I just noticed that Matti Eklund’s paper “What Vagueness Consists In” is on the ‘forthcoming papers’ list at “Philosophical Studies”:http://www.kluweronline.com/issn/0031-8116/contents. As is traditional, I’ll try to honour his paper by coming up with as many counterexamples as I can. I stole borrowed one of them from “Jonathan Ichikawa”:http://ichikawa.blogspot.com/2004_03_01_ichikawa_archive.html#107902380173158917. This all gets very long, and I got a little footnote-crazy (if you’re going to use footnotes you really should exploit their full comic potential) so it’s all below the fold.
Oversimplifying greatly, Matti thinks the vagueness of a term _t_ consists in the fact “that the vagueness of an expression consists in it being part of competence with the expression to be disposed to accept a tolerance principle for it.” (29)[1] What’s a tolerance principle? That’s part of what’s at issue, but it’s something like the following:
bq. (#) If a is t, then a’ is t.
where a is from the right range and ‘ is some kind of successor relation along a suitably chosen sequence. Here’s a couple of examples (both mine, and both cited in the paper) that illustrate these two points.
“Early thirties” is vague, but no principle like (#) is right for “early thirties”. That’s because it is (more or less) sharply bounded at the lower end. But Matti says it is vague because when you take someone[2] who’s near the upper boundary of ‘early thirties’, and let ‘ be _one day older than_, we are disposed to accept (#). The lesson is that (#) can’t be taken to be bound by a universal quantifier, but only by a restricted quantifier restricted to those around the vague boundary.
“Small integer” is vague, but if we let ‘ be _0.1 larger than_, then (#) is clearly false. 5 is a small integer, but 5.1 is not. Matti says that’s OK, because we can restrict attention to the case where ‘ is the relation _1 larger than_, and we are disposed to accept (#).
There’s a third point about (#) that we should mention here, and which I think is tacit in Matti’s discussion. Even if a is actual, there is no need that a’ be actual. (If a’ is a quantificational phrase, there is no need that anything satisfy the quantifier.) When someone says we are disposed to accept conditionals like (B), no one cares whether anyone has exactly 123456 hairs.
(B) If any person with 123455 hairs is bald, any person with 123456 hairs is bald.
Those qualifications in place, we can start running through some counterexamples. The first one is basically Jonathan’s.
1. Better than X
We can generate one-place predicates easily enough out of comparative relations. From _is a better hitter than_ we can get the predicate is a better hitter than “Roberto Clemente”:http://baseball-reference.com/c/clemero01.shtml. This is a vague predicate; “John Olerud”:http://www.baseball-reference.com/o/olerujo01.shtml is a borderline case. (I know that this claim might surprise some baseball afficianados, but I think it’s true. Clemente has a slightly longer career than Olerud, and a fractionally higher OPS+, but Olerud’s OPS is more OBP-heavy which is the more important constituent.[3])
At first this might look like a good case for Matti. Here’s how we might try to show that _is a better hitter than Roberto Clemente_ satisfies something like (#). Let Olerudn be a (possible) hitter just like John Olerud (with the same number of plate appearances, hits, home runs, etc.) but with _n_ more walks. It’s pretty clear that Olerud-400 is not a better hitter than Clemente, and Olerud400 is better than Clemente. Still, the following Sorites argument looks intuitively compelling.
P1. Olerud400 is a better hitter than Clemente
P2. For all _n_, if Olerudn is a better hitter than Clemente, then Olerudn-1 is a better hitter than Clemente.
C. Olerud-400 is a better hitter than Clemente.
P2 is false, but it is intuitively compelling in just the right way for Matti’s account to work. But the problem is that it doesn’t generalise. If the intuitive compulsion of (instances of) P2 is what makes _is a better hitter than Clemente_ vague, then we should expect (L) to be intuitively acceptable, where Clementen is defined as we defined Olerudn above.
(L) If Clemente1 is a better hitter than Clemente, then Clemente0 is a better hitter than Clemente.
But (L) is clearly false, as the antecedent is obviously true and the conclusion obviously false.
Could it be argued that Clemente1 is not one of the people to which this instance of (#) should apply? It’s hard to see how. That he’s a merely possible person is not a decisive consideration, as argued above. And he’s in the right kind of area of possibility space – batters around the same quality as Clemente, that the tolerance conditional should apply – unlike the cases of the just-turned-30 year old who’s near the sharp boundary of _early thirties_. Short of saying that the tolerance conditional need not apply where it doesn’t apply, we can’t see a principled way of saying that the vagueness of _is a better hitter than Clemente_ consists in the intuitive appeal of P2 even though (L) is simply not at all intuitively appealing.
