# From vague parts to vague identity

(Update: as Dan notes in the comment below, I should have clarified that the initial assumption is supposed to be that it’s metaphysically vague what the parts of Kilimanjaro (Kili) are. Whether we should describe the conclusion as deriving a metaphysically vague identity is a moot point.)

I’ve been reading an interesting argument that Brian Weatherson gives against “vague objects” (in this case, meaning objects with vague parts) in his paper “Many many problems”.

He gives two versions. The easiest one is the following. Suppose it’s indeterminate whether Sparky is part of Kili, and let K+ and K- be the usual minimal variations of Kili (K+ differs from Kili only in determinately containing Sparky, K- only by determinately failing to contain Sparky).

Further, endorse the following principle (scp): if A and B coincide mereologically at all times, then they’re identical. (Weatherson’s other arguments weaken this assumption, but let’s assume we have it, for the sake of argument).

The argument then runs as follows:
1. either Sparky is part of Kili, or she isn’t. (LEM)
2. If Sparky is part of Kili, Kili coincides at all times with K+ (by definition of K+)
3. If Sparky is part of Kili, Kili=K+ (by 2, scp)
4. If Sparky is not part of Kili, Kili coincides at all times with K- (by definition of K-)
5. If Sparky is not part of Kili, Kili=K- (by 4, scp).
6. Either Kili=K+ or Kili=K- (1, 3,5).

At this point, you might think that things are fine. As my colleague Elizabeth Barnes puts it in this discussion of Weatherson’s argument you might simply think at this point that only the following been established: that it is determinate that either Kili=K+ or K-: but that it is indeterminate which.

I think we might be able to get an argument for this. First our all, presumably all the premises of the above argument hold determinately. So the conclusion holds determinately. We’ll use this in what follows.

Suppose that D(Kili=K+). Then it would follow that Sparky was determinately a part of Kili, contrary to our initial assumption. So ~D(Kili=K+). Likewise ~D(Kili=K-).

Can it be that they are determinately distinct? If D(~Kili=K+), then assuming that (6) holds determinately, D(Kili=K+ or Kili=K-), we can derive D(Kili=K-), which contradicts what we’ve already proven. So ~D(~Kili=K+) and likewise ~D(~Kili=K-).

So the upshot of the Weatherson argument, I think, is this: it is indeterminate whether Kili=K+, and indeterminate whether Kili=K-. The moral: vagueness in composition gives rise to vague identity.

Of course, there are well known arguments against vague identity. Weatherson doesn’t invoke them, but once he reaches (6) he seems to think the game is up, for what look to be Evans-like reasons.

My working hypothesis at the moment, however, is that whenever we get vague identity in the sort of way just illustrated (inherited from other kinds of ontic vagueness), we can wriggle out of the Evans reasoning without significant cost. (I go through some examples of this in this forthcoming paper). The over-arching idea is that the vagueness in parthood, or whatever, can be plausibly viewed as inducing some referential indeterminacy, which would then block the abstraction steps in the Evans proof.

Since Weatherson’s argument is supposed to be a general one against vague parthood, I’m at liberty to fix the case in any way I like. Here’s how I choose to do so. Let’s suppose that the world contains two objects, Kili and Kili*. Kili* is just like Kili, except that determinately, Kili and Kili* differ over whether they contain Sparky.

Now, think of reality as indeterminate between two ways: one in which Kili contains Sparky, the other where it doesn’t. What of our terms “K+” and “K-“? Well, if Kili contains Sparky, then “K+” denotes Kili. But if it doesn’t, then “K+” denotes Kili*. Mutatis Mutandis for “K-“. Since it is is indeterminate which option obtains, “K+” and “K-” are referentially indeterminate, and one of the abstraction steps in the Evans proof fail.

Now, maybe it’s built into Weatherson’s assumptions that the “precise” objects like K+ and K- exist, and perhaps we could still cause trouble. But I’m not seeing cleanly how to get it. (Notice that you’d need more than just the axioms of mereology to secure the existence of [objects determinately denoted by] K+ and K-: Kili and Kili* alone would secure the truth that there are fusions including Sparky and fusions not including Sparky). But at this point I think I’ll leave it for others to work out exactly what needs to be added…

### 8 responses to “From vague parts to vague identity”

1. Dan López de Sa

Hi Robbie!

