What Is the Value of Vagueness?

Abstract

Classically, vagueness has been considered something bad. It leads to the Sorites paradox, borderline cases, and the (apparent) violation of the logical principle of bivalence. Nevertheless, there have always been scholars claiming that vagueness is also valuable. Many have pointed out that we could not communicate as successfully or efficiently as we do if we would not use vague language. Indeed, we often use vague terms when we could have used more precise ones instead. Many scholars (implicitly or explicitly) assume that we do so because their vagueness has a positive function. But how and in what sense can vagueness be said to have a function or value? This paper is an attempt to give an answer to this question. After clarifying the concepts of vagueness and value, it examines nine arguments for the value of vagueness, which have been discussed in the literature. The (negative) result of this examination is, however, that there is not much reason to believe that vagueness has a value or positive function at all because none of the arguments is conclusive. A tenth argument that has not been discussed so far seems most promising but rests on a solely strategic notion of function.

Full text

What Is the Value of Vagueness?
by
DAVID LANIUS
Karlsruher Institut fur Technologie
Abstract: Classically, vagueness has been considered something bad. It leads to the Sorites para-
dox, borderline cases, and the (apparent) violation of the logical principle of bivalence. Neverthe-
less, there have always been scholars claiming that vagueness is also valuable. Many have pointed
out that we could not communicate as successfully or efficiently as we do if we would not use
vague language. Indeed, we often use vague terms when we could have used more precise ones
instead. Many scholars (implicitly or explicitly) assume that we do so because their vagueness has
a positive function. But how and in what sense can vagueness be said to have a function or value?
This paper is an attempt to give an answer to this question. After clarifying the concepts of vague-
ness and value, it examines nine arguments for the value of vagueness, which have been discussed
in the literature. The (negative) result of this examination is, however, that there is not much rea-
son to believe that vagueness has a value or positive function at all because none of the arguments
is conclusive. A tenth argument that has not been discussed so far seems most promising but rests
on a solely strategic notion of function.
Keywords: vagueness, value of vagueness, indeterminacy, sorties paradox, function of vagueness
1. Introduction
Almost since the beginning of the philosophical debate on vagueness, most peo-
ple have seen it as a problem. It leads to the Sorites paradox, borderline cases,
and the (apparent) violation of the logical principle of bivalence. In jurisprudence,
for instance, it has widely been regarded as a threat to the rule of law.
Most theories of vagueness give up on certain logical principles in order to
accommodate the vagueness of natural language in an effort to save logic from
paradox and inconsistency. And, even from a layperson’s perspective, it is doubt-
ful what function vagueness could possibly have in everyday communication.
After all, clarity is an important goal to both speakers and listeners in most con-
versations. There is prima facie reason to treat vagueness as a problem.
Nevertheless, there have also been surprisingly many people claiming that
vagueness is valuable. Among them are Max Black, Ludwig Wittgenstein, Frie-
drich Waismann and, more recently, Manfred Pinkal, Nora Kluck, Manfred
Krifka, Matthew J. Green, and Kees van Deemter. They (explicitly or implicitly)
THEORIA, 2021
doi:10.1111/theo.12313
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assume that in many cases we use vague terms when we could have used more
precise ones because their vagueness itself has a positive function.
1
While there is a number of contributions in the philosophy of law (e.g.,
Sorensen, 2001; Endicott, 2005; Asgeirsson, 2015; Lanius, 2019;
Asgeirsson, 2020), game theory (e.g., Lipman, 2000; de Jaegher, 2003; de Jaegher
and van Rooij, 2011), as well as computational linguistics (van Deemter, 2009,
2010; Green and van Deemter, 2011; Green and Deemter, 2013; Green and
Deemter, 2019) that discuss in some way or other the value of vagueness, the legal
community is concerned with the use of vagueness and other forms of indetermi-
nacy specifically in the law and most of the contributions by game theorists and
computer linguists try to explain in general why language has evolved to be vague
in the first place. This paper, in contrast, deals with the question of why individual
language speakers use vague terms in ordinary conversation when they could have
used more precise ones instead. Unfortunately, except for the work by Green and
van Deemter, the arguments for this claim being true are rarely made explicit. In
this paper, I will discuss nine arguments that can be reconstructed from the litera-
ture in favour of the claim that vagueness is valuable in the specified sense.
2. Setting the Stage
Two general remarks are in order before we can dive into the discussion of the
arguments. First, let us get clear about what vagueness is. A proper grasp of the
concept will allow us to differentiate it from other —related —phenomena. Sec-
ond, let us get clearer about what it means to say that vagueness is valuable, has
a positive function, or is useful.
2
Both clarifications are, I take it, necessary to
precisely state and properly evaluate the arguments.
2.1 What is vagueness?
Vagueness is usually explained by reference to paradigmatically vague terms such
as “blue”,“tall”,or“heap”. Moreover, three characteristics have been accepted in
the literature as essential to vagueness. According to Rosanna Keefe (2000,
pp. 6–7), vague expressions are commonly understood to: 1) admit borderline
cases, 2) be susceptible to the Sorites paradox, and 3) (apparently) lack sharp
boundaries. These three criteria are, however, rarely accepted as individually nec-
essary and collectively sufficient conditions for vagueness. Sometimes they are
treated simply as symptoms or signs of vagueness.
1 A term is “precise”iff it is not vague.
2 Please note that I will use the phrases is “valuable”, has a “positive function”, and is “useful”inter-
changeably and nontechnically in this paper. See section 2.2.
2DAVID LANIUS
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.

Consequently, there is no commonly accepted definition of vagueness. An
influential way to understand vagueness has been put forward by H. Paul Grice,
however:
To say that an expression is vague [...] is [...] to say that there are cases (actual or possible) in
which one does not know whether to apply the expression or to withhold it, and one’s not knowing
is not due to ignorance of the facts. (Grice, 1989, p. 177)
Cases in which an expression does neither clearly apply nor clearly not apply,
while the reason for this unclarity is not ignorance of the facts, are commonly
called “borderline cases”.
3
Higher-order unclarity about borderline cases is gener-
ally seen as the source of Sorites susceptibility. That is, vague terms are suscepti-
ble to the Sorites paradox because they do not (appear to) demarcate a (sharp)
boundary between those cases in which an expression does clearly apply and
those cases in which it does not clearly apply. There are borderline cases of bor-
derline cases as well as borderline cases of borderline cases of borderline cases,
and so forth.
Due to this property of vague terms, they give rise to the Sorites paradox:
(1) 1000,000 grains of sand is a heap of sand.
(2) A heap of sand minus one grain is still a heap.
(3) Thus: One grain of sand is a heap.
The premise (2) captures what has been coined the tolerance (by Crispin
Wright) or boundarylessness (by Mark Sainsbury) of vague terms. In the now
famous words of Wright (1976/1997, p. 156), “[w]hat is involved [...] is a cer-
tain tolerance [...], a notion of a degree of change too small to make any differ-
ence, as it were.”As Sainsbury (1997, p. 257) describes it, a vague term is
boundaryless
in that no boundary marks the things which fall under it from the things which do not, and no
boundary marks the things which definitely fall under it from the things which do not definitely do
so; and so on. Manifestations are the unwillingness of knowing subjects to draw any such bound-
aries, the cognitive impossibility of identifying such boundaries, and the needlessness and even
disutility of such boundaries.
Part of what “heap”means is that the addition or subtraction of a single grain
cannot make a difference to whether some arrangement of grains constitute a
heap. If someone denies this, we have reason to think that they do not understand
3 Epistemicists attribute the unclarity in question to ignorance of linguistic facts and thus seem not to
accept Grice’sdefinition of “borderline cases”. They could, however, accept it by interpreting “facts”as
solely those facts which concern the object of the expression and context of utterance —ruling out the
linguistic facts that determine the expression’s exact extension. I will leave this question open in the
remainder of this paper.
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3WHAT IS THE VALUE OF VAGUENESS?

