I read Patrick Greenough’s vagueness paper in Mind, and I was a little underwhelmed. So I wrote up a reply. Normally I would put this on a separate page, but it’s short enough to include here. I don’t yet have the citations done, but you can probably tell what they will all be. The tone is a little snarky right now. I’m not quite sure why that happened. I think I was just trying to be brief and I came off sounding terse. That will be corrected in future drafts, unless I find I like snarky. If anyone gets the joke in the third paragraph I shall be pleased and impressed. Oh, and I decided for a stylistic variant to state some facts as if they were facts, rather than as if they were points of contention. I think this comes off as rudeness towards the end of the first and penultimate paragraphs, so I might change that too.

Without further ado…

Patrick Greenough’s paper Vagueness: a Minimal Theory (all page referencess to this paper) aims to provide a theory-neutral account of vagueness. Greenough wants to identify what vagueness
is before we get to an analysis of it. This looks like a quixotic project. It
is hard to imagine such a theory-neutral identification of causation, or moral
justification, or recession, or indeed anything much of theoretical interest.
One difficulty is that Greenough assumes that we can tell whether a term is
vague prior to having a theory of vagueness. So Soames’s example smidget
(Soames 1989) is not vague, and nor are scientifically indeterminate terms like
mass in Newtonian physics (Field 1974). But in hard cases like these the
best thing to say might be that the term is vague iff it is an instance of the
nearest natural kind to our confused pre-theoretic concept of vagueness. In
this case a theory‑neutral identification of cases of vagueness will
necessarily include some misidentification. This is not just a theoretical
possibility. Despite the fact that these terms don’t look vague, since
vagueness really just is semantic underdetermination, and these terms are
semantically underdetermined, they are vague. Pre-theoretic intuitions here are
mistaken, and the value of a ‘minimal theory’ that captures them is
questionable.

Greenough argues that vagueness is
epistemic tolerance. Oddly, he starts the argument by noting that he’s only
going to concentrate on ‘one-dimensional’ vague terms. This is odd because (a)
it is hard to believe that vagueness in one-dimensional vague terms like tall
or old is a different thing to vagueness in multi-dimensional vague
terms like genius or rockstar; and (b) his analysis of
one-dimensional vagueness relies crucially on a property that one-dimensional
vague terms have in virtue of being one-dimensional, rather than on a property
they have in virtue of being vague. Still, the first dimension contains enough
hidden complexity to show that vagueness should not be identified with
epistemic tolerance.

Greenough’s definition of epistemic
tolerance takes some unpacking. Let S be a declarative sentence that we
want to investigate for vagueness. Assume that whether S is true in a “depends only on the value v(a) taken by some discretely or continuously varying parameter v
in a ” (240). This is the assumption that S
is one-dimensional. Let t be a variable “which ranges over possible modalities or ‘truth-states’” (263). So
the value for t might be something simple like true, or determinately
true or something more complicated like determinately not determinately not true, or determinately
determinately not determinately true, or determinately not not true. And let
s be a speaker who knows the meaning of S (if this is possible
without knowing its truth-conditions) and the value of v(a) in all situations a. And let
c be some small value. Then S is epistemically tolerant iff

"t"a"b if |v(a)-v(b)|<c and in a s
knows that S is t then it
is not the case that in b s
knows that that s is not-t. (263)

I have three objections to this as account
of vagueness; two of these are fairly trivial but one I think is serious.

The first trivial objection is that this
theory implies, falsely, that S is vague if all terms in S are
rigid designators. So 7 is prime is vague, because in any
situation a , it is true, and determinately true,
and determinately determinately true, and so on. In all situations b such that |v(a)-v(b)|<c (or that |v(a)-v(b)|>c),
it is not known in b that 7
is prime is not true, or not determinately true, and so on. Presumably the
theory is meant to apply to sentence radicals, like x is prime, rather
than to complete sentences like 7 is prime. Indeed, all of Greenough’s
examples involve such radicals (save when he is explicitly talking about
predicates). So I will assume the theory to be so interpreted.

