## Tuesday, March 17, 2009

### From Borderline Tables to Count Indeterminacy

Imagine assembling a two-piece table: the top (T) is being affixed to the base (B). There are points in the assembly process at which T and B are just beginning to be fastened together and at which, intuitively, it is vague whether they compose anything. Ted Sider’s (riff on Lewis’s) argument from vagueness purports to show that (despite appearances) there can’t be borderline cases of composition. Here’s the argument: If it could be indeterminate whether some things compose something, then it could be indeterminate how many things there are (e.g., whether there are just two things – T and B – or three things – T, B, and a table). But there can’t be count indeterminacy. So there can’t be borderline composition. And (moreover) if there can’t be borderline composition, then composition must be unrestricted.

There are ways of blocking the argument, but they’re pretty nasty (e.g., nihilism, sharp cut-offs, ontic vagueness, relativism). So many prefer just to accept the conclusion, that composition is unrestricted (“universalism”). Some don’t even think they’re biting a bullet here because (for one reason or another) they think that universalism is innocuous.

I want to convince you that universalists aren’t out of the woods yet. Let’s grant that T and B definitely compose *something*: a mereological fusion (or MF for short). But surely T and B are, at the very least, a borderline case of composing a *table*. Even universalists should admit that. But here’s the rub. No table is identical to any MF. Tables can survive the annihilation of certain of their parts; MFs can’t. So if T and B don’t compose a table, there are three things: T, B, and the MF. If they do compose a table, there are four things: T, B, the MF, and the table. Since they’re a borderline case of composing a table, they’re a borderline case of composing something other than an MF. In which case it’s indeterminate whether there are three or four things. In which case there’s count indeterminacy. In which case the argument from vagueness fails.

Friends of the argument from vagueness need to find some way to block this argument from borderline tables to count indeterminacy. And they need to find a way of doing this that doesn’t undercut the argument from vagueness. As far as I can tell, friends of the argument from vagueness have two options. Both involve resisting the move from T and B’s being a borderline case of composing a table to their being a borderline case of composing something other than an MF. And both involve finding something that definitely exists and is definitely composed of T and B and that itself is a borderline case of being a table.

Option #1: They definitely don't compose anything other than an MF. Here the idea is that there is definitely only one thing composed of T and B – namely, the MF – and the MF itself is a borderline case of being a table. To get this response to work, you’re going to need some way of defusing the sort of Leibniz’s Law argument I gave above for the distinctness of MFs and tables. Here are some of the tasty options: you can deny that tables can survive the loss of parts, you can say (a la Burke) that the original MF ceases to exist when its parts come to be arranged tablewise, or (like Lewis and Sider) you can go for a counterpart-theoretic account.

Option#2: They definitely do compose something other than an MF. Here the idea is that there is a further thing composed of T and B which (unlike the MF) has a “tablish” modal profile, but it’s nevertheless indeterminate whether this further thing is a table, e.g., because it’s indeterminate whether its parts are sufficiently stuck together to count as a table. There are unprincipled ways of taking this line, e.g., by saying that exactly one modally table-like entity conveniently springs into existence as soon as the grey area begins, but let’s set those aside. The only principled way of taking this line (as far as I can tell) is to accept bazillionthingism (a.k.a., plenitude, explosivism, absolutism), on which there are a bazillion things, with different modal profiles, occupying the region that’s filled by T and B. In that case, there isn’t count indeterminacy. There are exactly a bazillion and two things: T, B, and the bazillion things composed of them.

So friends of the argument from vagueness are going to get saddled with some sort of non-innocuous commitment: either bazillionthingism or else one of the revisionary packages needed to block the Leibniz’s Law arguments. Some (e.g., Lewis and Sider) have already chosen their poison. But nobody escapes unscathed.

1. My own inclinations are with Lewis and Sider here, but there seems to be at least one option you've neglected. For some people believe in constitution, and while they might think composition is mereological, they probably don't think constitution is. So these people would accept option #1 in one sense—because they reckon T and B compose just one thing, a mereological fusion T+B. But they think the MF T+B, when T and B are properly arranged, will constitute a table without being identical to a table, because 'constitution is not identity'. So they also accept option #2 in a sense.

