Wolfgang Schwarz

Soon after I discovered this comment, in July 2005, I sent an e-mail to Wolfgang S. with a reason for abandoning this 'standard defintion'. It, together with a principle and an assumption that are difficult to deny, entails a contradiction. A revised definition of 'fusion' does not have this problem. Perhaps Wo will include my response somewhere on his site.

The comment I sent yesterday went up on the page immediately. Today I'll try to paste the original.

I have just come across your Monday, 09 December 2002 entry entitled ?Things and Fusions?. I have some comments.

You say the standard definition of ?fusion? is

(Def 1) X is the fusion of Y1, Y2, . . . iff all of Y1, Y2, . . . are parts of X and no part of X is distinct from all of Y1, Y2, ?

Then you state a consequence of Def 1:

If X is not the fusion of Y1, Y2, . . . then either one of Y1, Y2, . . . is not a part of X or some part of X does not overlap Yl, Y2, ?

If think that another consequence of Def 1 is more interesting:

If either one of Y1, Y2, . . . is not a part of X or some part of X does not overlap Y1, Y2, . . . then X is not the fusion of Y1, Y2, . . .

This leads to questions about what happens when something loses a part. I will approach issues of this sort by another route.

Consider the following principle:

(Existence) If X is the fusion of Y1, Y2, . . ., then X exists iff Yl, Y2, ? all exist.

And also the following assumption:

(Corruption) There are cases in which something X is the fusion of Y1, Y2, . . . according to Def 1 and X is also the fusion of Z1, Z2, . . .according to Def 1 and Y1, Y2 , . . . all continue to exist although Z1, Z2, . . . do not all continue to exist.

Existence and Corruption together entail that there are cases in which something exists and does not exist. And that really is a contradiction. Here is a little example:

B is a big cube composed of eight non-overlapping medium cubes, M1, M2, etc. The medium cubes are each composed of eight non-overlapping small cubes, S1, S2, etc. There are 64 non-overlapping small cubes in all.

According Def 1, B is the fusion of the M?s and also a fusion of the S?s. Now suppose that the small cubes are scattered far and wide, and that none of them is destroyed. According to Existence, B still exists. But scattering the small blocks destroy the medium cubes. So according to Existence, B does not still exist. B still exists and does not still exist, a theoretically uncomfortable consequence.

So much the worse, I say, for Def 1. Denying Existence undermines extensional mereology. Denying Corruption invites questions like ?So according to your view, if all the atoms that there once the parts of Socrates? body still exist, then Socrates? body still exists? Isn?t metaphysics full of surprises??

Let?s try instead a replacement for Def 1.

(Def 2) For all Y1, Y2, . . ., X is the fusion of Y1, Y2, . . . iff all of Y1, Y2, . . . are parts of X and no part of X is distinct from all of Y1, Y2, ?

Def 2, so far as I can see, is consistent with Existence and Corruption. If so, that is a good reason to prefer it to Def 1.

Nothing stands in the way of our adopting the axiom

(Composition) For all Y1, Y2, . . ., there is a fusion of Y1, Y2, . . .

I have no objections to Composition. It does not entail that a cat is a fusion, although it does entail that if cats have parts, there are fusions of these parts.

Hi, many thanks for the explanations!

Unfortunately, I don't quite see the difference between (Def 1) and (Def 2). It seems to me that (Def 2) is just (Def 1) with some of the implicit quantifiers made explicit.

Re your example, I guess what you describe is not a big cube composed of 64 small cubes that are scattered far and wide, but rather a big cube that *at t1* is composed of 64 small cubes which *at t2* are scattered far and wide. That is, the small cubes are parts of the big cube at t1, but not at t2. Then to derive the contradiction, I think you actually need

Existence') If X is the fusion of the Ys at t1 then for any t2, X exists at t2 iff the Ys exist at t2.

But that doesn't strike me as very plausible.

(As a fourdimensionalist, I think that besides the relativized notions "part at t1" etc. you employ, there are also absolute notions in terms of which the relativized ones can be defined. But I believe (Def 1) is unproblematic for both notions.)