So the upshot of the Weatherson argument, I think, is this: it is indeterminate whether Kili=K+, and indeterminate whether Kili=K–. The moral: vagueness in composition gives rise to vague identity.

I take it that here by ‘vagueness in composition’ you mean vagueness in mereological statements, and by ‘vague identity’ you mean vagueness in identity statements, right? In order to get from there to the “metaphysical” claims assumptions about determinacy in reference of the relevant expressions are required. The standard line has been to claim that ‘Kili’ is indeed indeterminate in reference. If I understand it right, the form of your suggestion is to claim that the Evans’ argument can be blocked granting determinacy in reference to ‘Kili’ by charging ‘K+’ ad ‘K–’ to be indeterminate in reference, is this right?

I couldn’t quite follow how the details go, sorry. You say:

Let’s suppose that the world contains two objects, Kili and Kili*. Kili* is just like Kili, except that determinately, Kili and Kili* differ over whether they contain Sparky.

I’m not sure how to understand ‘Kili*.’ I take it that we are assuming that ‘Kili’ determinately refers to one particular entity, that so does ‘Sparky’, and that nonetheless the statement ‘Sparky is part of Kili’ is indeterminate. But then it is not clear to me how things can be exactly like this also with respect to ‘Kili*’ but still be the case that determinately Kili and Kili* differ over whether they contain Sparky.

2. Robbie

Yeah: but the standard line is that the referential indeterminacy is due to semantic indecision (lack of conventions about how to use the words, for example). My thought (defended in the paper I cite) is that certain kinds of metaphysical indeterminacy can generate referential indeterminacy, and so block the Evans argument in the standard way.

I think I can put the thought in a neutral way. Let’s suppose that there’s no referential indeterminacy in “Kili” or “Sparky”. Then we can formulate the claimed metaphysical vagueness in composition in a way that you should find acceptable. Weatherson’s argument gets us to another claim—it’s indeterminate whether Kili=K+—which configures exactly the same “determinacy” operator. And the key point is that if “K+” is referentially indeterminate, this is quite consistent, in particular, the Evans argument fails.

I suppose, using your way of talking, you could summarize the point like this: metaphysically indeterminate composition-facts entails indeterminate identity statements, but not metaphysically indeterminate identity.

On your query about the argument. Think of the following sort of situation: it’s indeterminate whether particle x is spin-up or spin-down, and indeterminate whether particle y is spin-down or spin-up, but determinate that they have opposite spins. That is exactly the sort of situation that a logic of “determinacy” should allow us to formulate. And something analogous is in play here: it’s indeterminate whether Kili contains Sparky, and indeterminate whether Kili* contains Sparky, but determinate that exactly one of Kili and Kili* contains Sparky.

On reflection, maybe some more setup was needed to explain how I’m thinking of these things!

3. Dan López de Sa

Thanks for that, it really helps me!

… metaphysically indeterminate composition-facts entails indeterminate identity statements, but not metaphysically indeterminate identity.

I see now better the structure: (i) assuming ‘Kili’ (as well as ‘Sparky’ and ‘is a part of’) is determinate in reference, vagueness in the relevant mereological statements suffice for the corresponding metaphysically indeterminate facts; (ii) Weatherson’s argument gets one from vagueness in mereological statements to vagueness in identity statements; but (iii) the argument from there to metaphysically indeterminate identity is blocked via the indeterminacy in reference of ‘K+’ and ‘K-’.

As to the latter, I still have some doubts—probably connected to the further setup you mention and to the last thought in the original post concerning the worry that “the “precise” objects like K+ and K- exist, and perhaps we could still cause trouble.” Assume that ‘Kili’ determinately refers to a particular entity which is such that it is indeterminate whether Sparky is one of its parts (and for simplicity assume also that it is not like this for all entities distinct from Sparky). Then it seems also that the following entities exist:

(+) Kili plus Sparky

(-) Kili minus Sparky

where ‘plus’ and ‘minus’ stand for the relevant mereological operations. But if they do, then we can introduce ‘K+’ and ‘K-’ to determinately refer to these (determinately different) entities. Am I wrong? Or is the though that facts about the existence of Kili* prevent the entities in (+) or (-) to exist?