Just before T and B are properly arranged, it is borderline whether T+B constitutes a table. Whether constitution theorists can say something better than 'there are many things which exist and are constituted by the MF T+B and some of which are sort of good candidates to be a table because they have the right modal profile, and maybe one or more of which are perfect candidates' (I take it this is bazillionism in their mouths) is not clear to me. Maybe they can say there is one thing constituted by the MF which is a borderline case of a table? (This is the option dubbed 'unprincipled', but I'm thinking it is maybe more principled for the constitution-theorist?) But it does seem like it might open other avenues than those provided by the options already on the table, no?

2. Hey Dan,

Fun post! Two related points.

First, I'm not sure I understand how you're using 'mereological fusion'. You say:

"No table is identical to any MF. Tables can survive the annihilation of certain of their parts; MFs can’t."

I would have thought that to say that a thing, y, is a mereological fusion is just to say that for some things, the xs, y is a fusion of the xs. But to say that y is a fusion of the xs is just to say that y is composed of the xs (i.e. that each of the xs is part of y, and that no part of y fails to overlap at least one of the xs). But if so, then I am a mereological fusion--I'm certainly composed of some things. Nonetheless, I can survive annihilation of some of my parts. So at least some mereological fusions can survive the annihilation of some of their parts. Moreover, notice that on this (standard) understanding of 'mereological fusion', *every* table is trivially a mereological fusion--every table is certainly composed of some things. And if, as you say, tables can survive annihilation of certain of their parts, then so can at least some fusions (i.e. those that happen to be tables).

So anyways, I'm not sure how to understand your claim that no table is a mereological fusion, or your claim that mereological essentialism is true of mereological fusions. Nor do I see how would they provide the basis for an ad-hominem against the universalist: why would she be committed to these claims?

Second, from the point above I'm not clear why Option #1 would be unpalatable: there's exactly one thing composed of T and B, such that it is sometimes indeterminate whether that thing satisfies a certain condition (i.e. being a table), depending on the distance and position of T and B relative to one another (in my lingo, that thing is an expansion of T and B). If tables are fusions, tables can lose some of their parts, and hence some fusions can lose some of their parts, then I don't see why you need to go for any options you mention in order to keep Leibniz's Law.

(By the way, Option #1 looks even more plausible when you think of it alongside other cases of vagueness. Consider Michael Jordan, who is now both bald and tall. There has always been a single person, Michael Jordan, even if at times it was indeterminate whether that person satisfied certain conditions, i.e. being bald and being tall).

3. Hi Antony,

I think you’re right that constitutionalism is going to lead friends of the argument from vagueness either to bazillionism or else (what I’ve called) an unprincipled version of option #2. (Though I’m not sure I see how it can be a version of option #1: even though constitution isn’t mereological, it’s still the case that if the xs compose y, and y constitutes z, then y and z will have the same parts, in which case z is composed of the xs as well.)

Maybe you’re right that I was too quick to dismiss all nonbazillionist treatments as unprincipled. Here’s a principled(-enough) constitutionalist view. Let the Ks be the kinds of constituted things the constitutionalist recognizes; let’s suppose that there are fewer than a bazillion of them; and let’s suppose that there are nonarbitrary reasons for recognizing the Ks and not the plenitude of other kinds. Here’s a principle: some xs compose something other than an MF just in case, for some K, the xs are nearly as good as K-ish. Here’s what it is for some xs to be nearly as good as K-ish: it’s for the xs to have everything that it takes to compose a K *with the one possible exception*, namely, that the physical arrangement of the xs doesn’t have to meet the conditions for their composing a K – it’s enough that it not be determinately false that they’re arranged the way they’d need to be in order to compose a K.