I think that Existence entails Existence':

If X is the fusion of the Ys at t1 then for any t2, X exists at t2 iff the Ys exist at t2.

You say this doesn't strike you as very plausible. Why not? What counterexamples do you have in mind? Counterexamples will have forms (a) or (b):

X is the fusion of the Ys at t1,
(a) X exists at some t2 although not all the Ys exist at t2;
or (b) X does not exist at some t2 although all the Ys exist at t2.

If you think that fusions gain or lose parts, what do you think about sets? Do they gain or lose members? If so, what are the criteria of set identity? If not, why do fusions differ in the analogous respect? Is this a diffrence between sets and fusions that the founders of mereology intended? And what are the criteria of a fusion identity if Existence' is false?

I hope that Def2 provides a definition of fusion that, unlike Def1, does not entail that everything is a fusion of all its parts. I have already given a reason for trying to avoid this consequence.

We can understand Def1 to have various universes of discourse. These we can always make explicit by introducing new predicates. I refer back to my example of the small, medium, and big cubes.

(DefS) For all small cubes S1, S2, . . . S63, S64, X is the fusion of S1, S2, . . .iff all of S1, S2, . . . are parts of X and no part of X is distinct from S1, S2, . . .

Suppose that there is a fusion X that satisfies DefS. Suppose also that:

(M1) For all small cubes S1, S2, . . . S63, S64, there is a medium cube of which it is a part.

It does not follow from DefS and M1 that X is a fusion of all the medium cubes of which any of S1, S2, . . . are parts. S1, S2, S65, and S66 might all be parts of a medium cubes that is not part of X (although of course it overlaps it).
Now suppose that:

(M2) Every medium cubes has eight of S1, S2, . . . as parts.

It does not follow from DefS and M2 that X is a fusion of medium cubes. M2 does not entail that every S1, S2. . . . is a part of some medium cube.

I want Def2 to allow in my cube example that there is a fusion X of S's and also a fusion Z of M's without thereby entailing that X = Z. I do not think that I have demonstrated that Def2 succeeds in this respect. If it doesn't, I would look for a replacement that does.

Why resist the view that X = Z? Again, in brief: The small cubes can outlast the medium cubes. So the fusion of small cubes can outlast the fusion of medium cubes. If the fusion of the small cubes is identical to the fusion of the medium cubes, then the fusion of small cubes can outlast itself.

I think I see what you're getting at.

You have an atemporal and non-substantial notion of fusions: Whenever some things exist, there is a fusion of these things, and this fusion exists exactly as long as the original things do. Let's call the relation between the original things and such fusions "parthood*".

Fusions never gain or lose parts*. By contrast, ordinary things constantly gain and lose parts, and they often cease to exist even if their parts are still around. So ordinary things are not fusions of their parts. In other words, the parts of ordinary things are usually not part* of these things.

Parthood, unlike parthood* (and set-membership), is time-relative: things are parts of other things only at specific times. We can define a corresponding time-relative notion of fusion: something X is a fusion+ of the Ys at t iff all the Ys are part of X at t and for all parts x of X at t there is some y among the Ys such that x and y share a part at t. (More or less as in Def.1.) So things are always fusions+ of their parts. They are also always fusions of their parts*. But usually they are neither fusions of their parts nor fusions+ of their parts*.

The big cube is a fusion+ of both the medium cubes and the small cubes, but it is not a fusion of them, for it ceases to exist before they do. For the same reason, the fusion of the small cubes (X) is not identical to the fusion of the medium cubes (Y) (even though some current fusion+ of the small cubes is identical to some current fusion+ of the medium cubes), and the medium cubes are not fusions of the small cubes.

If that make sense to you, then I think what you're looking for is a definition of "fusion" in terms of "part". Since "fusion" is definable in terms of "part*" (by Def.1), it suffices to define "part*" in terms of "part". I'd suggest that X is part* of Y iff for any time t, if either X or Y exists at t, then X is part of Y at t.