the meaning of “heap”. That is, vague terms are tolerant to small changes (or, in
other words, they are boundaryless).
4
Although it is controversial what properties are essential to vagueness, we
could surely say that vagueness has a positive function if at least one of the three
characteristics has a positive function. If it can be shown that being susceptible to
the Sorites paradox, lacking sharp boundaries, or admitting borderline cases has
positive effects, then we can also justifiably say that vagueness has a value. May-
be vagueness has a value only if all three properties are combined or due to some
other property of vague terms that is (more fundamentally) connected to vague-
ness. We will look at this possibility when examining the arguments in due
course. What is certain, however, is that it is not enough to show only that some-
times we prefer vague terms when we could have used more precise ones instead;
we need to show that we do so because of their vagueness (however understood).
2.2 Value, function, utility
But what can it mean that vagueness has a positive value,function,oruse? People
who have claimed that vagueness has a value have rarely explicated what having
a value means or even in what sense vagueness might possibly have a value.
Clearly, when asking what value vagueness might have, we are concerned with
instrumental value. The instrumental value of something is the value that it has in
virtue of the fact that it is a means to something desirable. Desirability can be
understood very broadly as anything that is desired by someone. If Mary desires
to manipulate her audience into believing that she is a goddess, then whatever is
a means to fulfilling that desire has instrumental value to her. Usually, desirability
is understood somewhat narrower: more normatively. Something has instrumental
value only if it is a means to the common good, to some rational goal, or to
something that is objectively desirable.
5
We have an intuitively clear grasp on what it would mean for tools such as
pens to be instrumentally valuable. A pen has instrumental value if it helps us to
accomplish tasks such as writing or drawing. We desire to write a letter, and the
pen is a suitable means to that end. This is due to some properties such as, for
example, its handiness or having a functioning ballpoint, but not due to others
such as, for example, its having a certain colour.
A pen might even be valuable as a hair clip or a tool for scratching. Again, this
is arguably more due to its handiness and less due to its colour. It is indispensable
4 Sainsbury already suggests that boundarylessness itself be a feature —something that has utility, in
contrast to “the disutility of boundaries.”However, like so many others, he does not offer an explanation
of how boundarylessness might be useful or an argument why this be the case.
5 Most scholars who argue for the value of vagueness, arguably, have such narrower concepts of value.
What happens when we broaden our concept of value, will be discussed in section 5.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
4DAVID LANIUS

that the pen has some colour, but this does not make the colour a means to the
end of pinning up one’s hair. In contrast, a pen can have aesthetic value, too. With
regard to this end, its colour may very well be the property that is functional to
further it. Its colour may help to accomplish that we feel pleased when looking at
the pen.
How are things when examining the potential value of vagueness? How can
borderline cases, Sorites susceptibility, or the lack of boundaries be useful? What
might these properties help to accomplish?
Please note that it does not suffice to have an explanation for why there is
vagueness in our language. Of course, we also want to know why we speak vague
languages. But giving such an explanation is not the same as providing reasons
for the claim that vagueness has a value. Equating this presupposes an
impoverished (and fairly uninteresting) concept of value. We would not say that
something is valuable if it is an unavoidable side effect of something that has a
value. Something has a value if it helps us to achieve something.
This is why it is paramount to differentiate between vagueness and other prop-
erties of vague terms. As Roy Sorensen points out:
“Vague”has a sense which is synonymous with abnormal generality. This precipitates many
equivocal explanations of vagueness. For instance, many commentators say that vagueness exists
because broad categories ease the task of classification. If I can describe your sweater as red, then
I do not need to ascertain whether it is scarlet. This freedom to use wide intervals obviously helps
us to learn, teach, communicate, and remember. But so what? (Sorensen, 1997)
If we want to know whether vagueness is valuable, we need to ascertain that it is
not something else that helps us to achieve the goal in question; that vagueness
is not simply a byproduct of successful communication, but that it contributes to
its success; that we have not mistaken the vagueness of the vague term “red”
(that it allows for borderline cases, is susceptible to the Sorites paradox, and that
it lacks boundaries) with its generality (that it can be equally applied to many
things).
Of course, there are also people who claim that some linguistic property that
they call “vagueness”has value. In most cases, it becomes immediately clear that
these people do not mean vagueness in the sense captured by Keefe’s three char-
acteristics when they argue for the “value of vagueness”—but some other prop-
erty of (contingently) vague expressions (such as their generality or ambiguity).
6
Other people, however, list paradigmatic examples of vague terms, use Keefe’s
three characteristics of vagueness, and cite the philosophical literature on the
Sorites paradox and borderline cases. These people claim that vagueness in the
6 Most economic literature on the “value of vagueness”actually deals with the value of generality. For
instance, Hadfield (1994), Staton and Vanberg (2008), or Choi and Triantis (2010).
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5WHAT IS THE VALUE OF VAGUENESS?

