In their Vague Parts and Vague Identity, 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.

- Assume, for reductio, that s is part of K.
- Then K and K+ coincide.
- So K and K+ are identical.
- This contradicts our earlier result that K and K+ are not identical.
- So s is not part of K.
- Hence K and K- coincide.
- So K and K- are identical.
- 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.

I’m afraid I haven’t yet read any of the papers you mentioned – but they sound really interesting! Am I right in thinking the principle that seems to be doing the work is the principle that, determinately, two things are identical iff they have the same overlappers. So it’s vague whether x and y have the same overlappers iff it’s vague whether x=y – I guess this shows it can never be vague whether two things ovelap exactly the same things (although I don’t see how this rules out vague parthood altogether.)

“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.

I didn’t quite follow this argument. I assume you generated c with unrestricted fusion? It seems to me to be completely compatible with unrestricted fusion as it is normally stated

that (determinately) there is a fusion of a and b but indeterminate which thing it is (b or c.) (Obviously it would also be indeterminate whether this fusion was a vague existent or not.)

“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.”

If you’re thinking of c as some kind of rigid designator (over precisifications), if a doesn’t exist neither does c (although the description “the fusion of a and b” determinately refers and denotes b on some precisifications and c on others.)

I guess it’s more complicated than I thought to specify the reference of c. Perhaps we can stipulate that it’s the dfusion of a and b where the principle of unrestricted dfusion is: for any phi there is a x such that

determinatelyfor any z, z overlaps x iff z overlaps a phi. If you’ve dubbed c this way then it’s vague whether c refers.Then I guess it gets even more confusing if you can have non-existent parts. It seems you can hold onto unrestricted fusion and dfusion with vague existence without any vague parthood if you allow non-existent parts. If you don’t allow it then your argument rules out the principle of unrestricted dfusion although you can keep unrestricted fusion. Or do I get this wrong?

Hi Brian,

Thanks for the shout out! And interesting thoughts. Just a quick query—-was there a particular NC logic you were thinking of running the argument above in? The use of reductio might be dodgy in some of the non-classical logics that are flying around for indeterminacy, for example——and be regarded as tantamount to LEM. (Take a logic with double negation elim and de Morgan. Assume for reductio ~(Av~A). Get a contradiction by de Morgan. Apply reductio to get ~~(Av~A). DNE gives Av~A.) So while your argument looks, say, intuitionistically ok, I was worried that various many-valued logics aren’t going to sustain it.

Sorry if I’m missing something crucial…

FWIW, I have worries similar to Andrew’s about the argument against vague existence.

The worry I had that is that it’s open to the friend of indeterminate existence to say that “c” is indeterminate in reference—-indeterminate between b (if a doesn’t exist) and the fusion of b and a (if a exists). Indeterminacy over what exists generates referential indeterminacy, due to the way the term in question is introduced.

Given this, I’m not sure about the following move: “b and c are distinct, since one but not the other determinately is wholly distinct from a”. It looks to me from the definitions, you get the wide-scoped claims: D[a is not part of b]; ~D[a is not part of c]. But if “c” is indeterminate in reference, then applying Leibniz law on this basis to argue for the distinctness of b and c is dodgy—-like applying in a context containing a non-rigid designators under a modal operator.

This is essentially the same kind of worry that, in the joint paper, motivates the move to the narrow-scoped-determinate formulation of universal fusion, else you get similar issues in the argument against vague parts. (I talk about this sort of stuff some more in my PQ paper “multiple actualities).

Again, I might have misread the intended form of the argument—-apologies if so…

Uck. Sorry for the typos.