The second trivial objection is that even
with this restriction, we still over-generate vagueness. The sentence radical x
is self-identical turns out to be vague for just the same reason. It is not
immediately obvious how to avoid this problem. We could say that S
is vague only if it takes different truth values for some values of x.
This is certainly the most obvious response to this problem. This will rule out
x is self-identical being vague, but it might rule out too much. Define a new predicate virtuous such that x is virtuous iff x is
self-identical and 7 is small. (I assume 7 is a penumbral case of smallness.)
Then by my intuitive lights, x is virtuous is vague, but its truth-value
is independent of x. Perhaps there is some other technical fix possible
here, so I will not press this point.

The serious problem concerns predicates
that are true of objects in a range, and one of the borders of the range is
sharp while the other is vague. Such predicates are bound to be a little
artificial, but to a good approximation early thirties, as in Shane
Warne is in his early thirties, is such a predicate. I assume that one
enters ones early thirties precisely on one’s thirtieth birthday, but it is a little vague when one leaves this era. (Precisely when one turns thirty is a little unclear, which complicates matters. I will assume a slightly artificial
language where we precisely measure ages by how many rotations of the sun have
passed since the subject’s umbilical cord was cut.) At thirty-three, as he now
is, I assume Mr Warne is still in his early thirties. I would say, though some
would not, that even when he turns thirty-four this September, he will still be
in his early thirties. But from then on matters get a little dicey. Will he
still be in his early thirties at thirty-five? At thirty-six? Thirty-seven?
Thirty-eight? Probably not by then.

Note that as defined, if S is x
is in his early thirties, it is not epistemically tolerant. Let a be a situation where x has just turned thirty, and b a situation immediately before this happens. If s knows all
the details, she knows S is true in a and not true in b . But a and b differ minimally, so S is not
epistemically tolerant. But it is, intuitively, vague. I think Mr Warne will be
a penumbral instance of early thirties for a while to come. If you
disagree, insert some other predicate that has the feature that it is sharply
bounded at one end, and vaguely bounded at another. It is implausible that
there could be a conceptual restriction on languages that they not contain such
a term. And such terms are clearly vague despite being epistemically tolerant.

Such problems recur when we reach for higher
dimensions. If whether or not x is F depends on a number of
factors, some of them precisely specified and others vaguely specified, then x
is F will be epistemically intolerant despite being vague.

Greenough has a formal argument that the
kind of case I have described, an epistemically intolerant term with borderline
cases, is impossible. Since the argument concludes that early thirties
is impossible it must, one supposes, be mistaken. Actually, once we have the
example in front of us, it is not too hard to find the mistake. The argument is
an attempted reductio on the hypothesis that there is a borderline case m
of F even though F is epistemically intolerant. If F is
intolerant, there is an x such that Fx is knowable while Fx´
is unknowable (where x´ differs minimally from x). There are a
number of cases to consider depending on the ordering relation between x,
x´ and m, but we will focus on just the one that matters, where x´
is 29.99 years old, x is 30.01 years old, and m is right around
the end of his early thirties. The key premise is that if m is older
than x, and x is knowably in his early thirties, then m is
knowably in his early thirties. This is obviously false. It looks plausible in
the context of the paper because the focus is on predicates like old or tall
that do not change truth value twice as you move along their underlying
scale. Anyone older than an old person is old. Take old to be the
paradigm for F, and it might look plausible that if x < m
then for all y if s knows that y is F then she can
know that y+1 (or someone one ‘more’ than y in the relevant
dimension) is F. This is just what Greenough does assume (sentence 14 on
page 271). But it is false for early thirties.

There is a deeper problem with the use of
epistemic concepts in a ‘minimal theory’ of vagueness. It is contentious that
typical borderline cases generate ignorance. Cian Dorr (ms) has argued that in
borderline cases of Fness, it is just as indeterminate whether someone
can know that the object is F as whether the object is, in fact, F.
If this is right then even the minimal theory, which that indeterminacy implies
ignorance, goes to far. I won’t go into the arguments in too much detail here,
partially because I want to keep this short and partially because they aren’t
my arguments. But if Dorr is right then there really is no hope for completely
identifying vagueness in a theory-neutral way. As Greenough argues, we cannot identify vagueness with any non-epistemic phenomenon without begging the question against the epistemicist, and as Dorr argues we cannot identify vagueness with any epistemic phenomenon without misidentifying it.

This should not be a depressing
conclusion. As said, there’s no way to isolate instances of most the concepts
we use in philosophy, or economics, or psychology, or biology, or linguistics,
or any other social science, in a theory neutral way. But social science
progresses anyway.