philosophical sense has a positive function. In what follows, I will examine argu-
ments for this claim only.
3. Nine Arguments for the Value of Vagueness
As a matter of fact, we often use language in ways less determinate than possible.
In particular, we often use vague terms when we could have used more precise
ones instead. Why do we so often voluntarily opt for vague terms? Do we and
should we do so because of their vagueness —or do other forms of indetermi-
nacy (e.g., ambiguity, generality, or indeterminacy in implicatures) or even
entirely different features of these terms have positive functions only? What does
the vagueness of vague terms do (independently from their other properties)?
We are looking for arguments that have approximately this structure:
(1) X is desirable.
(2) Vagueness is a suitable (or indispensable) causal means to bring about X.
(3) If vagueness is a suitable (or indispensable) causal means to bring about
something desirable, then vagueness has instrumental value.
(4) Thus: Vagueness has instrumental value.
When do we use the vagueness of vague terms? To show that using vague lan-
guage is a suitable means to some desirable end is clearly not sufficient to show
that its vagueness has instrumental value. If we can show that being susceptible
to the Sorites paradox, lacking sharp boundaries, admitting borderline cases, or a
combination thereof is a suitable means to some desirable end, however, we can
be certain that vagueness has instrumental value. Let us now examine the nine
most promising arguments for the value of vagueness that can be reconstructed
from the literature.
3.1 The argument from communicative success
Some philosophers have argued that vagueness does not preclude successful com-
munication and thus cannot be problematic.
7
They reject the Fregean ideal of a
perfect, precise, and unambiguous language. As Wright (1975, p. 325) puts it,
“we have long since abandoned the Frege-Russell view of the matter. We no lon-
ger see the vagueness of ordinary language as a defect.”Indeed, there are hardly
any philosophers who defend this ideal anymore. Most linguists and philosophers
of language today explicitly hold the view that ordinary language is fine as it
7 Historically, this argument has been advanced in some form or other by, for example, Black (1937),
Hempel (1939), Wittgenstein (1953/2009), Peirce (1960), or Quine (1960).
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
6DAVID LANIUS

is. It is close to uncontroversial that vagueness is not necessarily a problem. But
what does this tell us about its value?
The most famous version of the argument from communicative success might
be attributed to Rohit Parikh.
8
Parikh (1994) game-theoretically shows that even
if borderline cases actually arise and unclarity results, as long as the conversa-
tional purpose is not (seriously) obstructed, vague messages can be useful.
According to his interpretation of vagueness, vague terms have an extension
which is slightly different for different language users such that there is disagree-
ment between them in some cases (sc. borderline cases). Based on this under-
standing of vagueness, he shows that vague messages can succeed in promoting
the conversation’s goal —even under perfect alignment of interests.
Consider the following situation: Ann wants Bob to bring her the topology
book from the library. She only tells him that it is blue. Due to its vagueness, the
phrase “blue books”does not have a single extension, but is divided into blue
A
books (within Ann’s semantic) and blue
B
books (within Bob’s semantic); the
extensions of the homonymous terms “blue
A
books”and “blue
B
books”merely
overlap. As long they do so sufficiently (i.e., if there are not too many borderline
cases), Bob obtains sufficiently useful information to efficiently find the correct
book and bring it to Ann. If Bob searches all blue
B
books first and then the others
(the less and less blue ones from his perspective), he will find the correct book
with higher probability more quickly than if he has to search all books more or
less randomly. Bob will thus save time by searching all blue
B
books, even though
this is not (strictly speaking) what Ann intended.
Parikh’s representation captures an important aspect of vagueness. What is
problematic about it, however, is that language users usually do not unconsciously
have idiosyncratic idioms which they assume to be precise. It is possible for lan-
guage users to explicitly disagree in borderline cases on whether the term cor-
rectly applies or to explicitly agree that they face a borderline case in which they
should suspend their judgement about the term’s application. Vagueness does not
(merely) consist in incompletely overlapping extensions between idiolects. But if
we accept that this captures at least an important aspect of vagueness (and I think
that we should), we can conclude with Parikh that this property of vague terms is
useful inasmuch as it does not preclude successful communication.
Do we also have an appropriate argument to show that vagueness has a value?
Parikh’s argument establishes the claim that borderline cases do not preclude suc-
cessful communication. It does not establish, however, the claim that borderline
cases are a suitable causal means to bring about successful communication.
Parikh convincingly shows that communication in vague terms can be valuable in
8 In fact, it has been attributed to him by Kluck (2014, p. 83).
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
7WHAT IS THE VALUE OF VAGUENESS?

the sense that it is better than no communication at all. However, it is also clearly
the case that Ann and Bob would communicate even better if the extensions of
blue
A
books and blue
B
books would overlap entirely, that is, if the terms used
were precise.
9
Parikh presupposes that vagueness is a necessary feature of natural
language, which it arguably is, and, based on this presupposition, argues that nat-
ural language can be used to successfully exchange information despite its vague-
ness. Parikh cannot —and does not even try to —explain why vagueness is
valuable in a more interesting sense. Unless we are willing to say that something
is valuable if it does not preclude something valuable, we cannot draw the con-
clusion that vagueness is valuable.
This is also the case for more elaborated scenarios like the ones discussed by
van Deemter (2009, pp. 624ff.). He argues that the vagueness of vague terms
facilitates search. Van Deemter asks us to imagine a situation in which a diamond
has been stolen from an emperor by one of his 1,000 eunuchs, and the only wit-
ness can describe the thief just with the expression “tall”. Due to its vagueness,
van Deemter argues, the emperor can rank the eunuchs, searching them from
tallest to less tall, and thus save time (instead of searching all the eunuchs ran-
domly). I take it that, also in such scenarios, communication would be even more
successful if the terms used would be precise (and that the vagueness here is prac-
tically unavoidable due to limited information). But even if that was not the case,
the property facilitating search is not borderline cases, Sorites susceptibility, or
boundarylessness. Van Deemter even mentions himself that it is the gradability
of the term “tall”that is needed to make sense to rank the eunuchs in the first
place. But gradability is not the same as vagueness. We can imagine a term
“tall*”that is gradable and allows for ranking eunuchs from tallest*to least tall*
but does not allow for borderline cases, is not susceptible to the Sorites, or has
precise boundaries (by, for instance, introducing a sharp cut-off point at the 500th
eunuch that is precisely 170 cm tall).
How precise must our expressions be for communication to be successful?
Consider the following passage from Wittgenstein. He asks us to take into
account the conversation’s purpose and shake off any ill-advised longings for the
ideal of exactness:
Now, if I tell someone: “You should come to dinner more punctually; you know it begins at one
o’clock exactly”—is there really no question of exactness here? After all, one can say: “Think of
the determination of time in the laboratory or the observatory; there you see what ‘exactness’
means”“Inexact”is really a reproach, and “exact”is praise. And that is to say that what is inex-
act attains its goal less perfectly than does what is more exact. So it all depends on what we call
“the goal”. Is it inexact when I don’t give our distance from the sun to the nearest metre, or tell a
9 This has also been shown formally. See Lipman (2009, 2000).
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8DAVID LANIUS