2. Tiny Number
Just like _small integer_ is vague, so is _tiny integer_. But the following principle is _very_ implausible for _tiny number_.
(T) If _n_ is a tiny number, so is _n_+1.
It should be obvious what the problem is here. Because we’re only talking about integers, the successor relation in (#) has to move in big enough steps to take us from one candidate member of the extension to the next. I.e. it has to move in steps of at least size 1. But although ‘tiny’ is vague, steps of size 1 are pretty big relative to the size of its borderline. So (T) looks fairly excessive – 4 looks very different with respect to tininess than does 3.
This problem should generalise. All you need is a predicate that (a) is know to only applies in a narrow range, so its borderline is going to be small, but (b) only applies to things that are separated by a large discrete space, so the successor relation will have to involve fairly large steps. In any case there will be no candidate meaning for ‘ that makes (#) true.
3. Vehicle
Scott Soames argues that some vague terms are not Sorites-susceptible. His preferred example is _vehicle_. _Vehicle_ is vague because, as Hart famously noted, it is vague whether skateboards are vehicles. That’s all. No need for a Sorites sequence here in order to show that it is really vague. And certainly no need for a candidate meaning for ‘ that makes (#) intuitively appealing when _t_ = _vehicle_.
Strictly speaking, I think Soames is wrong and _vehicle_ is Sorites-susceptible, but I don’t think this ultimately rescues Matti’s theory. I’ll argue for each conjunct separately.
My old new Beetle was clearly a vehicle. It’s counterpart that was a duplicate in every respect except for the presence of one atom somewhere in the vehicle was also clearly a vehicle. That car’s counterpart that was a duplicate in every respect except for the presence of one atom somewhere in the vehicle was also clearly a vehicle. And so on for quite a while. But eventually one of the counterparts of counterparts of … is not a vehicle. If nothing else, the last atom is not a vehicle. So here’s the Sorites sequence. And here’s the way to defend (#) for vehicle. Let ‘ be the relation _duplicate of except for the presence of one atom_, and it’s intuitively appealing that if _a_ is a vehicle, so is _a’_.[6] That’s why, contra Soames, _vehicle_ is not Sorites susceptible.
But why doesn’t that help Matti? Because the vagueness of _vehicle_ doesn’t just consist in the presence of such sequences. It also consists in the fact that skateboards are borderline cases. We can make this more vivid by imagining a resolutely commonsensical metaphysician. Our commonsensical metaphysician believes that only when some things constitute something that fall under a commonsense sortal do they constitute anything at all. So, in my alleged Sorites sequence, it is clearly true that whenever I’m pointing at anything, I’m pointing at a vehicle. There’s no Sorites sequence from vehicles to non-vehicles here, but rather a gappy sequence that consists of some vehicles, then some attempts to refer to a non-entity, then perhaps an atom at the end. If that was the correct metaphysical picture, which of course it isn’t, then Soames would be right and _vehicle_ would not be Sorites susceptible. And there’d be no way to interpret (#) as intuitively appealing with _t_ = _vehicle_. But _vehicle_ would still be vague, because skateboards would still be vehicles. So the vagueness of _vehicle_ does not consist in the temptation of speakers towards tolerance.
4. Very Hamlet
So here’s a very strange feature of Matti’s paper. At the start he notes that there’s a dispute about vague names. The dispute is over whether such a name determinately refers to a vague object, or whether it indeterminately refers to some precise object or not. The latter is correct, but that’s not the point here.
The point is that there’s this literature on vague names, so we should have an account of what the vagueness of names consists in. But Matti’s theory only applies to vague predicates. Even if it’s the correct account of what the vagueness of verb phrases consists in (and I think there’s three strikes against it), it isn’t a complete account of what vagueness consists in.
Worse still, it’s _really_ implausible that the vagueness of noun phrases consists in something different to what the vagueness of verb phrases consists in. But if we try to put a noun phrase in place of _is t_ in (#), the result isn’t even syntactically well formed.
It doesn’t really add anything to the point, but let me note some examples of other vague terms.
Adverbs can be vague. There’s no (known) precise meaning for _very_, for example.
Quantifiers can, on many theories, be vague. (Not on Kent’s theory though!) If quantifier domain restriction is syntactic, then it can be vague what the semantic contribution of _every_ in _Every boy wanted to win_ is.