That’s principled enough for me. My worry now, though, is that this strategy will undercut the argument from vagueness. In order to avoid count indeterminacy, there has to be a determinate point at which T and B begin to compose something other than an MF. So there are sharp cut-offs with respect to whether they compose something other than an MF. But (I say) if you’re willing to bite that bullet, then you’d have no grounds for endorsing Sider’s premise that you there can’t be sharp cut-offs with respect to whether some things compose anything at all. (Perhaps this is best treated as a challenge: to provide some reason for allowing the one sort of sharp cut-off but not the other.)

4. Hi Raul,

Yes, this was pretty unclear. What I had in mind by “mereological fusions” were composites that have their parts essentially and exist as long as their parts do. Maybe I should have called them “classical MFs”. A tacit premise of my argument was that whenever there are some things, they compose a classical MF; and I assumed that universalists would grant this.

Now you ask: why should the universalist grant that? Universalism just says that some things always compose *something*, but it’s silent with regard to what sorts of things these composites are. That’s right. Call the view that some things always compose a classical MF “Classical Universalism”. So one way for universalists to resist my argument is to reject Classical Universalism.

But what are the alternatives? Here’s one. Before T and B come to be arranged tablewise, they compose exactly one thing (call it O1). By the time they have come to be arranged tablewise, O1 has ceased to exist and a new composite has come into existence (call it O2). And in the grey area, too, there’s exactly one composite but (on pain of sharp cut-offs) it’s indeterminate which of O1 and O2 is there at that time. But it’s determinate that at most one of the two is there.

This is the sort of Burke-style view that I was referring to under option #1. And it’s a pretty nasty view. For what sort of thing is O1? Something that can survive having its parts (T and B) taken to opposite sides of the room, and can survive having T balanced on top of B, but can’t survive having T screwed onto B? That’s weird. Or is it something that can't survive any change in the position of its parts? That kind of positional essentialism, in the words of Peter van Inwagen, is "infinitely worse, and never has the phrase 'infinitely worse' been used more appropriately"!

5. Dan,

Let me see if I got it right. Let's say that y is a *weak* fusion of the xs iff y is composed of the xs (this is what I was taking to be the standard defnition of 'fusion'). And let's say that y is a *strong* fusion of the xs iff for every world at which y exists, y is composed of the xs at that world, and for every world at which the xs exist, y is composed of the xs at that world (this is what I think you mean by 'classical fusion'). When you say in your response to my comment above that you "assumed that universalists would grant this", are you saying that you assumed that universalists would accept that any things have a strong fusion, and not only a weak one?

It'd strike me as odd if universalists (and Lewis and Sider in particular) actually accepted that, given other theses they tend to hold. For instance, many universalists believe that composition is unique (i.e. that weak fusions are unique), as well as that there are no co-located material things. But if a universalist believed that any things have a strong fusion, then she'd have to both reject uniqueness of weak fusions and accept co-located objects. For consider my body. Everyone but the nihilist would agree that my body is a weak fusion of my extremities, torso, and head, since it is composed of them. Now suppose that there is a strong fusion, x, of my extremities, torso, and head; call it 'x'. Like my body, x would be a weak fusion of my extremities, torso, and head (by the definition of weak and strong fusions). But x would not identical to my body (by Leibniz's Law, given the difference in their modal profiles). Moreover, x would be co-located with my body (given inheritance of location).

So anyways, I'm not sure that most universalists would actually accept the strong view you're assuming they would accept. I doubt that Lewis and Sider would grant you that in addition to my body, there is a material thing co-located with it and composed of the same parts.

In any case, independently of whether most universalists actually believe that any things have a strong fusion, consider just the view that any things have a weak fusion, coupled with the claim that some weak fusions can survive annihilation of some their parts (i.e. the weak fusions that happen to be tables, dogs, etc). Such a view has all the unintuitive consequences that moderates about composition dislike, e.g. that there's something composed of my nose and the Eiffel tower. Nonetheless, for the reasons I suggested in the comment above, such a view can go for Option #1 and leave Leibniz's Law intact without having to appeal to any of the unappealing strategies you mention. So I think that this is a view that moderates about composition would like to reject, but that is immune to your argument.