joiner the width of a table to the nearest thousandth of a millimetre? No single ideal of exactness
has been envisaged [...]. (Wittgenstein, 1953/2009, p. 88)
Wittgenstein’s point is that the goal of perfect precision is not a generally reason-
able one. There is no sense in which someone is completely punctual. But also
for most other scales there is usually a certain point from which onward precise
measurement becomes increasingly difficult or outright impossible, such as the
measurement of position and momentum of particles due Heisenberg’s Uncer-
tainty Principle. As a consequence, it is futile to strive for absolute precision in
many cases.
10
As Charles Peirce is supposed to have said, “vagueness is no more to be done
away with in the world of logic than friction in mechanics.”
11
Accordingly, the
purpose of most conversations can be furthered by using vague terms. Conversa-
tions demand different “ideals of exactness”depending on their purpose. What
counts as perfectly precise in one conversation, is unacceptably vague in another.
Just as with Parikh’s claim about communicating successfully with vague lan-
guage, there is nothing to say against this claim. It is as convincing as modest.
Communication would obviously be hindered by always telling the time exactly
to the nanosecond. Doing it would achieve nothing. Indeed, it would violate the
conversational maxims of quantity and relation to always telling the time exactly
to the nanosecond or to describe the blueness of a book in terms of its wave-
length. Even though a message informing about the exact time would be more
precise, it would also be more informative than required and, as a result, less rele-
vant for the purposes of the conversation. There are some contexts in which the
highest possible precision is pursued, as in science or logic. But the standard of
precision is adjustable. It depends on the object and purpose of the conversation.
As such, in virtually all contexts of ordinary conversation it is significantly lower.
More importantly, the fact that we usually do not —and should not —tell the
time to the nanosecond or describe colours by their wavelength spectrum does
not entail that there is value in the vagueness of vague language.
3.2 The argument from stability
Some have seen in an argument by Manfred Pinkal (1995) evidence for the claim
that vagueness is valuable. He argues that a coarse-grained degree of granularity
can facilitate stability in judgement. Perhaps this means that also vagueness (due
to the fuzziness in the terms’extension and the lack of (sharp) boundaries) can
10 This is in part due to the imperfection of measurement. At least for continuous scales like time one
can add always another decimal place —exacting ever more precise measurement.
Cf. Swinburne (1969, p. 288).
11 Cited in McNeill and Freiberger (1993, p. 136).
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9WHAT IS THE VALUE OF VAGUENESS?

facilitate stability in judgement. Let us have a look at his argument and the con-
cept of granularity.
Granularity is the concept of breaking down a description into smaller parts
(or granules). A coarse-grained description contains only a few parts and is thus
less informative than a more fine-grained one. For instance, one can describe
someone by saying that she is a person (coarse-grained description) or by saying
that she is a self-confident, 190-cm tall, and brown-haired Polish citizen (more
fine-grained description). Most parts in the descriptions can be decomposed fur-
ther into finer levels of granularity, for example, that the person’s hair is long,
thick, and parted on the left.
Now, if we were to describe someone by her precise height measured to the
tenth decimal place, the description would need to be adjusted on a regular basis.
The same is true if we describe her by the number of hairs on her head. Other-
wise, it would not fit the subject who is slightly changing in height and number
of hairs every day. Because we cannot simply perceive the height of a person or
the number of hairs exactly how they are, it is better to be less precise.
Consider the following two utterances:
(E1) “Anna is tall.”
(E2) “Anna has a height of 190cm.”
The utterance in (E1) is vague. However, if used as an answer to the question of
how tall Anna is, the utterance in (E2) normally does not convey something pre-
cise either but rather that Anna has a height of 190 cm plus or minus a few
nanometres or millimetres. If the degree of granularity is sufficiently coarse-
grained, (E2) could even communicate that Anna has a height of 190 cm plus or
minus a few centimetres. Strikingly, the expressions used in (E2) are perfectly
precise. What is vague is the implicit content.
12
This is what has been called
pragmatic vagueness.
13
Thus, it is better to utter the vague utterance in (E1) or at
least the pragmatically vague one in (E2) instead of the precise utterance in
(E3) “Anna has a height of 191.02 cm.”
For pragmatic reasons, a more coarse-grained degree of granularity is typically
not possible for (E3). The judgements expressed by (E1) and (E2) are more stable
because they are not susceptible to such small changes. It seems that vagueness
facilitates this stability in judgement.
14
12 I use the notion of implicit content as Bach (1994) introduced it.
13 See, for example, Endicott (2000, pp. 50–53).
14 For instance, de Jaegher and van Rooij (2011) or Kluck (2014) explicitly claim this.
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10 DAVID LANIUS

We can now try to explicate the argument from stability:
(1) Stability in judgement is desirable.
(2) The lack of (sharp) boundaries is a suitable means to achieve stability in
judgement.
(3) If the lack of (sharp) boundaries is a suitable causal means to bring about
something desirable, then vagueness has instrumental value.
(4) Thus: Vagueness has instrumental value.
At first glance, this looks like a promising candidate of an argument to establish
the value of vagueness. Unfortunately, however, a coarse-grained degree of granu-
larity is not identical to a lack of (sharp) boundaries. Stability is primarily facili-
tated by the generality of the expressions used.
However, generality must be differentiated from vagueness, as pointed out in
section 2.2. A higher generality entails that the utterance is true in more (possi-
ble) states of affairs. The utterances in (E1), (E2), and (E3) will all be accepted as
true (or appropriate) if Anna has a height of 191.02 cm. The utterances in
(E1) and (E2) will, however, also be accepted as true (or appropriate) if Anna has
a height of 190.98 cm. In contrast, if the speaker utters (E3), she can be reason-
ably criticized if Anna’s height differs even by a millimetre from 191.02 cm. The
lack of (sharp) boundaries is not what makes such less than fully precise utter-
ances useful. It is their generality. Thus, Pinkal does not establish anything about
the value of vagueness when he argues that sometimes a more coarse-grained
degree of granularity is useful. He establishes that generality can be valuable.
3.3 The argument from perception
It has also been argued that vagueness is useful because it adequately represents
our perceptual experience.
15
The unclarity caused by vagueness is, so the argu-
ment goes, a perfectly accurate reflection of the vague language’s object. Frie-
drich Waismann (1965, p. 210) illustrates this claim with the example of rain:
The picture of the rain I see is blurred. [...] I could not say of any exact description —e.g. for a
description mentioning an exact number of raindrops —that it describes my experience exactly.
Waismann’s argument can be reconstructed as the argument from perception in
the following way:
(1) It is desirable to adequately represent our perception.
(2) Our perception is blurred.
15 See, for example, Russell (1923), Wittgenstein (1953/2009), or Waismann (1965) for this claim.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
11WHAT IS THE VALUE OF VAGUENESS?