Whatever syntactic category _if_ falls into, it can be vague. At least, if something like Lewis’s theory of subjunctive conditionals is correct, and the correct similarity metric on worlds on different occasions of usage can be different, and the similarity metric gets ‘carried into’ the semantics of individual sentences through _if_, then I’m pretty sure _if_ will be vague. There’s a lot of assumptions there, but they’re collectively plausible. I have no idea how something like (#) should be extended to say what the vagueness of _if_ consists in.
And I’d bet “sui generis terms”:http://itre.cis.upenn.edu/~myl/languagelog/archives/000614.html can be vague, though I’ve got no !@#$%^& idea how to modify (#) to account for _them_.
On that note…
fn1.. Why haven’t I linked to the paper yet? Because it’s incredibly hard to find it! Here’s the most efficient strategy I’ve found. Go to the “Philosophical Studies”:http://www.kluweronline.com/issn/0031-8116/contents site. In the sidebar there is a link to ‘Articles in Advance’. Click it. Now sacrifice a sheep and a goat to your preferred deity. Scroll down and down until you see *What Vagueness Consists In*, Matti Eklund. Click on the small PDF button. Pray. The paper should now open.
The prayer stage may be optional, but some experimenting suggests that the sacrifice stage is not.
If you want to _save_ the paper, it’s a little trickier. Here’s the best I could do. Try saving the paper from Acrobat. If it works, good. It won’t though. Instead email the paper to yourself. (Or your best friend. Or someone who’s been spamming you – I find academic PDFs are very good for cheap DOS attacks.) Go to your email and open the attachment. Now try saving it, and ignore Acrobat when it says something is about to go wrong. There, now you have your very own Matti Eklund preprint. I’ll be surprised if anyone tries this strategy, and I bet it won’t work for them.
fn2. I ran through about ten different ideas for which of my friends I should link to as an illustration of ‘upper end of early thirties’, but I decided none of them might appreciate it.
fn3.. What on earth is OPS+? I hear you ask. Well, OPS, as you probably know, is OBP (On-Base-Percentage) plus SLG(SLugGing percentage). OPS is a pretty good crude measure of batting effectiveness, though it would be better to have something like 1.5*OBP+SLG, because a point of OBP is worth more than a point of SLG. (That’s why the point in the text about Olerud’s OPS+ being OBP-heavy matters; OPS+ undervalues him a touch.)
To get OPS+ from OPS, you do two things. First, you adjust the OPS for the park the hitter plays in. Then you divide by league average (to account for differences in scoring rates between eras) and multiply by 100. So an OPS+ of 130 means (roughly) that Olerud has been 30% more effective than the average hitter over the course of his career.[4]
I should note that using a rate stat like OPS+ somewhat overvalues current players over past players because it doesn’t factor in the ‘decline phase’. But, (a) it’s not obvious how to account for this in Clemente’s case, and (b) Olerud is already well and truly in his decline phase, so this isn’t a big factor here. I hope he has a comeback this year though – in his 156 games against the not-Red Sox.
fn4. Did you know that you can do footnotes to footnotes in “Textile”:http://www.bradchoate.com/mt-plugins/textile? I didn’t either.[5]
fn5. But what I really meant to say is that looking at Olerud’s record, he’s a pretty plausible Hall of Famer. He’ll never make it in, but an OPS+ of 130 over a 15 year career is pretty good, and I think he’s an historically good fielding first baseman. His range is I think above average, though this is hard to tell from 1B stats, but he’s an incredible throw-receiver. There’s this curious fact that wherever he goes, the middle infielders suddenly start to look incredible. Robbie Alomar became the guy you give the 2B Gold Glove to every year the first year he played with Olerud in Toronto. Rey Ordonez got his (vastly overrated) fielding rep when playing with Olerud in New York. And he had that incredible error-free streak when Olerud was there. (1999 Mets with Olerud – 33 infield errors; 2000 Mets without Olerud – 83 infield errors. Some of that is random noise, but still.) And in Seattle, suddely Bret Boone, _Bret Boone_, is considered a Gold Glove 2B. I don’t think anyone should go into the Hall for 1B defence, but it should count, and since his rate stats suggest he’s already a borderline 1B, I think that should put him over the top. In reality, I’ll be surprised if he stays on the ballot more than a couple of years.
fn6. If this is written up for something more serious (if lower readership) than this blog, I need to clean up what I mean by ‘ throughout. I keep using it as if it’s a function, but then I have it defined by non-functional relations. It’s pretty obvious in context what’s going on, but it’s as sloppy as all get out.