## Sider on Unrestricted Composition

May 20, 2008

A few weeks back I read Siders book on four-dimensionalism (I should have done that ages ago, but better late than never). Anyway something struck me as wrong about his argument for unrestricted composition from vagueness, so I thought I’d post it here.

The original argument, due to Lewis, goes as follows. Suppose there was a restriction on when things have a fusion. Then that restriction will most likely be vague (e.g. when the objects are sufficiently close, when they constitute a living object, etc…) So suppose there are cases where it is vague whether composition occurs, e.g. suppose that it is vague whether x is a fusion of the xx‘s. Then the following sentence must be vague

1. $\forall y(y \circ x \leftrightarrow \exists z(z \circ y \wedge z \prec xx))$

Now this sentence could only be vague if (i) the logical constants ($\exists , \wedge, \prec, \neg$) are vague or (ii) parthood is vague. But that is intolerable in either case.

Sider claims to have improved on Lewis’s argument by dispensing with premise (ii). According to Sider, if there could be a vague case of composition, then it would be indeterminate how many things there are. Since stating how many things there are (in finite) cases can be done in purely logical vocabulary, (and denying ‘vagueness in the world’) this contradicts (i), the only premise Sider claims to need.

My problem is that it’s not clear to me at all why vague cases of composition will commit us to indeterminacy over how many things there are. For example couldn’t there be three things, a, b and c, such that a and b are atoms, and it is indeterminate whether c is an atom or c is the fusion of a and b? So we have an indeterminate case of composition, but no indeterminacy in the number of things.

[To spell it out a bit more carefully – on one precisification of the proper parthood relation no object has any proper parts – a, b and c are all atoms. On the other precisification a and b have no proper parts again, but c has two proper parts – a and b. On all precisifications there are exactly three objects, yet only on some but not all precisifications is c the fusion of a and b.]

Of course, Sider might object that parthood can’t be a vague relation. But then we are back where we started. Lewis’s argument needed this as a premise – while Sider claimed to do away with it. (That said, I guess when you look at specific proposals, they often will commit you to indeterminacy in the number of things. E.g. suppose you said that things have a fusion just in case they are sufficiently close. If there are only two atoms, and it’s indeterminate whether they’re close, it looks like we are going get indeterminacy in the number of things.)

1. For example couldn’t there be three things, a, b and c, such that a and b are atoms, and it is indeterminate whether c is an atom or c is the fusion of a and b? So we have an indeterminate case of composition, but no indeterminacy in the number of things.

Hi Andrew,

I’m not sure I see that. Suppose a and b are blocks that are roughly contiguous. How could there be a precisification of the closeness or contiguity relation (supposing this is what is required for composition) under which there is a distinct atom c? Why wouldn’t the alternatives be that, on some precisifications, a+b = c, so there are three objects and on other precisifications there are only a and b? I guess I’m wondering how a precisification of the spatial relations between a and b might yield another simple, c.

2. Hi Mike, thanks for you comment.

I’m taking it that it is sufficient to refute Sider’s claim (that vagueness in composition entails vagueness in the number of things) by providing a case where it is determinate how many things there are, but indeterminate whether fusion occurs. That’s all I’m trying to do.

That said, if you look at the end of my post I did say essentially what you just said (I said: “If there are only two atoms, and it’s indeterminate whether they’re close, it looks like we are going get indeterminacy in the number of things.”)

So I still think that Siders objection is a problem for specific answers to the special composition question – (but then again so is Lewis’s.)

3. Sorry – I see your question now. I’m just stipulating that parthood is such that it is vague whether a and b have a fusion. This, I think, is sufficient to undermine Sider’s claim that he didn’t need the premise that parthood could not be vague.

Anyway I’m not saying this particular kind vagueness in parthood could arise from vagueness about their spatial locations. Although I’m thinking you might be able to conjure up a scenario such that on every precisification of closeness on which the \$xx\$’s don’t have a composition, the \$yy\$’s do, and vice versa (and these are the only borderline cases of composition.) I’ll post something if I make any progress on this!

4. […] Sider on time and vagueness. Even if you disagree with it everyone ought read Sider’s book IMO. […]

5. Predilection says : I absolutely agree with this !

6. Somehow i missed the point. Probably lost in translation 🙂 Anyway … nice blog to visit.

cheers, Microscope!

7. What is the difference between unrestricted composition, restricted composition, and nihilism?