6. Raul,

I guess I thought most universalists would reject extensionality. Here’s some evidence: universalism is the most common view of composition, and pluralism is the most common response to the puzzles of material coincidence. Here’s more evidence: nonmetaphysicians frequently treat it as a mark of good philosophical sense when I tell them I think the statue and clay are distinct, but they get all bent out of shape when I try to resist universalism (and start raising worries about slippery slopes, imaginary linguistic communities, etc.).

Also, I think there’s tremendous pressure for universalists to accept classical universalism. Think about van Inwagen’s dilemma in Material Beings chapter 8. Let the As be the simples that now compose me. The As composed something a hundred years ago. What sort of thing was that? Dilemma: (i) it’s something that can survive arbitrary rearrangements of its parts, or (ii) it’s something that can’t survive *any* rearrangement of its parts. But (says PVI) the former leads to distinct coincident objects and the second leads to “positional essentialism” (which is “infinitely worse” than mereological essentialism). I would add a third horn, which came up in my reply to Antony: (iii) it’s something that can survive any rearrangements of its parts except those that make it interesting (e.g., tablewise arrangements, personwise arrangements).

I always assumed that Lewis, et. al., would go for (i) – and, thereby, accept classical universalism – but deny that this leads to distinct coincidents. “But it and my body coincide now, and it but not my body can survive arbitrary rearrangements of parts, so aren’t they distinct by Leibniz’s Law?” Not quite, say these counterpart theorists. Indeed, the strong fusion can survive these changes and, indeed, my body can't survive them. But there’s a concealed equivocation in the Leibniz’s Law argument, an equivocation between two different counterpart relations. [[And, because these counterpart theorists are lurking around, I’m reluctant to follow you in building commitment to transworld individuals into the characterization of “strong fusions” or classical universalism.]]

7. Dan and Raul,

I like this discussion. But, I wonder if some of the same points that Dan makes can be made with weaker assumptions. Let's consider the things that compose the table now, the table bits. At one time those things were widely scattered across the universe and they will be widely scattered again in the future. Now suppose Universalism is true. Then, those things composed something long ago when they were scattered. Either the thing they composed is the same as the table they compose now or it is not. If it is, then, surprisingly, nothing comes into existence when we put the table together. But, something does come into existence when we put the table together. So, whatever the table bits composed long ago is distinct from the table. But, if the thing the table bits composed long ago is distinct from the table, then either that thing continues to exist along with the table (in which case we have the counting problem that Dan introduced) or it went out of existence when the table came into existence (in which case we have another kind of vagueness problem). Either way the Universalist is stuck with some problems he might not have realized he had.

Now, I don't think anything said here assumes that there are strong fusions. But, it still gets the Universalist into the kind of trouble that I think Dan was hoping to get him into.

-Joshua

8. Hi Joshua,

I suspect that whether the kind of case you describe makes trouble for the universalist will depend on her views on persistence, change, and whether things have properties and stand in relations simpliciter or relative to times. (And I suspect that it would not make trouble for her if she were a perdurantist and thought that things change by having different temporal parts at different times, which have properties and stand in relations simpliciter. If I'm right about this, then the kind of case you consider would be at best problematic for certain combinations of views, not for universalism per se.) But let me suggest something in the spirit of Option #1 in response to this kind of case, while trying to remain neutral on issues about persistence, change, etc.

