I just wrote a short paper responding to some suggestions by Nick Smith in his paper Vagueness as Closeness. The title is rather utilitarian, and I haven’t checked over the paper greatly (e.g. to check I haven’t left off anyone from the thanks list) but it seems good enough to blog post.

## 2 Replies to “New Paper!”

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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

simpliciterif it non-vacuously satisfies Closeness across theentire domain of discourse, and is

in partvague if it is not vaguesimpliciter, but there is

somesubset of the domain of discourse overwhich it non-vacuously satisfies Closeness. Even apart from your problem,

however, this will not do, because there

isa subset of the domain ofdiscourse over which intuitively

precisepredicates such as ‘isgreater-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

everyF-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

knowwhether some things A aredegree-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 lateis true to degree 0.6 just before quarter past, anddegree 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 lateistrue to degree 0.6 just before quarter past, and degree 0.8 just after) is

granted, then the predicate will be

in part vagueby my definition, andby 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

knowthat the degree of truth of the claim is now sufficiently high toconfidently 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

veryandif, I don’t have anyintuitions about their vagueness or lack thereof. I have intuitions about

‘very tall’ and so on, but no intuitions either way concerning whether

‘very’

by itselfis vague. I would only be troubled if there were somewords 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 generalis therelationship 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.

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.