The antisymmetry of temporary parthood

Sider suggests this four-dimensionalist characterization of “temporary parthood” in terms of temporal parts:

1. Necessarily, x is part of y at t iff x and y each exist at t, and x’s temporal part at t is part of y’s temporal part at t.

Somehow I never noticed something that is probably pretty obvious to people who work on this: temporary parthood defined thus is not antisymmetric (where a relation R is antisymmetric provided that xRy and yRx implies x = y). For consider myself, x, and my temporal part, x_t, at t. Then x_t ≠ x, but x_t is a part of x and x is a part of x_t, assuming plausibly that the temporal part of x_t at t is x_t (otherwise, we get a weird infinite regress where each temporal part at t has a distinct temporal part at t).

Is this a problem? I think so. We do after all seem to have an ordinary notion of parthood at a time, and that notion is antisymmetric: two distinct objects can’t be mutually parts of each other at the same time. If you are a part of a club at t, the club isn’t a part of you at t, even if you are the only member at that time.

We can avoid the above temporal part counterexample to asymmetry by modifying (1) to be:

2. Necessarily, x is part of y at t iff x and y each exist at t, and x’s temporal part at t is part of y’s temporal part at t, and y is not a proper part of x.

But that’s not good enough. For consider non-instantaneous temporal parts, such as myself from time a to time b (both inclusive): x_[a,b]. Then x_[0,3] and x_[1,4] have the same temporal part at t, and so by (2) they are still mutually parts of each other at t, since neither is a proper part of the other simpliciter.

But perhaps we can argue that temporary parthood is not antisymmetric. Consider this story:

• t₁: you have a lump of unbaked clay

• t₂: the clay has been formed into a statue

• t₃: the statue has just been gilded

• t₄: the gilding has just been removed

• t₅: a piece of the clay has been pinched off in order to improve the appearance of the statue’s nose.

Consider the cloud of molecules of the original lump of clay. This cloud persists through the whole story. Initially, it does not form a statue, and at the end it forms more than the statue (since the pinched off bit is a part of the cloud but not the statue then). It seems clear that:

1. The cloud is a part of the statue at t₃, with the gold being the other part at t₃

But the cloud did not become a temporary part of the statue when it was gilded. Thus:

2. The cloud is already a part of the statue at t₂.

Conversely:

3. The statue is a part of the cloud at t₅.

But the statue does not become a temporary part of the cloud when a bit is pinched off. Thus:

4. The statue is a part of the cloud at t₄.

But there is surely no difference in the temporary relationships between the statue and the cloud at t₂ and t₄, so:

22. The statue is a part of the cloud at t₂.

But since the statue is not the cloud, (ii) and (v) imply that parthood at t₂ is not antisymmetric.

If we have a strong intuition that temporary parthood should be antisymmetric, the above example suggests that we should have a restricted view of composition, one on which at least one of the cloud and the statue does not exist.