Consider the following picture: T and B compose exactly one thing, and it's always the same thing, no matter how apart T and B may be scattered. And depending on the distance and position of T and B relative to one another, that thing sometimes determinately falls under the extension of 'is a table', sometimes it determinately doesn't, and sometimes it is indeterminate whether it does. What's wrong with this picture? You are probably thinking: "on this picture, we don't get something new--something we didn't already have--when we put T and B together. But we do get something new--something we didn't have before--when we put T and B together." However, notice that on the picture I'm suggesting there were no tables before T and B were attached together: the extension of 'is a table' was empty. And after T and B were attached together, we did have a table: the extension of 'is a table' was no longer empty. So, on this picture, we *do* get something new--something we didn't already have--when we put T and B together, i.e. a table. Of course, we don't get something new in the range of our unrestricted quantifiers; that is, we don't get something we didn't already have in our domain of quantification, irrespective of whatever predicates it may fall under. But that doesn't seem to be required in order to accommodate the intuition that we get something we didn't already have when we attach T and B together. All that seems to be required is that we get a new table. And for that it is sufficient that we get something new in the extension of 'is a table'. And the picture I'm suggesting does allow for that.

9. Hi Dan,

I'm having a hard time seeing how you intend to direct this argument against counterpart theorists like Lewis and Sider. You seem to be thinking that adopting counterpart theory is a cost that we should avoid if we can. But why should Lewis see it this way? Counterpart theory is just his view, and he has independent motivations for holding it. If it can solve this puzzle, that's another consideration in its favor. So the arguments you're providing just work in Lewis's favor: count indeterminacy is bad, and all the other options for avoiding it are bad. Only Lewis escapes unscathed.

10. Hi Dan,

You say,

"So if T and B don’t compose a table, there are three things: T, B, and the MF. If they do compose a table, there are four things: T, B, the MF, and the table. Since they’re a borderline case of composing a table, they’re a borderline case of composing something other than an MF. In which case it’s indeterminate whether there are three or four things. In which case there’s count indeterminacy. In which case the argument from vagueness fails."

I'm a bit confused by this. As you present the argument from vagueness, all that it is intended to show is that there can't be a borderline case of composition, or else there would be an undesirable count indeterminacy. Isn't this exactly what you've shown?

You seem to be suggesting that any set of material objects can compose at least kinds of things, a mereological fusion and a persisting object. You seem to take your argument as showing: if you're not a universalist about both kinds of composed item, then you'll face count indeterminacy.

I understand that both options might be undesirable, but I don't see how this is to be used to rebut the argument. Do you want to make room for the possibility of count indeterminacy?

11. UT Austin Represent!

Derek: I wouldn’t say that the argument is directed against counterpart theorists like Lewis and Sider. It’s directed against those people who both accept universalism and who go for a constitutionalist response to the puzzles of material coincidence. Alternatively, it’s meant to show that these awful ways of resisting the Leibniz’s Law arguments (like counterpart theory) aren’t optional for friends of the argument from vagueness.

Bryan: Right, I don't take myself to have identified any flaw in the argument from vagueness. The point is just that even some universalists (not including Lewis and Sider) should be interested in blocking the argument from vagueness and, in particular, in the various ways of coming to terms with count indeterminacy.

Raul: Your view is more revisionary than you let on. Consider my body. You’re saying that it will still be around after the bomb drops and my atoms go flying across the universe. I don’t find your decrazification strategy (“we’re really only having the intuition is that it’s no longer *a body*, not that it no longer exists”) any more plausible that van Inwagen’s (“we’re really only having the intuition that there are atoms arranged tablewise, not that there are tables”). Better, I think, just to admit that you’ve endorsed a deeply counterintuitive Option-1-style response.

12. Forgive a stranger and amateur, but I read "But there can’t be count indeterminacy." and wonder "Why not?".

I read your scenario straightforwardly as "there's a T, and a B, and the MF TB, and sometimes a table and sometimes not a table and sometimes it's vague whether there is a table or not". And I don't find myself recoiling from anything there.

Who is against count indeterminacy, and where can I find them opposing it?