4. Robbie

Yeah. I’m a bit conflicted about how to describe these things. “indeterminate identity in virtue of metaphysical indeterminacy” is the best neutral phrase I could come up with. (The intuitive point is the reference relation has two ends: so referential indeterminacy might either be due to us not settling things enough, or the world not settling things enough).
To figure what’s going on with your definitions, let’s consider a minimal case. Suppose we have three particles, a and b and c, and suppose there exists an object Z, which has a and b as parts, and it is indeterminate whether it has c as a part; and Z*, which has a and b as parts, and it is indeterminate whether it has c as a part (Z and Z* are related as Kili and Kili* are). Also supopse there are fusions X=b+c, Y=a+c.

Then I think the mereological principle of universal fusion is satisfied. Question is: is the whole of mereology satisfied? If so, what’s gone wrong when we try to write down names like “Z+c” and the like?

My temptation is to think that the axioms are determinately satisfied, but that the phrase “Z+c” is referentially indeterminate between Z and Z*. It refers to whichever includes c, but it is indeterminate which object satisfies that descriptive condition. And that’s effectively the diagnosis of what’s going on with the more complex case like “K+”.

5. Dan López de Sa

Exactly, I think I can now sharpen my worry. Given that both ‘Z’ and ‘c’ determinately refer to two things, then (given the relevant background mereological assumptions) there is one entity that is a sum of the two, and we can introduce a name that determinately refers to it, say ‘Z+c.’ The case is such that it is indeterminate whether Z contains c. Hence, as you say, it is indeterminate whether ‘Z+c’ refers to Z. But this is not because ‘Z+c’ is indeterminate in reference: it is determinate that ‘Z+c’ refers to Z+c. It is just indeterminate whether Z+c is Z—where this is a metaphysically indeterminate identity of the familiar (problematic?) kind.

Mutatis mutandis, it seems to me, for your original case. Assume that it’s indeterminate whether Kili contains Sparky, and indeterminate whether Kili* contains Sparky, but determinate that exactly one of Kili and Kili* contains Sparky. You said:

What of our terms “K+” and “K-“? Well, if Kili contains Sparky, then “K+” denotes Kili. But if it doesn’t, then “K+” denotes Kili*. Mutatis Mutandis for “K-“. Since it is is indeterminate which option obtains, “K+” and “K-” are referentially indeterminate…

I don’t see how the last step follows. Insofar as I can see, we have every reason to suppose (given the background) that the stipulations on ‘K+’ and ‘K-’ secure determinate reference. True, if Kili contains Sparky, then K+ is Killi, and as a result ‘K+’ denotes Kili; and if it doesn’t, then K+ is Kili*, and as a result ‘K+’ denotes Kili*. Mutatis mutandis for ‘K-.’ But even if it is indeterminate which option obtains, ‘K+’ and ‘K-’ need not be referentially indeterminate.

The only way I can see of preventing this line of thought would be to hold that there are two (determinately specifiable) entities, to hold that they compose, but to claim that one can not introduce an expression that determinately refers to the relevant sum. But this is an option which doesn’t seem very plausible to me. Is the line you are prepared to take or I just got something wrong?

6. Robbie

(1) Determinately, there is one entity that is the sum of Z and c.

(2) there is one entity that is determinately a sum of Z and c.

I think you need (2) to make your point (otherwise it’s still open that “Z+c” is indeterminate in reference).

I think, in your final paragraph, the problem is that it’s determinate that Z and c compose, but there’s no entity such that it’s determinate that they compose it. So the expression “the relevant sum” is referentially indeterminate.

It is a bit mind-bending, but I think it’s all perfectly coherent.

7. Robbie

Of course, a natural observation at this point is that the vague objects aren’t helping very much in resolving the problem of the many, since we’ve got exactly as many of them as we would have of the precise ones! But I think the case is intrinsically interesting (and I favour a different solution to the POM anyway).

8. Dan López de Sa

Oh, yes, you’re totally right: I was assuming that if you have two (determinately specifiable) things, then there is something that is determinately a sum of the two. (Maybe it is ultimately coherent to deny this, but it looks to me to be a bit of a cost.)

Very interesting discussion, I learned a lot, many many thanks for it!