(3) Vagueness is a suitable (or maybe even indispensable) causal means to
adequately represent blurred perception.
(4) If vagueness is a suitable (or maybe even indispensable) causal means to
bring about something desirable, then vagueness has instrumental value.
(5) Thus: Vagueness has instrumental value.
Due to our blurred perception of the world a vague language is more useful than
a precise one because the former corresponds to our experience, whereas the lat-
ter simply misses the point.
Wittgenstein makes a similar argument:
The moment we try to apply exact concepts of measurement to immediate experience, we come up
against a peculiar vagueness in this experience. But that only means a vagueness relative to these
concepts of measurement. And, now, it seems to me that this vagueness isn’t something provi-
sional, to be eliminated later on by more precise knowledge, but that this is a characteristic logical
peculiarity. If, e.g., I say: ‘I can now see a red circle on a blue ground and remember seeing one
a few minutes ago that was the same size or perhaps a little smaller and a little lighter,’then this
experience cannot be described more precisely. (Phil. Remarks, p. 263)
But the argument continues:
Admittedly the words “rough,”“approximate”etc. have only a relative sense, but they are still
needed and they characterise the nature of our experience; not as rough or vague in itself, but still
as rough and vague in relation to our techniques of representation. This is all connected with the
problem “How many grains of sand make a heap?”(ibid.)
Inga Bones (2020, p. 7) thinks that “what Wittgenstein is getting at […] is that
compared to more sophisticated means of measurement and representation —
such as, for example, spectroscopic analysis —the nature of our experience is
rough, and this roughness is reflected in the vagueness or ‘fuzziness’of the
expressions of ordinary language.”However, it is worthwhile to look closer at
what is going on when we use vague language to describe our “fuzzy”or
“blurred”experiences. We may have a blurred picture of the rain (or of a red cir-
cle), and there is certainly not a number of raindrops that we directly perceive.
However, even if we grant for the sake of the argument that there is no more
exact description of our experience (which I doubt), the fuzziness involved is
not the right kind of fuzziness that relates to vagueness in the philosophical
sense. There may be unclarity about the number of raindrops, about their size,
and about other aspects of the weather conditions, but what we see is
clearly rain.
There seems to be an ambiguity and confusion about the term “fuzziness”.
Vagueness concerns fuzziness in the extension of the term “rain”; it does not con-
cern the fuzziness Waismann and Wittgenstein talk about. The phenomenon that
we perceive may more or less clearly qualify as rain, but this does not depend on
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
12 DAVID LANIUS

any fuzziness in our perception. It depends on the fuzziness in the term’s exten-
sion —on there being borderline cases and maybe on the boundarylessness and
Sorites susceptibility of the term. Evidently, borderline cases, boundarylessness,
and Sorites susceptibility can be there when perception is rather exact; we can
count the number of sand grains; and we may even know everything there is to
know about a particular arrangement of grains and it might still turn out to be a
borderline case of “heap.”
16
Conversely, our perception can be blurred, while at
the same time the term clearly applies to the phenomenon experienced —some-
thing that quite generally is the case when we use terms such as “rain”to clearly
refer to the rainfall that we perceive more or less blurredly.
3.4 The argument from memory
Perhaps we adequately represent our perceptual experience due to the generality
and broadness —and not the lack of (sharp) boundaries —of language. But our
cognitive limitations with respect to memory do seem to require the use of vague
language: vagueness is useful because it suits the way we store and retrieve infor-
mation —it suits the way our memory works.
17
Consider the following scenario
by van Deemter (2010, p. 263):
Suppose I told you that 324,542 people perished in some cataclysmic earthquake. For a while,
you might remember the exact death toll. In the longer term, however, details are likely to be
corrupted (if you remember the wrong number) or lost: the next day, you may only recall that the
victims numbered in their hundreds of thousands; a year later, you may only remember that there
were many.
Van Deemter concludes that human memory retains only the gist of information
due to evolutionary economization of memory space. At first glance, this really
seems like a promising argument to establish the value of vagueness. And,
indeed, something seems to draw (most of) us to vague terms such as “many”
when our memory is fading. What is it about “many”that we tend to prefer it to
other comparatively easily truth-assessable but perfectly precise expressions?
It seems that two cognitive mechanisms described by Alan Baddeley (2007)
support van Deemter’s conclusion. First, there is semantic coding by which only
the meaning of an expression is retained, whereas the expression itself is forgot-
ten. Second, there is chunking by which a handful of items are clustered together
as a unit. Chunking is clearly connected to generality. We lump together specific
information into general “chunks.”This is perhaps what is also going on in the
case of perception. If details are corrupted or lost, one can choose a more general
16 As an anonymous reviewer pointed out to me, this is very much at the core of the arguments con-
cerning inexact knowledge by Williamson (1992, 1994).
17 See Waismann (1953/1978), van Deemter (2010), Kluck (2010), or Kluck (2014) for this claim.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
13WHAT IS THE VALUE OF VAGUENESS?

expression instead of a more specific one. Even if you cannot remember whether
there are hundreds or merely tens of thousands people, you are usually still able
to say whether there are more than 1,000 (or more than 10,000, etc.).
Semantic coding, however, could be the source of vagueness. Maybe we really
think “vaguely.”Maybe, it is easier for our memory to retain a fuzzy image than
a clear-cut one. Let us assume for the sake of the argument that this is the case.
Then we can explicate the argument from memory:
(1) It is desirable to effectively memorize and remember information.
(2) Semantic coding is a suitable causal means to effectively memorize and
remember information.
(3) Vagueness is a necessary consequence of semantic coding.
(4) If vagueness is a necessary consequence of some suitable causal means to
something desirable, then vagueness has instrumental value.
(5) Thus: Vagueness has instrumental value.
Maybe our brains are such that we cannot but think and speak vaguely. Maybe
vagueness is a necessary feature of human thought and language. If one does not
have sufficient information and wants to be truthful, one must be less precise than
generally desirable. But the reason is not that the lack of (sharp) boundaries helps
one to stay truthful or to effectively memorize and remember information.
Chunking and semantic coding facilitate this, and vagueness may be a necessary
consequence of these mechanisms.
However, it is not enough that the vagueness of vague terms is a necessary
consequence of their other (actually valuable) properties for it to be valuable. As
Hrafn Asgeirsson (2015, p. 426) argued, “value only ‘transmits’from ends to
means but not to necessary consequences of those means.”That is, whereas
semantic coding facilitated by a vague and general term may be valuable in virtue
of being a means to effectively memorize and remember information, its vague-
ness (even though it might be a necessary consequence of semantic coding) is
not. The argument goes through only if we assume an extremely diminished sense
of value because only then premise (4) can be upheld.
3.5 The argument from metaphysics
Something very similar can be said for another argument that might be made for
the value of vagueness. Because the world is fuzzy and we want to correctly and
adequately represent the world, we should use vague language.
(1) It is desirable to correctly and adequately represent the world.
(2) The world is fuzzy.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
14 DAVID LANIUS