13. Hi Craig,

The short (and not quite accurate) answer is that if you accept count indeterminacy, you’re going to have to reject the linguistic theory of vagueness and, moreover, you get stuck with the worst kind of ontic vagueness: vague existence. Here’s a sketch of the argument against count indeterminacy:

(1) There’s count indeterminacy only if some numerical sentence lacks a determinate truth value.
(2) A numerical sentence lacks a determinate truth value only if some expression in it is vague.
(3) An expression is vague only if it has multiple precisifications.
(4) No expression in any numerical sentence can have multiple precisifications.
(5) So there can’t be count indeterminacy

A numerical sentence is a sentence of the following sort which says that there are exactly n concrete objects, for some number n (in this case, two): ‘∃x∃y(Cx & Cy & x≠y & ∀z(Cz → (x=z v y=z)))’.

To give up (3) is, arguably, to give up the linguistic theory of vagueness. There are other ways of blocking it, e.g., you might deny (2) on the grounds that numerical sentences lack determinate truth values for reasons having nothing to do with vagueness. I take it that’s what Amie Thomasson would say. Or you might deny (4), which (arguably) is going to commit you to some form of relativism. Or you might go for ontic vagueness and vague existence but reject (2) rather than (3): no expression needs to be vague in order for a sentence to lack a determinate truth value as a result of vagueness. That's what Katherine Hawley would say. Me too.

Check out Sider’s book *Four-Dimensionalism*, pp. 127-130, and his "Against Vague Existence". Also Dan Lopez de Sa, "Is 'Everything' Precise". And I think Merricks gives count indeterminacy some nasty looks in his "Composition and Vagueness".

14. Hi Dan,

I know you were just sketching the argument, but there's a question about how to categorize the options for denying premises that's been bugging me.

The status of (3) is the thing I'm interested in. I've never been too convinced it carves a distinction between linguistic/non-linguistic analyses of vagueness. I do see that some paradigm linguistic views endorse (3). E.g. "semantic indecision" views seem exactly spelled out in terms of multiple classical interpretations of our language compatible with all the meaning-fixing facts.

But e.g. I reckon that the best way to understand what metaphysical/ontic vagueness would be will end up endorsing (3) as well (My own view would be that someone who thinks it's metaphysically indeterminate how many things there are should defend count-indeterminacy by denying (4), though I don't think this makes them a "relativist". Rich Woodward has a nice paper on this.)

I also don't see off the bat why non-precisificational ideas about indeterminacy like degree-theory and non-classical truth value gap approaches should automatically count as "ontic" as opposed to "linguistic". To qualify this: I don't see the reason why *just* adopting one of these views makes you not a linguistic theorist. (I can imagine thinking that e.g. the problem of the many seems way more threatening when you've got something like Tye or Machina's semantic machinery to play with rather than supervaluations---but let's set that aside).

Here's the general picture of linguistic indeterminacy for a non-classical truth value gapper I'm thinking of: reality can be described classically using Lewisian/Siderian perfectly natural vocab. But then there are these people running around with patterns of assent and dissent involving the term "bald". E.g. maybe they dissent from both "Harry is bald" and "Harry is not bald", for middling hairy Harry, without endorsing either. That makes a prima facie case for truth-value gaps; but so far, the supervaluationist version of the semantic indecision view can agree. However if our hypothetical community also takes the same attitude to "Harry is bald or he isn't"---dissenting from it, and its negation, without endorsing either---then that's something that fits with non-classical gaps better than supervaluational gaps. If someone thought that this was the way *English* works, and also that the prima facie case doesn't get trumped by other considerations, then I'd think of them as having a linguistic theory of vagueness.

Now, I do think there's something to the thought that if we think that count-statements are indeterminate, with quantifiers unrestricted, something in this sketch goes wrong. In particular, the starting point of a world that's classically describable in the most fundamental terms seems a non-starter. But that worry doesn't seem to have much to do with precisifications or their absence. It's interesting that the Sider argument, with its reliance on a premise about precisifications, doesn't bring out what goes wrong for a non-classical gappy theorist.

(I guess what I'd be sympathetic to is the thought that if you deny (3), and looking at the options out there, it looks like you're going to not think that all classical tautologies are (perfectly) true. And that might be enough of a "bad thing" for many to grant that premise---I think the issue is more about the logic/shape of semantics than the *kind* of indeterminacy in play (ontic vs. linguistic). But even claims like this are dangerous---Weatherson's "truer" theory is many-valued rather than precisificational, but the way he sets it up he gets classical logic out).

