Weak supplementation says that if *x* is a proper part of *y*, then *y* has a proper part that doesn’t overlap *x*.

Suppose that we are impressed by standard counterexamples to weak supplementation like the following. Tibbles the cat loses everything but its head, which is put in a vat. Then Head is a part of Tibbles, but obviously Head is not the same thing as Tibbles by Leibniz’s Law (since Tibbles used to have a tail as a part, but Head did not use to have a tail as a part), so Head is a proper part of Tibbles—yet, Head does not seem to be weakly supplemented.

But suppose also that we don’t believe in unrestricted fusions, because we have a common-sense notion of what things have a fusion and what parts a thing has. Thus, while we are willing to admit that Tibbles, prior to its injury, has parts like a head, lungs, heart and legs, we deny that there is any such thing as Tibbles’ front half minus the left lung—i.e., the fusion of all the molecules in Tibbles that are in the front half but not in its left lung.

Imagine, then, that there is a finite collection of parts of Tibbles, the *T*s, such that there is no fusion of the *T*s. Suppose next that due to an accident Tibbles is reduced to the *T*s. Observe a curious thing. By all the standard definitions of a fusion (see SEP, with obvious extensions to a larger number of parts), after the accident Tibbles is a fusion of the *T*s.

So we get one surprising conclusion from the above thoughts: whether the *T*s have a fusion depends on extrinsic features of them, namely on whether they are embedded in a larger cat (in which case they don’t have a fusion) or whether they are standalone (in which case their fusion *is* the cat). This may seem counterintuitive, but artefactual examples should make us more comfortable with that. Imagine that on the floor of a store there are hundreds of chess pieces spilled and a dozen chess boards. By picking out—perhaps only through pointing—a particular 32 pieces and a particular board, and paying for them, one will have bought a chess set. But perhaps that particular chess set did not exist before, at least on a common-sense notion of what things have a fusion. So, one will have brought it into existence by paying for it. The pieces and the board now seem to have a fusion—the newly purchased chess set—while previously they did not.

Back to Tibbles, then. I think the story I have just told shows that if we deny weak supplementation and unrestricted fusions also suggests something else that’s really interesting: that the standard mereological relations—whether parthood or overlap—do not capture all the mereological facts about a thing. Here’s why. When Tibbles is reduced to a head, we want to be able to say that Tibbles is more than its head. And we can say that. We say that by saying that Head is a proper part of Tibbles (albeit one that is not weakly supplemented). But if Tibbles is more than his head even after being reduced to a head, then by the same token Tibbles is more than the sum of the *T*s even after being reduced to the *T*s. But we have no way of saying this in mereological vocabulary. Tibbles *is* the fusion or sum of the *T*s when that fusion is understood in the standard ways. Moreover, we have no way of using the binary parthood or overlap relations to distinguish the how Tibbles is related to the *T*s from relationships that *are* “a mere sum” relationship.

Here is perhaps a more vivid, but even more controversial, way of seeing the above point. Suppose that we have a tree-like object whose mereological facts are like this. Any branch is a part. But there are no “shorn trunks” in the ontology, i.e., no trunk-minus-branches objects (unless the trunk in fact has no branches sticking out from it). This corresponds to the intuition that while I have arms and legs as parts, there is no part of me that is constituted by my head, neck and trunk. And (this is the really controversial bit) there are no other parts—there are no atoms, in particular. In this story, suppose Oaky is a tree with two branches, Lefty and Righty. Then Lefty and Righty are Oaky’s only two proper parts. Moreover, by the standard mereological definitions of sums, Oaky is the sum of Lefty and Righty. But it’s obvious that Oaky is more than the sum of Lefty and Righty!

And there is no way to distinguish Oaky using overlap and/or parthood from a more ordinary case where an object, say Blob, is constituted from two simple halves, say, Front and Back.

What should we do? I don’t know. My best thought right now is that we need a generalization of proper parthood to a relation between a plurality and an object: the *A*s are jointly properly parts of *B*. We then define proper parthood as a special case of this when there is only one *A*. Using this generalization, we can say:

Head is a proper part of Tibbles before *and after* the first described accident.

The *T*s are jointly properly parts of Tibbles before *and after* the second described accident.

Lefty and Righty are jointly properly parts of Oaky.

It is not the case that Front and Back are jointly properly parts of Blob.