Category Archives: Vagueness

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…

Seduction and the sorites

Consider a red-yellow sorites sequence. Famously, “There is a red patch right next to a non-red patch” looks awful. But deny it (assert its negation) and you have the major premise of the sorites paradox. Plenty of theorists want to say that the “sharp boundary” sentence turns out to be true. They then face the burden of saying why it’s unacceptable. Call that the burden of explaining the seductiveness of the sorites paradox.

There is a fair amount of discussion of this kind of thing, and I have my own favourites. But in reading the literature, I keep coming across one particular line. It is to explain, on the basis of your favoured theory of vagueness, why we should think that each instance of the existential is false. So, theorists explain why we’d be confident that this isn’t a red patch next to a non-red patch, and that isn’t a red patch next to a non-red patch. And so on throughout the series.

However, there’s something suspicious about that strategy. Consider the situation that generates the preface “paradox”. Of each sentence I write in my book, I’m highly confident that it’s true. But on the basis of general considerations, I’m highly confident that there’s some sentence somewhere in it that’s false.

Suppose we accept that, of each pair in the sorites series, we have grounds for thinking that the red/non-red boundary is not located there. Still, we have excellent general grounds (e.g. a short logical proof, from obvious premises using apparently uncontroversial principles) for the truth of the existential claim that the boundary is located somewhere. So far, it looks like we should be something like the preface situation. We should be comfortable with the existential claim that there is a cut-off somewhere (/there is an error somewhere in the book) while disbelieving each instance, that the cut-off is here (/the error occurs in this sentence).

But, of coures, the situation with the sorites is strikingly not like this. Despite the apparently compelling general grounds we can give for the truth of the existential, most of us find it really hard to believe.

The trouble is this: the simple fact that each instance of an existential appears false does not in general lead us to believe that the existential itself is false (the preface situation illustrates this). So there must be something special about the sorites case that makes the move seem compelling in this case. And I can’t see that the authors that I’ve been reading explain what that is.

(A variation on this theme occurs in Graff Fara’s “Shifting sands”. Roughly, she gives a contextualist(-ish) story about why each instance asserting that the cut-off is not here will be true. She then says that it is “no wonder” will count universal generalization (the major premise of the sorites) as true.

But again, it’s hard to see what general pattern of inferring this falls into (remembering that it has to be one so compelling that it survives confrontation with a short proof of the truth of the existential). After all, as I look around my room, the following are successively true: “my chair is currently visible” “my table is currently visible”, “my cabinet is currently visible” etc. I feel no temptation to generalize to “all of the medium sized objects in my room are currently visible”. I have reasons to think this general statement false, and that totally swamps my tendancy to generalize from the various instances. So again, the real question here is to explain why something similar doesn’t happen in the sorites. And I don’t see that question being addressed.)

Against against against vague existence

Carrie Jenkins recently posted on Ted Sider‘s paper “Against Vague Existence“.

Suppose you think it’s vague whether some collection of cat-atoms compose some further thing (perhaps because you’re a organicist about composition, and it’s vague whether kitty is still living). It’s then natural to think that there’ll be corresponding vagueness in the range of (unrestricted) first order quantifier: it might be vague whether it ranges over one billion and fifty five thing or one billion and fifty six things, for example: with the putative one billion and fifty-sixth entity being kitty, if she still exists. Sider thinks there are insuperable problems for this view; Carrie thinks the problems can be avoided. Below the fold, I present a couple of problems for (what I take to be) Carrie’s way of addressing the Sider-challenge.

Sider’s interested in “precisificational” theories of vagueness, such as supervaluationism and (he urges) epistemicism. The vagueness of an expression E consists in there being multiple ways in which the term could be made precise, between which, perhaps, the semantic facts don’t select (supervaluationism), or between which we can’t discriminate the uniquely correct one (epistemicism). (On my account, ontic vagueness turns out to be precisificational too). The trouble is alleged to be that vague existence claims can’t fit this model. One underlying idea is that multiple precifications of an unrestricted existential quantifier would have to include different domains: perhaps precisification E1 has domain D1, whereas precisification E2 has domain D2, which is larger since includes everything in D1, plus one extra thing: kitty.

But wait! If it is indeterminate whether kitty exists, how can we maintain that the story I just gave is true? When I say “D2 contains one extra thing: kitty”, it seems it should be at best indeterminate whether that is true: for it can only be true if kitty exists. Likewise, it will be indeterminate whether or not the name “kitty” suffers reference-failure.

Ok, so that’s what I think of as the core of Sider’s argument. Carrie’s response is very interesting. I’m not totally sure whether what I’m going to say is really what Carrie intends, so following the standard philosophical practice, I’ll attribute what follows to Carrie*. Whereas you’d standardly formulate a semantics by using relativized semantic relations, e.g. “N refers to x relative to world w, time t, precification p”, Carrie* proposes that we replace the relativization with an operator. So the clause for the expression N might look like: “At world w, At time t, At precisification p, N referes to x”. In particular, we’ll say:

“At precisfication 1, “E” ranges over the domain D1;
At precisification 2, “E” ranges over the domain D1+{kitty}.”

In the metalanguage, “At p” works just as it does in the object language, binding any quantifiers within its scope. So, when within the scope of the “At precisification 2” operator, the metalinguistic name “kitty” will have reference, and, again within the scope of that operator, the unrestricted existential quantifier will have kitty within its range.

This seems funky so far as it goes. It’s a bit like a form of modalism that takes “At w” as the primitive modal operator. I’ve got some worries though.

Here’s the first. A burden on Carrie*’s approach (as I’m understanding it) will be to explain under what circumstances a sentence is true. usually, this is just done by quantification into the parameter position of the parameterized “truth”, i.e.

“S” is true simpliciter iff for all precisifications p, “S” is true relative to p.

What’s the translation of this into the operator account? Maybe something like:

“S” is true simpliciter iff for all precisifications p, At p “S” is true.

For this to make sense, “p” has to be a genuine metalinguistic variable. And this undermines some of the attractions of Carrie*’s account: i.e. it looks like we’ll now the burden of explaining what “precisifications” are (the sort of thing that Sider is pushing for in his comments on Carrie’s post). More attractive is the “modalist” position where “At p” is a primitive idiom. Perhaps then, the following could be offered:

“S” is true simpliciter iff for all precisification-operators O, [O: “S” is true].

My second concern is the following: I’m not sure how the proposal would deal with quantification into a “precisification” context. E.g. how do we evaluate the following metalanguage sentence?

“on precisification 2, there is an x such that x is in the range of “E”, and on precisification 1, x is not within the range of “E””

The trouble is that, for this to be true, it looks like kitty has to be assigned as the value of “x”. But the third occurence is within the scope of “on precisification 2”. On the most natural formulation, for “on precisification 2, x is F” to be true on the assignment of an object to x, x will have to be within the scope of the unrestricted existential quantifier at precisification 1. But Kitty isn’t! There might be a technical fix here, but I can’t see it at the moment. Here’s the modal analogue: let a be the actual world, and b be a merely possible world where I don’t exist. What should the modalist say about the following?

“At a, there is an object x (identical to Robbie) and At b, nothing is identical to x”

Again, for this to be true, we require an open sentence “At b, nothing is identical to x” to be true relative to an assignment where some object not existing at b is the value of “x”. And I’m just not sure that we can make sense of this without allowing ourselves the resources to define a “precisification neutral” quantifier within the metalanguage in reference to which Sider’s original complaint could be reintroduced.