Classical mereology, atomism, temporal parts, and unrestricted diachronic fusions

In this post, I aim to answer the question of the conditions under which in an atomic world with classical mereology we have unrestricted minimal diachronic fusions, as Sider claims we always do. I will need to make some additional plausible assumptions.

My assumptions:

1. Classical (atemporal) Mereology.

2. Parthood-at-t is reflexive and transitive, and x is a part at t of y only if x and y both exist at t.

3. Atomism: Every object has a simple part.

4. Parthood Bridge A (from temporal to atemporal parthood): If y is a part of x at all the times at which y exists, then y is a part of x.

5. (Weak) Parthood Bridge B (from atemporal to temporal parthood): If y is a part of x while y exists at exactly one time and x also exists at that time, then y is a part of x at that time.

6. Fusion Bridge: If z is a fusion of the set S of objects, then z exists at a time t if and only if at least one of the objects in S exists at t.

7. Temporal Fusion Parthood: If x is a fusion-at-t of U and y is a fusion-at-t of V while U is a subset of V, then x is a part of y at t.

Note that, assuming (2), Temporal Fusion Parthood follows from Sider’s condition:

8. PO: If x and y exist at t, but x is not a part of y at t, then x has some part at t that does not overlap y at t.

For suppose in the setting of Temporal Fusion Parthood we don’t have x a part of y at t. Then there is a part w of x at t that does not overlap y. We must then have w overlapping at t some member z of U by the fusion-at-t condition, and hence w must overlap y at t (since y is a fusion of a superset of U), a contradiction.

Further, note that (1)–(4) imply:

9. Weak Temporality: If there are two objects, every object exists at at least one time.

For suppose y exists at no time. Then y is a part of x at every time at which y exists, for any x. Hence there is an object that is a part of every object, which violates Classical Mereology if there are two objects (e.g., use fusion and supplementation).

Say that an admissible function is a function f defined on some non-empty domain consisting of times such that f(t) is a non-empty set of objects existing at t. A minimal diachronic fusion of f is an object that exists at all and only times in the domain of f which is the fusion-at-t of f(t) for all such times in the domain.

Say that an object is temporally extended provided it exists at two or more times.

Theorem: Under the above assumptions, every admissible function has a minimal diachronic fusion if and only if there are no temporally extended simples.

Corollary: Under the above assumptions, the following are equivalent:

1. For any non-empty set T of times and any object x, there is a T-temporal part of x

2. For any time t and any object x, there is a {t}-temporal part of x

3. There are no temporally extended simples.

Here, a T-temporal part of x is a minimal diachronic fusion of f_T, x where f_T, x has domain T and f_T, x(t) = x for all x. The Theorem shows (iii) implies (i), and (i) trivially implies (ii). To see that (ii) implies (iii), suppose x is a temporally extended simple, and let t be one of the times at which x exists. A t-temporal part of x would have to exist at t and only at t, and would be a part of x at t, and hence would be a part of x by Parthood Bridge B, which would violate the simplicity of x.

Now, a philosophical comment. These results shows that given classical mereology, atomism, and some plausible assumptions, the crucial question for Sider’s argument is whether there are temporally extended simples. But this question side-steps all the stuff in his paper about vagueness and continuity. Sider’s continuity argument does nothing to establish that there are no temporally extended simples.

And a technical comment: I’d like to find a good way to weaken Fusion Bridge.

Proof of Theorem: Note that the Theorem is trivial if there is only one object, so we can assume every object exists at a time by Weak Temporality.

Suppose x is a temporally extended simple. Let f have as its domain a non-empty proper subset S of the times at which x exists, and let f(t) = x for all x. Suppose y is a minimal diachronic fusion of f. Then by Parthood Bridge A, y is a part of x. Since x is simple, y = x. But since y exists at a proper subset of the times at which x exists, this is a contradiction.

Conversely, suppose there are no temporally extended simples. Given an object y that exists at a time t, I claim that y is a fusion-at-t of all of y’s simple parts that exist at t.

For by Classical Mereology and Atomism, y is a fusion of all of y’s simple parts. By Fusion Bridge, at least one of y’s simple parts exists at t since y exists at t. Suppose z is one of these simple parts. Then z exists at t and only at t as there are no temporally extended simples. Hence, by Parthood Bridge B, z is a part of y at t. Next suppose that z and y overlap-at-t. Let x be a common part at t. Then x is a fusion of its simple parts by Classical Mereology and Atomism, and hence has a simple part w that exists at t by Fusion Bridge since x exists at t. Then w is a part of y at t by Parthood Bridge B (since w exists only at t). Then w is a part of y at t as well as of z at t (by transitivity of parthood-at-t), and so z overlaps w at t, while w is a simple part of y that exists at t.

Now suppose f is admissible. Let S be the set of all simples x such that there is a t and a y ∈ f(t) such that (a) x exists at t and (b) x is a part of y. By Classical Mereology, let z be the fusion of S.

I claim that z is minimal diachronic fusion of f. First, note that z exists at all and only the times in the domain of f. Let t be a time in the domain of f. Fix any object y in f(t). By Classical Mereology and Atomism, y is a fusion of simples. By Fusion Bridge, at least one of these simples, say x, exists at t. Then x is in S. Hence z exists at t by Fusion Bridge. Conversely, suppose that t is a time not in the domain of f, and suppose z exists at t. By Fusion Bridge, one of the simples in S exists at t. But each of the simples in S exists at a time in the domain of f. Thus, that simple must be temporally extended, and we assumed there aren’t any such.

Hence, we have shown that z at exists at all and only the times in the domain of f. It remains to show that at any time t in the domain of f, z is a fusion-at-t of f(t). Fix any y that is in f(t). We must show that y is a part of z at t. Now, z is a fusion-at-t of its simple parts existing at t, and likewise y is a fusion-at-t of its simple parts existing at t. But any simple part of y that exists at t is a simple part of z by the construction of z, so by Temporal Fusion Parthood, y is a part at t of z.

Next, suppose y overlaps z at t. Thus, y and z have a common part x at t. Then x is a fusion of its simple parts by Classical Mereology and Atomism, and hence there is a simple part w of x that exists at t by Fusion Bridge. Then w is a part of z. Since z is the fusion of S, there must be some member of S that overlaps w, and since w is simple, w is a member of S. Hence w is a simple part existing at t′ of some u in f(t′). Since w is not temporally extended, t = t′, and so w is a simple part existing at t of some u in f(t). By Parthood Bridge B, w is a part of u at t, and likewise w is a part at t of x. Hence w is a part at t of y by transitivity of parthood-at-t, and so y and u overlap at t, which completes our proof.