(3) Vagueness is an indispensable causal means to correctly and adequately
represent a fuzzy world.
(4) If vagueness is an indispensable causal means to something desirable,
then vagueness has instrumental value.
(5) Thus: Vagueness has instrumental value.
Premise (2) is, of course, highly controversial. But even if we would accept such
an ontology, the argument would still run into the same difficulties as the argu-
ment from memory and the argument from perception. There is an underlying
ambiguity of “fuzzy”that makes the arguments invalid. The fuzziness induced by
vagueness is simply not the same as the (assumed) fuzziness in perception, mem-
ory, or the world. Vagueness is not concerned with fuzziness in the object, but
with fuzziness in the extension of the term.
A term such as “mountain”is vague because it is susceptible to the Sorites par-
adox and there are (possible) borderline cases of mountains. It, so to speak, fuzz-
ily refers to some objects. The term can, however, also be used to (clearly) refer
to fuzzy things in the world. When referring to the mountain Zugspitze with the
phrase “this mountain”, we do not care whether the mountain begins at the camp
at an altitude of 1,000 meter, at the tree line at 2,000 meter, or anywhere in
between. But this fuzziness in the objects themselves has nothing to do with the
fact that there are borderline cases of mountains in the world.
In order to maintain premise (3), one would have to say that the world is fuzzy
not because there are fuzzy objects in it, but because there are things in it that we
fuzzily refer to with our vague language. Then, however, the argument begs the
question because the world cannot be considered fuzzy without reference to our
already existing vague concepts: Vague concepts are needed to correctly and ade-
quately represent a world that is fuzzily divided by these concepts.
3.6 The argument from learning
Some philosophers have hinted at the possibility that vagueness could improve
the learning of language. Maybe it is easier to learn vague terms than to learn pre-
cise ones? In fact, children first learn vague expressions and only later more diffi-
cult precise ones such as number words and mathematical operators. In general,
children learn expressions by reference to prototypes.
18
They learn the expression
“red”by reference to fire trucks, roses, and sunsets. They do not care about
boundaries between red things and non-red things.
Nora Kluck (2014) claims that vagueness plays the central role in the learning
strategy of overgeneralization. Consider a child who does not yet know the
18 See, for example, Rosch (1973) and Rosch et al. (1976).
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
15WHAT IS THE VALUE OF VAGUENESS?

correct expression for some object. She could either remain silent, thus giving
away even the chance to be understood; or, alternatively, she could overgeneralize
making herself at least potentially understood. Kluck claims that vagueness facili-
tates overgeneralization because it allows the child to freely move within the bor-
derline area. She can use a term that only borderline applies if she does not know
a better term which would clearly apply. Thus, she argues, overgeneralization is a
communicative strategy in language learning relying on the use of vagueness.
And this means that vagueness has a value. Let us explicate the argument from
learning in this way:
(1) Borderline cases help children to learn language due to
overgeneralization.
(2) If borderline cases help children to learn language due to over-
generalization, then vagueness has instrumental value.
(3) Thus: Vagueness has instrumental value.
Unfortunately, premise (1) is highly questionable. Children primarily over-
generalize in clear cases of non-application. Vagueness is neither conceptually
dependent on the phenomenon of overgeneralization nor empirically correlated.
Whereas overgeneralization might make it slightly easier for children to learn cer-
tain terms, borderline cases do not play any role here because children can over-
generalize by applying expressions to cases to which they clearly do not apply.
That is the point of overgeneralization.
More strikingly, the value of overgeneralization is highest when the child’s
audience can figure out that she wanted to clearly apply the expression —despite
it semantically not clearly or even clearly not applying. Borderline cases here are
least disadvantageous when their main effect is mitigated; they are not disadvan-
tageous when we can figure out that the term is intended to apply even though it
is in fact (semantically) a borderline case. Hence, also the argument from learn-
ing fails to show that vagueness has instrumental value.
3.7 The argument from cognitive costs
Maybe overgeneralization does not require vagueness but somehow vagueness
makes our language processing easier, does it not? There is reason to believe that
vague messages can be processed more easily than precise ones and, for that rea-
son, cause less cognitive costs. It is arguably easier to pronounce the term “many
people”than “324,542 people.”It is arguably also easier to hold the thought in
mind and to learn the term. Examples such as this one are often used in an
attempt to demonstrate that vagueness is valuable because it makes it easier for
us to use language. Let us explicate the argument from cognitive costs as follows:
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
16 DAVID LANIUS

(1) Vague language is (cognitively) easier to process than precise language.
(2) If vague language is (cognitively) easier to process than precise language,
then vagueness has instrumental value.
(3) Thus: Vagueness has instrumental value.
Premise (1) is uncontroversial: most people have a harder time to process mathe-
matical equations than statements about tall people or blue books. But can we
really infer from this that it is vagueness that makes it (cognitively) easier to pro-
cess these utterances?
It appears that we generally use language according to some form of the mini-
max principle, that is, we are as unclear and inarticulate as possible in order not
to waste energy, while at the same time as clear and articulate as necessary in
order to still get the message across. Hence, the goal of avoiding unclarity has to
be balanced with the goal of not wasting energy. This principle is a reformulation
of the so-called false Zipf’slaw.
19
If that is true, any formal proof for the value of
vagueness based on cognitive costs becomes trivial. The agents’expected utility
increases simply because costs decrease.
20
As Manfred Krifka (2009) points out, it is often beneficial for both speaker
and listener to use descriptions of the world at a more coarse-grained degree of
granularity. It is both linguistically and cognitively easier to process the utterances
in
(E1) “Anna is tall.”
and
(E2) “Anna has a height of 190 cm.”
than the utterance in
(E3) “Anna has a height of 191.02 cm.”
Cognitive costs seem to rise with degree of granularity. A coarse-grained degree
of granularity allows for loose interpretation even of precise terms. In general,
the more fine-grained the degree of granularity, the less pragmatically vague the
utterance can be interpreted. If the speaker uses several decimal places, there is
no room for the hearer to interpret her utterance loosely. The reason is the
19 It is not to be confused with Zipf’s law in mathematical statistics. Cf. Zipf (1949).
20 In the classical model by Spence (1974), sending a message is costly by itself and one could
increase the costs for precise messages by stipulation.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
17WHAT IS THE VALUE OF VAGUENESS?

conversational maxim of quantity: the speaker would not have added the informa-
tion if it were not required.
Conversely, the lack of decimal places indicates (in combination with other fac-
tors) that a few minutes more or less do not matter. This allows for pragmatic vague-
ness. However, the use of less precise terms and loose interpretation do not imply
that it is the vagueness of the less precise terms that accounts for less cognitive costs
when processing them. An imprecise term such as “tall”is not only vague but also
general; that is, “tall”has a larger extension than “a height of 191.02cm.”Amore
coarse-grained degree of granularity allows thus for more generality —a property of
language which clearly does make information processing easier (and also helps, as
we noted, to adequately represent our inexact perceptual experiences and memories).
Whereas it might be both informationally and economically best to use a vague
term, its vagueness does not become actual in most situations. Importantly, any
gains in processing vague terms are lost when actually confronted with borderline
cases. Then we have to think twice about what could have been meant by the
vague utterance. That is why vague terms are usually not used when confronted
with borderline cases; it would hinder the purposes of communication. Rather,
they are used to apply to clear cases. It might thus be the case that the costs of
using precise terms outweigh the expected costs of vague ones because the likeli-
hood of actually encountering borderline cases is negligible. The risk of running
into borderline cases is merely tacitly assumed.
Moreover, it is not entirely clear whether vagueness reduces cognitive costs
even in clear cases. The experiments by Green and van Deemter (2011, p. 5) do,
according to the authors’own conclusion, “more to cast doubt on the cost reduc-
tion hypothesis than to confirm it.”They think that other linguistic properties of
vague terms are actually effective in reducing the cognitive costs. Consider again
replacing the term “many”with the expression “more than 1,000”when reacting
to van Deemter’s earthquake scenario. What is done by replacing “324,542 people”
with “more than 1,000”is pruning away details, which may not be available any-
more (due to a fading memory) or which may simply be irrelevant. The same effect
is present when going from “324,542 people”to “many.”The vague term “many”
is not only more general than the precise “324,542 people,”it is also shorter, gram-
matically less complex, and devoid of mathematical concepts. These properties
alone explain why “many”is cognitively less costly than “324,542 people.”
More recent and impressively careful experiments by the same authors seem to
replicate these results. Although Green and van Deemter (2019, p. 83) found that
instructions containing vague words tended to be processed more rapidly and
more reliably by hearers than their precise equivalents, they could not rule out
that “the observed benefits of vague expressions may be due to factors other than
vagueness: factors like avoiding numbers, permitting comparison tasks, and range
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
18 DAVID LANIUS