In short: the argument as stated seems to me worth taking seriously as a challenge to anyone who likes count indeterminacy and also likes precisificational theories of indeterminacy. But the latter distinction I think cross-cuts the linguistic/ontic issue.

15. Hi Robbie,

If I understand your proposal, I’ve toyed with a very similar idea (maybe the same idea) in another blog post: http://rationalhunter.typepad.com/close_range/2008/06/borderline-comp.html. The idea is to deny premise (2): the numerical sentences do lack a polar truth value as a result of vagueness, but there’s no expression in particular that’s to blame. I think this is something that Horgan and Potrc might want to say. They’ll presumably want the relevant numerical sentences (uttered outside the ontology room) to come out gappy, but they think reality can be described classically.

I suggest in the post that a crucial part of this strategy is denying that there’s any such thing as the proposition that ∃x∃y(Cx & Cy & x≠y & ∀z(Cz → (x=z v y=z))). If there were such a thing, there would be soritical indeterminacy that isn’t the result of its being indeterminate which things we’re talking about. I was assuming (following K Hawley) that that’s just what it is for there to be ontic indeterminacy. What do you think about that as a criterion? Do you think that a nonprecisificational view could be counted as a linguistic account if it allowed that not just sentences but also some propositions lacked a polar truth value?

Also, I’m curious to hear more about your preferred strategy, denying (4). Do you think the quantifiers have multiple precisifications?

16. Hi Dan,

Sorry for the slow reply---I should've checked back earlier!

The stuff you've written on this on Close Range is interesting---I do think that you're right to say that this debate looks different if we have certain kinds of anti-realist thoughts about non-sparse objects (which is what I take the heart of that proposal to be--e.g. following your response to Kenny).

I wasn't really thinking of anything that detailed in connection to the non-precisificational views. I had something a bit narrower and more boring in mind. Say Harry is a degree theorist (of the degree-functional kind). Harry doesn't have truck with "multiple precisifications" of borderline sentences. That's the sort of thing a supervaluationist would appeal to, and Harry is no supervaluationist. Rather, Harry thinks that borderline sentences are the ones that are true to an intermediate degree.

Qua degree theorist, Harry's happy with "vague properties" in a certain sense---just as abundant properties for a classicist can be modelled by functions from objects to 1 or 0, Harry appeals to functions from objects to the range [1,0], and thinks these count as "abundant properties"---he identifies them with baldness, tallness and the like. But that's not scary---after all, anyone who believes in sets and numbers believes in such things! The same goes for vague propositions. Classicists identifies propositions with functions from worlds to {1,0}, but Harry thinks that what plays the proposition role are maps from worlds to the range [1,0]. In each case Harry just thinks these entities (which all parties believe in) have a certain role as semantic values of expressions.

Harry furthermore agrees with Ted Sider about what reality contains (they'd agree on everything expressed in Ontologese, including e.g. universal composition and the like). Harry just happens to think that in the competition to find the most eligible interpretation fitting English speakers patterns of assent and dissent, a fuzzy interpretation does better than a classical or supervaluational interpretations.

My thought is (i) it's not clear that Harry doesn't have a semantic conception of vagueness; (ii) I really don't see any very significant sense in which he's buying into a "metaphysical" or "ontic" conception of vagueness. But he does reject (3)---so I'm thinking it's a little strange to think of rejecting (3) as a touchstone of buying into a scary "metaphysical" theory of indeterminacy.

Harry’s kind of indeterminacy is indeterminacy that persists even though there are quite definite semantic values for each expression in the relevant statements. So, re your suggested criterion, I guess it’d count as not appealing to “indeterminacy in what things we’re talking about”. But I'm inclined to resist---I don't a sense in which (as Katherine Hawley puts it) we’re here dealing with an indeterminacy whose “source” is non-linguistic.

