Thomson,

1
Thomson, Parthood and Identity Across Time

• Thomsons aim is to argue against the thesis of
  temporal parts.
• I
• Make a Tinkertoy house, H, and place it on a
  shelf
• (1) H the Tinkertoy house on the shelf at
  115.
• Appeal to the Leonard-Goodman Calculus of
  Individuals.
• Primitive is x D y (x is discrete from y).
• Define x lt y (x is a part of y) and x O y
  (x overlaps y) in the following way
• x lt y df (z)(z D y ? z D x)
• x O y df (?z)(z lt x z lt y)

2

• Add to this the following axioms 
• Identity (x y) (x lt y y lt x)
• Overlap  (x O y) ? (x D y)
• Fusion  (?x)(x ? S) ? (?y)(y Fu S)
• Fusion is defined as follows
• x Fu S (y)y D x (z)(z ? S ? y D z)
• But there is also the fusion principle, namely,
  that, if there is a member of S, then there is a
  unique thing that fuses the Ss.
• (?x)(x ? S) ? (E!y)(y Fu S)
• If all of the axioms are true, then the fusion
  principle is as well. Therefore, there must be a
  fusion of the Tinkertoys on the shelf.
• (2) W the fusion of the Tinkertoys on the
  shelf at 115.
• So, it would seem that
• (3) H W (I.e., the house is the fusion of
  Tinkertoys.)

3

• II
• Perhaps the fusion principle is too strong
  perhaps we should reject it. Still it seems fine
  to talk about the wood in your hands when you
  hold your Tinkertoys. So lets call that W.
  Therefore,
• (2) W the wood on the shelf at 115.
• And, so,
• (3) H W
• III
• Replace one piece of the Tinkertoy house with
  another. (Ship of Theseus problem.)
• Most of us would say
• (4) H is on the shelf at 145.
• But the conjunction of (3) with (4) entails
• (5) W is on the shelf at 145
• which is not true.

4

• But the conjunction of (3) and (4) entails
• (5) W is on the shelf at 145
• which is also not true.
• We must hold on to (4). So the problem is the
  equivalence of (3) and (3).
• Cartwright would suggest that there are temporal
  parts of the H and W. But what does that mean?
• IV
• What are the metaphysical theses underlying the
  view of temporal parts?
• (M1) If x is a temporal part of y, then x is a
  part of y.
• P ranges over places p is a point in space.
• T ranges over times t is a point in time.

5

• A definition of cross-sectional temporal part
  leads to the second metaphysical thesis.
• (M2) (T)y exists through T ? (?x)(x exists
  through T no part of x exists outside T
  (t)(t is in T ? (P)(y exactly occupies P at t ?
  x exactly occupies P at t))
• Now we need to ensure uniqueness.
• (M3) If x is a part of y and y is a part of x,
  then x is identical with y.

6

• The friends of temporal parts certainly also
  hold a kind of fusion thesis with respect to
  temporal parts. Thus,
• If x is a temporal part of z and y is a temporal
  part of z, then there is a z that fuses the set
  whose members are x and y.
• The final metaphysical thesis
• (M4) x is a temporal part of x.
• This seems innocuous but is in fact very strong.
• Question Do times have sharp boundaries?

7

• V
• This is a crazy metaphysic! But it is hard to
  give a proof for its falsehood.
• Why should we accept it? Advocates probably
  have two main motivations first, it seems to
  solve problems related to identity over time
  second, there is a spatial analogy that seems
  to make sense.
• Concerning the latter For homework, try
  breaking a bit of chalk into two temporal parts!
  (306b)
• The full craziness of this view comes out when
  we take the spatial analogy seriously.

8

• How exactly is H related to W?
• Parthood is surely a three-place relation, among
  a pair of objects and a time.
• Let us emend the Leonard-Goodman Calculus of
  Individuals.
• Primitive x is discrete from y at t ? xDy _at_
  t
• First Existence Principle if x does not exist
  at t, then there is no z such that z is a part of
  x at t ? x does not exist at t ? (y)(xDy _at_ t)
• Second Existence Principle if everything is
  now discrete from a thing, then that thing does
  not now exist ? (y)(xDy _at_ t) ? x does not exist
  at t

9

• This leads to the following x exists at t
  (y)(xDy _at_ t)
• So we introduce x E _at_ t (x exists at t) and
  then have the following
• x E _at_ t def (y)(xDy _at_ t)
• Parthood and Overlap are now defined as follows
• x lt y _at_ t def x E _at_ t y E _at_ t
• (z)(z D y _at_ t ? z D x _at_ t)
• x O y _at_ t def (?z)(z lt x _at_ t z lt y _at_ t)
• New overlap axiom 
• (CCI2) (x O y _at_ t) (xDy _at_ t)

10

• New identity axiom 
• (CCI1) (x y) (t)(x E _at_ t ? y E _at_ t) ?
• (x lt y _at_ t y lt x _at_ t)
• New fusion axiom 
• (CCI3) (?x)(x ? S x E _at_ t) ? (?y)(y Fu S _at_ t)
• So how does H relate to W?
• H lt W _at_ t W lt H _at_ t is true for all times
  between 100 and 130.
• Since H and W exist at times at which this is
  not true, H is not identical with W.
• More generally, a Tinkertoy house is made only
  of Tinkertoys, and Tinkertoys are bits of wood
  so, at every time throughout its life, a
  Tinkertoy house is part of, and contains as part,
  the wood it is made of at that time. (310a)