reduction”as well as granularity and the use of evaluative terms.
21
Green and van
Deemter (2019, p. 84) conclude that in most situations it seems to be “relatively
unimportant whether a given expression is vague”or not.
Although it arguably is cost-efficient to use vague terms, it is doubtful that
their vagueness is the reason for it. A reduction in cognitive costs with vague
terms cannot by itself show that vagueness is valuable, that is, that borderline
cases, Sorites susceptibility, or the lack of (sharp) boundaries are effective. Again,
it seems that vagueness is only a side effect of other actually valuable —in this
case, cognitive cost reducing —properties of vague terms. That is why premise
(2) is not warranted —at least not without further evidence.
3.8 The argument from measurement
It is (surprisingly often) argued that vagueness is valuable because precision is
inefficient. In particular, it is said that the use of precise terms requires unneces-
sary and inefficient measurement.
22
Vagueness, in contrast, makes ordinary lan-
guage expressions fit for everyday use. Without vagueness we would have to
measure all day long the number of grains, the height of people, and the wave-
length of light when all we want is to talk about heaps, tall people, and blue
books. The argument can be reconstructed as follows:
(1) The use of precise language requires unnecessary and inefficient
measurement.
(2) If the use of vague language prevents unnecessary and inefficient mea-
surement, then vagueness has instrumental value.
(3) Thus: Vagueness has instrumental value.
Let us assume for the sake of the argument that premise (2) be true and let us
focus on premise (1). It certainly would be inefficient to measure the number of
grains each time we want to determine whether some arrangement of grains is a
heap. But does the use of precise language really require measurement?
It does not. It is perfectly conceivable to have a completely precise language
without the need of any additional measurement. The replacement of the vague
term “blue”with the precise term “blue*,”the extension of which is determined
by an exact range in the wavelength spectrum would not require any new mea-
surements. Most of the time we use language such that it clearly applies. For that
reason, no measurement is necessary because we do not come sufficiently close
21 See also Green and van Deemter (2013, p. 5). Cf. Kennedy (2011) for the relation between vague-
ness and comparisons.
22 See, for instance, Kluck (2014). Cf. also the Arguments from Perception,Memory and Cognitive
Costs.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
19WHAT IS THE VALUE OF VAGUENESS?

to the vicinity of the (sharp) boundary —independently of the term’s vagueness
or precision. Hence, we can most of time use the precise term “blue*”just as we
use our ordinary language expression “blue”.
On one hand, it is correct that measurement is necessary if the case at hand is
close to the sharp boundary of a precise term and we want to be certain about its
application to it. On the other, if the case is close to the fuzzy boundary of a
vague term, we are necessarily ignorant about its application. Both precision and
vagueness thus result in ignorance if we are close to the boundary of the term’s
extension. But precision has the additional advantage that ignorance can be
resolved by measurement. Precision does not require measurement but makes it
possible. Thus, premise (1) has to be rejected.
Gottlob Frege’sanalogy of the hand is sometimes cited as support for a variant
of the argument from measurement:
The [...] shortcomings result from a certain softness and changeability of language which [..] is a
precondition of its viability and versatility. In this respect, language can be compared to the hand
which, despite its ability to adapt itself to most diverse tasks, does not suffice. We created artificial
hands, tools for special purposes which work in such precision in which the hand could not. And
by what is this precision facilitated? Exactly by the rigidness, the parts’unchangeability, the lack
of which makes the hand so versatilely skilled.
23
Natural language is “soft “and “changeable.”It is multifunctional just like the
human hand. In contrast, tools such as screwdrivers are rigid. They can only be
used for particular purposes —and the same is true for the tools of formal logic.
The precision of screwdrivers and formal logic is facilitated by their rigidity. Sup-
posedly, the multifunctionality of natural language is based on its vagueness
because it is precision that makes formal logic so useful for its purpose. However,
this is a non sequitur. Let me show why.
Frege’s analogy —understood in this way —can be reconstructed as the fol-
lowing argument:
(1) Formal languages are useful because of their precision.
(2) If formal languages are useful because of their precision, natural language
must be useful because of its lack of precision.
(3) Thus: Natural language is useful because of its vagueness.
23 Translated from Frege (1882, p. 52): “Die [...] Mängel haben ihren Grund in einer gewissen
Weichheit und Veränderlichkeit der Sprache, die [...] Bedingung ihrer Entwicklungsfähigkeit und
vielseitigen Tauglichkeit ist. Die Sprache kann in dieser Hinsicht mit der Hand verglichen werden, die
uns trotz ihrer Fähigkeit, sich den verschiedensten Aufgaben anzupassen, nicht genügt. Wir schaffen uns
künstliche Hände, Werkzeuge für besondere Zwecke, die so genau arbeiten, wie die Hand es nicht ver-
möchte. Und wodurch wird diese Genauigkeit möglich? Durch eben die Starrheit, die Unveränderlichkeit
der Teile, deren Mangel die Hand so vielseitig geschickt macht.”
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
20 DAVID LANIUS