Actually, as I read Katherine in her (really good!) paper “Vague existence”, what she says is that she’s argued elsewhere in favour of a particular semantic-indecision/precisificational account of linguistic vagueness. Given that, she feels she can characterize ontic vagueness as vagueness that isn’t epistemic and isn’t of the semantic indecision kind. Which seems fair enough (modulo concerns about exclusivity of the three options here). But it does depend on those arguments for knocking out non-precisificational linguistic theories working.

Oops, this has gone on too long. I wanted to say something about the deny (4) strategy---I'll put that in a separate comment.

17. Ok, on to precisificational "metaphysical" theories of indeterminacy.

Just a word about the general territory: precisificational accounts of metaphysical indeterminacy crop up from time to time (Akiba’s papers are the most explicit I know before the last few years). But my thinking about this was really kicked off from discussions with Elizabeth Barnes, whose dissertation looked at this stuff in depth (various papers are now coming out from it). Elizabeth and I have a joint paper setting out a view on how to fit certain things together here: http://www.personal.leeds.ac.uk/~phljrgw/wip/theoryofmetaphysicalindeterminacy.pdf

Here's a quick sketch of the view (pretending for a moment that I do think there’s metaphysical indeterminacy out there). Indeterminacy is something primitive and irreducible. Just as some people think we hit metaphysical rock-bottom when saying that such-and-such is possible (a certain kind of modalist) and some people think we hit bedrock when saying that such-and-such will occur (a certain kind of serious tenser), at bedrock, we just have to say that it's indeterminate whether or not A is part of B (or whatever your favourite example is). Slogan: "Indeterminacy" operators are part of the ideology of Ontologese.

So the metaphysics of indeterminacy is pretty short and sweet, and doesn't allude to precisifications at all. In fact, it’s pretty uninformative all around. However, the thought is that we can we can *use* the notion of indeterminacy to characterize a notion of precisification which proves useful (e.g. in spelling out the semantics and logic for a language containing “determinately”).

Let's suppose we can help ourselves to such a space of ersatz worlds (say, classically complete world-descriptions---each therefore taking a stance on whether or not A is part of B). The precisifications are those world-descriptions that don't determinately misdescribe reality. There can be multiple such precisifications---one that says that A is part of B, and one that says it isn't. It's then plausible to maintain that quite generally, it is indeterminate whether P iff there are precisifications on which P, and precisifications on which ~P. But there's nothing semantic or linguistic about the theory of vagueness we're shooting for here. So this is how I think metaphysical indeterminacy quite generally can be combined with precisifications.

I think vague existence raises some delicate issues, but the general approach will be the same as that just described. Indeterminacy is irreducible, and it might be that the rock-bottom description of reality has it that it's indeterminate whether there are exactly three things, or instead exactly four. That'll mean (if all goes smoothly) that there'll be multiple precisifications of a count-vague reality---one being an ersatz world that says there are exactly three things, another being one that says there are exactly four things.

But talking about precisifications of reality doesn’t answer the question about the precisifications of linguistic expressions such as quantifiers. To get that, we need to think about how semantics works out in this setting. Basically, the story is going to be bog-standard intensional semantics---expressions get extensions at worlds, and quantifiers have “domains” at worlds. Suppose w and u are our two precisifications of reality. There are the things that exist at w---let that set be X. There are the things that exist at u---let that set be Y. (X and Y are distinct, because, for example, they contain different numbers of entities). The natural thought about unrestricted “exists” is that its domain should be X at w and Y at u. So there’s a good sense in which X and Y are rival precisifications of the quantifier---they’re its extensions at rival precisifications.

Is this package somehow unstable? I don’t think it’s obviously so, but it does depend on some delicate issues in the metaphysics of modality. Specifically, it depends on what you think about erstaz possibilia, or how you intend to handle possiblia-talk. This is the focus of the paper I mentioned by Rich Woodward ---how the "domain problem" from the metaphysics of modality interacts with the vague existence issue at this point.