, E, F
 T2: A = B, D, E, F
Key point: at T2, D+E+F do not add up to
any kind of object. There is only A at T2.
I.e. composition is not universal: it is not the
case then whenever A and B exist, there
exists the object A + B.
Objection: This means nothing ever has as
a proper part something it was once
identical to:
 E.g. build an addition to a house
(without disturbing original parts) then
there is no such object as ‘the original
house’, only the new building.
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3. Funny persistence conditions: C did not
exist prior to A being joined to B. [Denies
4].
 I.e., C was created after A & B joined.
Since C and A never coincided, there is no
reason to suppose C = A.
Objection: Suppose I have a red brick
house and I build a blue addition to it.
 After the addition, a new object comes
into existence, one that is made up of
the parts of the original house but is not
identical to it.
 I.e., in adding only blue bricks, one
created a new red brick structure.
 Similarly: If one removes only the blue
bricks, it follows that the red portion of
the house ceases to exist and a new
red object comes to exist.
 E.g. if you lost all your body but your
head, your head would thereby be
destroyed!
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4. Coincidence: Distinct objects can
coincide mereologically. [Denies 6].
I.e., A and C are distinct and coincide at T1
and are distinct but don’t coincide at T2.
Put another way: C exists before B is
attached to A but is not identical to A.
Objection: If B becomes part of A, why
doesn’t it also become part of C?
 After all, at T1 A and C are the same in
every way!
 Why is it that only A gets bigger?
Perhaps A is an organism, C just a lump of
matter.
 But C has all the properties of an
organism at T1! Why isn’t it an
organism too?
Perhaps this is a primitive property of C?
 Theft over honest toil?
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5. Temporal parts: Objects (at least objects
that can gain/lose parts) are temporally
extended and consist of temporal parts.
Objects are 4-D space-time ‘worms’.
Hence, two objects can coincide in virtue of
sharing a temporal part that exists at that
time. [Denies 6].
A temporal part of O is ‘all’ of O at some
time or between two times.
E.g.: The ‘part’ of you that exists from the
start of this lecture to its end.
 The instantaneous part of you right now.
 The part of you from your second
birthday to yesterday.
The temporal parts of an object are located
at different times, but may overlap (like
parts of a road).
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How this solves the problem
A is an extended ‘worm’ that consists of C
(also an extended worm) and all the
temporal parts of B (worm) starting from T1.
A just is (timelessly) composed of C and the
later parts of B.
Objects do not really gain and lose parts on
this view.
 Rather: some of an object’s temporal
parts overlap with those of another
object, but some do not.
Time
C
B
A = shaded part (i.e. C + part of B)
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Atemporal parthood
Four-dimensionalists deny that ‘having a
part’ is a temporal notion. In other words:
 Not: X has Y as a part at T.
 Rather: X (timelessly) has a T-part that
(timelessly) includes Y.
X may have parts located at different times,
but these things are parts of X simpliciter.
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Resisting the argument
Four-dimensionalists resist step #6 in the
argument: A and C can overlap in virtue of
sharing temporal parts.
However, they accept an atemporal version
of #6: no two objects can mereologically
coincide simpliciter:
 Any given collection of temporal parts is
only one object.
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Objections to four-dimensionalism
First: if objects are composed of temporal
parts, then there are many more things in
the world than it seems:
‘You’ are composed of indefinitely many
temporal parts!
 Your seat in this classroom has been
filled by trillions of thinking entities since
you sat down!
Moreover, which temporal part are you?
 Perhaps you think you’ve been alive for
over a decade.
 So does the temporal part that exists
from one minute ago to now.
 So does the temporal part that exists
from yesterday to tomorrow.
 Etc.
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Second: Notice that on 4-Dism objects don’t
change overall; they just have parts whose
properties differ from each other, but as a
whole the object never changes.
But could the object have been different?
It seems that objects might have existed
even if they never gained a particular part.
 E.g. you might still exist even if you
didn’t have that pizza yesterday.
But on 4-Dism this seems impossible:
A = eat pizza
(Actual world)
B = No pizza
(Possible world)
It seems clear that A ≠ B.
So you wouldn’t be you had you not eaten
that pizza.
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Mereological essentialism
It seems that 4-Dism entails mereological
essentialism, namely:
If X is a part of Y, it is impossible for Y to
exist without having X as a part.
Is this solution worse than the problem?
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Appendix: Why not accept the
conclusion of the argument?
There appear to be two general ways to do
this…
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1. The revisionist view
Nothing persists through time.
When we say things such as:
(1) John was hungry at T1, now at T2 he is
full
What we mean is:
(2) A person called ‘John’ was hungry at
T1, a person called ‘John’ is full at T2,
and these two people stand in a
relation to each other.
Note:
 This relation is not identity: It is
something else: spatiotemporal
contiguity, psychological continuity, etc.
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2. The reactionary view
People are mereological simples: we have
no parts.
Since material objects seem to always
change their parts, this view suggests we
are immaterial.
Objection: Even if I am not this particular
body, right now this body exists (it won’t as
soon as it loses an atom).
 But can’t this body (which includes a
brain) think?
 Then why suppose there is something
else? I.e. why resist revisionary reply?
Possible reply: The body exists for too short
a period to think!
Possible rejoinder: it thinks in virtue of being
part of a sequence of momentary bodies
each of which forms part of a thought…
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