## 2 Replies to “New Paper!”

1. Wow! When I posted my paper with a request for comments, I never dreamed
of getting such a detailed and thought-provoking reply. Many thanks indeed.

Some responses to your three points.

(1) Vagueness without boundaries.

I have already been tinkering with my view, to get over some other problems
in this area. My original response to the Fine (tall or exactly four feet
in height)/Weatherson (early thirties) problem was to say that a predicate
is vague simpliciter if it non-vacuously satisfies Closeness across the
entire domain of discourse, and is in part vague if it is not vague
simpliciter, but there is some subset of the domain of discourse over
which it non-vacuously satisfies Closeness. Even apart from your problem,
however, this will not do, because there is a subset of the domain of
discourse over which intuitively precise predicates such as ‘is
greater-than-or-equal-to exactly four feet in height’ non-trivially satisfy
Closeness: the subset consisting of people less than three feet in height
or more than seven feet in height (thanks to an anonymous referee for this
point!).

My current thought is to avoid this problem as follows. We say that a set
S is F-connected iff for any two objects in S, either they are very similar
in F-relevant respects, or they can be linked together by a chain of
objects, all of which are in S, with adjacent members of the chain being
very close in F-relevant respects (this can be made precise, but not in
plain text!). We say that a set S is F-flat iff for every a and b in S,
‘Fa’ and ‘Fb’ are very similar in respect of truth. If a set is not F-flat
we say it is F-tilted. We now say that a predicate F is vague iff there is
some F-connected, F-tilted subset of the domain of discourse such that F
satisfies Closeness over this set. (To say that a predicate satisfies
Closeness over a set S is to say that it satisfies Closeness when the
initial quantifiers ‘for any objects a and b’ in the Closeness condition
are taken as ranging over S. Note that we do not now need to include the
‘non-vacuousness’ requirement.) A predicate is vague simpliciter iff it is
vague, and furthermore satisfies Closeness over every F-connected,
F-tilted subset of the domain of discourse. A predicate is in part vague
iff it is vague but not vague simpliciter. Thus ‘tall’ is vague
simpliciter, ‘is tall or exactly four feet in height’ is in part vague, and
‘is greater-than-or-equal-to exactly four feet in height’ is not vague.

Now for your case of F_170. I’m not quite sure how you intended the case,
i.e. what you mean by saying person A, who is 170cm in height, is tall but
not definitely tall. Option (a): ‘A is tall’ is true simpliciter/true to
degree 1/etc., but A is, say, in an epistemic penumbral region for ‘tall’.
(b) ‘A is tall’ is neither true simpliciter nor false simpliciter/‘A is
tall’ has a high degree of truth but it’s not equal to 1/etc. Considered
in turn:

(a) F_170 is not vague by my definition, but I am happy with that result.
To my intuitions at least, if everything is a degree-1 instance of F_170,
then F_170 is not vague, even if we cannot know whether some things A are
degree-1 instances of F_170.

(b) F_170 is ‘in part vague’ by my definition, and by my intuitions.

The only remaining issue is what happens when we get to n such that F_n has
a jump down in truth value when we cross height n, but such a small jump
down that ‘B is tall’ (B’s height is n or just under) and ‘C is tall’ (C’s
height is just over n) are very close in respect of truth. In this case
F_n is not vague by my definition, but I can live with that.

(2) Lumpy boundaries.

You say ‘If He’s late is true to degree 0.6 just before quarter past, and
degree 0.8 just after, that’s clearly a violation of Closeness. But this
seems perfectly compatible with the predicate being vague.’

I don’t see a problem here: If your assumption here (that He’s late is
true to degree 0.6 just before quarter past, and degree 0.8 just after) is
granted, then the predicate will be in part vague by my definition, and
by my intuitions. On the other hand, we might think about ‘late’ that your
assumption is not true: there is no jump in truth value. Rather, the
predicate is wholly vague—-the truth value of ‘He’s late’ changes
continuously as his arrival time increases—-but after quarter-past we
know that the degree of truth of the claim is now sufficiently high to
confidently assert ‘He’s late’. So quarter-past does not represent a jump
in truth value—-just a ‘safe point’ at which we can be quite confident in
asserting ‘He’s late’.

(3) Generality.

I say some stuff in the paper about noun phrases. I do think my view
handles these smoothly, but I will be saying much more about them in future
work (near future, I hope). As for very and if, I don’t have any
‘very tall’ and so on, but no intuitions either way concerning whether
‘very’ by itself is vague. I would only be troubled if there were some
words which intuitively seem akin to vague predicates, but which my
definition does not cover (e.g. if it only covered one-place predicates,
not many-place predicates). That would show my definition was worryingly
specific. But ‘very’ and ‘if’ don’t feel like that. The real question
about ‘very’ and vagueness seems to be: what in general is the
relationship between the vagueness of ‘F’ and the vagueness of ‘very F’.
My view has the resources to make sense of this question. But is there
really a question as to whether ‘very’ by itself is vague? I just don’t
feel any force to this question, so I’m not worried that my view does not
have the resources to make sense of it.

I think the point about vague pictures is very interesting. There’s an
important distinction here though. A picture is like a particular
utterance. When defining vagueness, most people focus on word types. It’s
one thing to ask what the vagueness of ‘bald’ consists in, another to ask
what the vagueness of a particular utterance ‘Bob is bald’ consists in. I
guess I would probably want to say that pictures can be
indeterminate—-there can be different, equally correct interpretations of
what a picture is telling us, i.e. different accounts of its content—-but
maintain this is different from vagueness. On your view of what vagueness
is, however, pictures could be vague in much (or exactly, depending on the
details) the same sense as linguistic items.

2. Just a few quick follow-ups on Nick’s points.

I wasn’t defining F170 in terms of degrees of truth. I just wanted 170 to be a height where an adult woman that height was tall but not definitely tall. I don’t know how you want to analyse that in terms of degrees of truth. (I know how I analyse it – as such a woman being tall and it being less true than 0=0 that she’s tall.) It seems a bad thing if this isn’t possible on a given analysis (because it seems it means you can’t have a theory of higher-order vagueness) and as soon as its possibility is shown I’ll have a way of creating the example.

That might (might – I’m really not sure how higher-order vagueness goes on numerical degree theories) be to say I agree with Nick’s second way of putting the problem in the last paragraph, except I do think it’s a problem.

I do think there’s a hard issue about whether ‘very’, and in particular ‘if’ is vague but maybe there’s not much more than issues of motivation here.

I have a fair bit more to say on lumpiness, but maybe I’ll leave them for an actual post.