In their “Vague Parts and Vague Identity”:http://www.personal.leeds.ac.uk/~phlejb/vague%20parts%20and%20vague%20identity.pdf, Elizabeth Barnes and Robbie Williams do some excellent work cleaning up, and noting the missing premises in, an argument of mine against vague parthood. In particular, they note that the argument I run could be improved in two ways.

• Let K be an object with s as a vague part. Let K- be the metaphysical difference between K and s, and K+ the metaphysical fusion of K and s. I supposed that K couldn’t be identical to K+ or to K-. They note that this should be argued for, not assumed, and offer an argument.
• They note that the Evansian argument I offer to complete the reductio makes a rather strong mereological assumption. It requires not just that for any things, it is determinate that they have a fusion, but that for any things, there is something that determinately is their fusion.

I agree with both points. I hadn’t noticed the potential scope ambiguity that the second observation turns on. Now that I have, I think I accept both disambiguations. But that is certainly a necessary move in the argument, and one that could be rejected.

Given that extra assumption, I think I was intending the argument that K is distinct from K+ and K- in a slightly different way. Here’s how I was thinking of K+ and K-.

• K+ is the determinate fusion of K and s.
• K- is the determinate mereological difference between K and s.

Now the argument that K is not identical to K+ or K- is a simple application of Leibniz’s Law. Since s is determinately a part of K+, determinately not a part of K-, and vaguely a part of K, it follows immediately that they are not identical.

Note that once we do things this way, we can get a reductio without appeal to any distinctively classical principles, as follows.

1. Assume, for reductio, that s is part of K.
2. Then K and K+ coincide.
3. So K and K+ are identical.
4. This contradicts our earlier result that K and K+ are not identical.
5. So s is not part of K.
6. Hence K and K- coincide.
7. So K and K- are identical.
8. This contradicts our earlier result that K and K- are not identical.

So we’ve generated a contradiction from just the assumption that s is a vague part of K, and the definitions of K+ and K-. Since Elizabeth and Robbie don’t believe the strong mereological principles that are used in the definition of K+ and K-, they won’t be overly moved by this reasoning. The target of this argument instead is someone who reads their paper and thinks that dropping classical logic, while accepting the existence of determinate fusions and differences, is consistent with vague identity.

Having said all this, I’m starting to wonder how strong an argument it is against metaphysical vagueness. I’m convinced by the Evans argument against vague identity. And I think a cleaned up version of the above argument shows that we cannot have vague parthood. A similar argument shows that there cannot be vague existence, as follows.

If a vaguely exists, let b be any distinct object, and c be the object that determinately has parts b and, iff it exists, a. Then b and c are distinct, since one but not the other determinately is wholly distinct from a. But if a doesn’t exist, then b and c coincide, and so are identical. Hence a doesn’t exist, and since it was arbitrary, all ‘vaguely existing’ objects don’t exist.

But that doesn’t cover all of the ways we might have metaphysical vagueness. In particular, it doesn’t rule out vague events. None of these arguments rules out the idea that there could be, say, a traffic jam for which it was vague whether a particular car was part of that traffic jam. If we have a plenitudinous ontology of events, then I imagine similar arguments to the ones discussed so far will lead to contradiction. (Let Herbie be the car that’s vaguely in the traffic jam T. Let T- be the event consisting of the traffic jam without Herbie, etc.) But I think the defender of vague events could simply deny that there are principles of event composition that are anywhere near as strong as the principles of classical mereology. (So in the Herbie example, there won’t be an event T-.)

I hope to write more of this in the near future. But for now I just want to note that there’s more to metaphysics than mereology, and even with a precise mereology, we might have some vagueness somewhere in the metaphysics.