## Wednesday, October 30, 2013

### The vagueness argument against restricted compositionality

Lewis and Sider have argued that if restricted compositionality is true—some but not all pluralities of two or more objects compose a whole—then there will be cases where it's vague how many objects there are. For instance, imagine two universes, A and B, each with the same finite set of n particles with the same intrinsic properties. But in A, the particles are neatly arranged into galaxies, trees, tables, buildings, etc. And in B there is just a blooming buzzing confusion. If restricted compositionality holds, then, assuming there are no immaterial objects, universe B has exactly n or at most n+1 objects—it's just too messy to have any cases of composition, except perhaps for the universe as a whole (that's why it might be n+1 rather than n). But A is much like our universe, and so we would expect lots of cases of composition, and hence the number of objects will be a lot more than n+1, say n+m for some large m. However, we can now imagine a continuous sequence of universes ranging from A to B, differing continuously in how the particles are arranged. As we move that continuous sequence, the number of objects will have to change from no more than n+m to n+1. But it is incredible that the object count should sharply change due to a very tiny shift in particle positions. Instead, the object count will at times be vague. But how many objects there are is a matter of which sentences using universal quantification, conjunction, negation and identity are true. But quantification, conjunction, negation and identity are not vague. So we have vagueness where we cannot have vagueness.

There may be some technical problems with the argument as I formulated it, given the assumption of no immaterial objects. Maybe we can't do without immaterial entities like God or numbers. One could reformulate the argument to restrict the counting to material entities, but "material" might actually be a vague term. Perhaps the best thing to do is to assume that these universes have no immaterial contingent entities, and then just count contingent entities. Contingency shouldn't be a vague matter, after all. The Aristotelian may balk at this. For it may well be that a necessary condition for a bunch of material entities to compose a whole that they have a form, and forms are immaterial but contingent. Maybe, though, "form" is not vague, and so we can just count the contingent non-forms.

But talking of forms suggests a more serious difficulty. If there are Aristotelian forms, then how many material objects there are may well not supervene on how material objects are spatiotemporally arranged and what intrinsic properties they have. For objects to come to compose a whole, there must come into existence a form. There is nothing absurd about there being sharp laws of nature specifying under which precise conditions a form comes into existence. There is no need for the laws of nature to be continuous (and the possibility of fundamental discreteness is empirically still open). Or perhaps God decides on a case-by-case basis whether to create a form. Then there is no vagueness as to how many material objects there are: the number of material objects equals the number of forms of material objects that in fact inform some matter (the souls of the resurrected are forms of material objects but temporarily fail to inform any matter). Of course in transitional cases we won't have much confidence whether some objects compose a whole, but that's just because we are unable to see forms except through their normal effects.

Heath White said...

The reply which immediately occurs to me is that the truth values of quantified statements certainly are vague. E.g. "there are exactly two bald philosophers having this discussion."

Alexander R Pruss said...

That's a statement that includes other terms with vague application: "bald", "philosopher", "having" and "this discussion". The vagueness does not seem to be rooted in the quantifiers.

Heath White said...

Right. My point was this: that bald philosophers are objects, and how many of them there are is vague, and this doesn't seem to be a problem.

Alexander R Pruss said...

Yeah, it's not a problem that it's vague how many objects of kind K there are, where K is a kind with vague boundary conditions. BUt it's a problem that it's vague how many Ks there are where K is a non-vague kind, e.g., contingent.

Tuomas said...

What's your view on Merricks' strategy regarding vague composition? What I have in mind is the idea that the emergence of new causal powers is a sign of composition. Merricks has this example about straws breaking a camel's back, but I think that particular example is pretty weak. Nevertheless, this can be developed into a more plausible argument.

If I'm allowed a self-reference, I suggested in a 2009 paper (http://philpapers.org/rec/TAHATV) that the case of a heap provides a good example: four grains of sand is plausible the least number required to compose a heap because of the new causal powers that emerge, e.g., the rise in potential energy that the pyramid shape allows.

Heath White said...

Suppose the universe begins as an amorphous mass of contingent particles. Gradually these particles begin to coalesce, at different rates, into bald philosophers. At any time before the endstage there will be a precise number of particles and a vague number of bald philosophers. So the total number of contingent objects will be vague. But I'm not seeing the difficulty in such a description.

Alexander R Pruss said...

I take it that it's part of your story that it's definite that the only objects are going to be particles and any bald philosophers there might be.

But then the number of objects will equal the number of particles plus the number of people. For it's definite that all bald philosophers are people, and no people are particles.

So any vagueness in your story has to do not so much with bald philosophers but with people. But one can't have vagueness about whether particles compose a person. They do so iff they have a personal soul.

Brian Cutter said...

Regarding the last paragraph: Couldn't we say something similar without introducing forms? You suggest two possibilities: (a) There are sharp laws of nature specifying when a form comes into existence. (b) God decides on a case by case basis whether to create a form. But we could adopt similar positions without introducing forms, i.e.: (a*) There are sharp laws of nature specifying when a new material object comes into existence. (So, instead of having the form: "Whenever some xs are related thusly, there comes to be a form F such that F informs the xs," we just have laws of the form "whenever some xs are related thusly, there comes to be an object y such that the xs compose y.") Or: (b*) God decides on a case by case basis when to create a new material object. So, instead of God's deciding to create a form F and put it in the "informing" relation to the xs, he just creates a new thing y, and puts the xs in the proper parthood relation to it.

Is there any reason to prefer (a) or (b) over their starred counterparts?

Heath White said...

Fine, leave people out of it.

Suppose the universe consists of air particles and sand particles. There are a definite number of each. The wind (air particles in motion) blows the sand around and they gradually form heaps. At some point they become definite heaps. But along the way they will be vague heaps. So the number of contingent entities will be the number of air particles plus the number of sand particles plus the number of heaps. As the latter is vague, the sum will be vague.

But this story isn't any more controversial than the existence (and instantiation) of vague predicates.

Alexander R Pruss said...

A nice argument against the existence of heaps. :-)

Heath White said...

*throws up hands*

:-)

Alexander R Pruss said...

Heath:

Lewis and Sider, on the other hand, will just say that your argument is just a version of their argument for unrestricted compositionality, except that it's based on more controversial entities, viz., heaps.

Heath White said...

As I understand it, the Lewis-Sider argument is

If unrestricted compositionality is false, the number of objects is vague
The number of objects is not vague
So, unrestricted compositionality is true.

I think the second premise is false, so we cannot draw any conclusions about un/restricted compositionality. The reason I think the second premise is false is that the number of objects depends on which sentences using the existential quantifier are true (here they are correct). But all such sentences will incorporate a predicate, and if the predicate is vague then the sentences will have vague truth values. That is just what a vague number of objects would be.

Alexander R Pruss said...

"all such sentences will incorporate a predicate"

Which predicate?

I can say that there are n objects with a sentence whose only predicate is identity.

William said...

The "n" in your sentence would be vague between, say, 7,121,543,290 and 7,121,543,312, where n is world population and the number changes as the sentence is read :).

Heath White said...

"I can say that there are n objects with a sentence whose only predicate is identity."

Hmmm. True. In that case I guess the thing to say is that the domain may be vague, that is, it may be vague whether we have an object here (e.g. a heap) or not. So quantification is vague, in effect.