Frege’s analogy does not aim at the vagueness of natural language, and it is diffi-
cult to see how Frege himself could have understood it in this way. His analogy
applies to other properties of natural language, which happens to contain lots of
vague terms. Natural language is certainly multifunctional due to its generality,
polysemy, and various pragmatic features. We can use the same terms for differ-
ent purposes because they are polysemous. We can use the same terms to express
different contents because we make implicature and implicitures. We use meta-
phors and irony to convey virtually any imaginable content with the same limited
number of symbols. Has vagueness anything to do with this? It is at least highly
doubtful that Frege thought it did.
Frege was right to claim that the exactness of formal languages stems from
their precision. Premise (1) is arguably correct. Also, premise (2) seems
warranted. Frege’s analogy suggests that formal and natural languages have com-
plementary functions. Formal language is useful because it is precise. Natural
language is useful because it is multifunctional. However, there is no explanatory
link between the effects of precision in formal languages and the effects of vague-
ness in natural language. Lack of precision does entail much more than vague-
ness. It certainly does not entail that the adaptability and efficiency of natural
language stems from borderline cases, Sorites susceptibility, or the lack of (sharp)
boundaries.
24
To draw the conclusion (3), an additional premise is needed. We
would need the premise that the usefulness of imprecision entails the usefulness
of vagueness —a premise that is untenable.
3.9 The argument from adaptability
A common argument for the value of vagueness is that it makes language more
flexible and adaptable. In particular, it has been argued that the gradual widening
of a term’s extension along a Sorites series over time, and by a significant group
of speakers can facilitate a change in meaning that satisfies the communicative
needs of a language community. We can reformulate it in the following way:
(1) Sorites susceptibility makes language more adaptable.
(2) If Sorites susceptibility makes language more adaptable, then vagueness
has instrumental value.
(3) Thus: Vagueness has instrumental value.
24 Although it is true that vagueness correlates with context sensitivity and precision correlates with
context insensitivity, there is no necessary connection. It is conceivable, as Åkerman and Gree-
nough (2010) show, that vague terms are completely context insensitive and equally conceivable that pre-
cise terms are context sensitive.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
21WHAT IS THE VALUE OF VAGUENESS?

This argument can be traced back to Waismann (1945) and his discussion of
open texture, which was subsequently adopted by many legal scholars to show
that the same (legal) terms can be used in varying circumstances and with flexible
and adaptable meanings.
25
The law needs to be adaptable and the open texture
(or vagueness) of its language facilitates this, the arguments goes.
Also, Renate Bartsch seems at first glance to support the argument from
adaptability:
Semantic norms carry the possibility of change with them. Because of this, we can adjust our lan-
guage to change in our physical and social world. If vagueness and context-dependence of mean-
ings were not part of the meanings of words, language would be a less efficient means of
communication [...]. (Bartsch, 1984, p. 372)
Bartsch claims that vagueness and context-dependence of meaning make lan-
guage more efficient by facilitating semantic change. Gradual change in meaning
allows us to adjust our language efficiently in a changing and complex world.
Kluck (2014) argues, based on Bartsch’s claim, that vagueness facilitates adapt-
ability of language. Kluck considers the gradual changes underlying the Sorites
susceptibility of vague terms as evidence for premise (1). Indeed, meaning does
change gradually, and this is the reason why we can create Sorites series from an
old meaning to a new one. But does this entail that it is the Sorites susceptibility
which facilitates the change?
On closer examination, also the argument from adaptability fails. Bartsch actu-
ally attributes the possibility of change to polysemy and other forms of context
relativity, explicitly denying any role of vagueness in the philosophical sense:
Vagueness due to gradualness does, to my knowledge, not play a role in semantic change, while
vagueness due to contextual indeterminateness of a relative term can give rise to metonymic rela-
tionships in the structure of meaning [...]. (Bartsch, 1984, p. 374)
Whereas it could still be possible that vagueness plays some role, other prop-
erties of language are clearly efficient to facilitate long-term adaptability —
as, for instance, the figurative use of words and their context relativity. We can
metaphorically use a term consistently in the same way until its metaphorical
meaning becomes part of the lexical meaning of the then polysemous term.
26
For some reason, it seems to be particularly difficult to keep vagueness and
standard relativity apart. Even van Deemter, who writes on the value of vagueness
with matchless accurateness, seems to oscillate between both phenomena. He is
25 See, for example, Bix (1991). Cf. also Lanius (2019, pp. 37–38).
26 In a similar way fails an argument by Barwise and Perry (1983) for the value of vagueness, who
conflated standard relativity with vagueness. This is discussed by Green and van Deemter (2019, p. 65).
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
22 DAVID LANIUS

quite explicit about his intention to talk about vague terms in the philosophical
sense that they “allow borderline cases.”Nevertheless, he then asks why people
make “such frequent use of words whose meaning is difficult to pin down”and
why it is that “their meaning varies so much from one context to the next”(van
Deemter, 2010, p. 2). The first question points to polysemy (a term’s property of
having multiple semantically related meanings), whereas the second aims toward
standard relativity or context sensitivity more generally. Van Deemter also
describes vagueness and context sensitivity as two sides of the same coin, arguing
that terms such as “indecent”or “vehicle”are vague “because context affects the
interpretation of these words in ways that are impossible to foresee: their
precisification depends on who is it that does the precisifying”(van
Deemter, 2010, p. 269).
Context, of course, affects the interpretation of vague words. But what
exactly is its relation to borderline cases, Sorites susceptibility, or the lack of
(sharp) boundaries? Surely, vague terms can be precisified differently by dif-
ferent persons in different contexts. But before they can precisify them, they
need to determine a context of application. This requires them to settle on a
particular meaning, sense, or standard of “indecent,”“tall,”“blue,”or “many.”
The impossibility to foresee in which ways context affects the interpretation of
these terms stems from their property of being context sensitive —not from
their vagueness.
As long as it cannot be shown how vagueness can help to achieve any of these
features of language, premise (1) is unwarranted. There is no positive argument
for the claim that Sorites susceptibility or the lack of (sharp) boundaries allow for
gradual change in meaning. Nor is there any reason to believe that it is vagueness
(and not polysemy, standard relativity, or the possibility of figurative speech) that
allows the desired flexibility in the application of vague terms. Hence, neither
Sorites susceptibility nor the lack of (sharp) boundaries entail that vagueness
plays an interesting role in facilitating the impressive flexibility and adaptability
of language that we set out to explain.
4. Preliminary Conclusions
We have gathered nine arguments for the value of vagueness that can be found in
or reconstructed from the literature. Unfortunately, it seems that they have a num-
ber of recurring problems. Let us sum up.
Some arguments fail due to a kind of equivocation. They conflate the bound-
arylessness of vague terms with the (presumed) fuzziness of the (perception of
the) world (as in the argument from perception and the argument from metaphys-
ics). However, there is no relation between the fuzziness of a cloud or of our
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
23WHAT IS THE VALUE OF VAGUENESS?

perception of the rain on the one hand and the vagueness of the terms “cloud”
and “rain”on the other hand. A cloud does not end at any particular point, and
there are water droplets that neither clearly belong to the cloud nor clearly do not
belong to it. But the boundarylessness of the vague term “cloud”is fundamentally
different, as it can clearly apply to any such object, while still entailing that there
are some objects in the sky that are neither clearly clouds nor clearly not clouds.
In fact, the term “cloud”could be perfectly precise while still adequately and
coherently referring to fuzzy objects.
Another problem is that in many cases there is simply no way around vague-
ness. Sometimes (as in the argument from communicative success and the
argument from perception) it is close to impossible to find a fitting precise
description that could be substituted for the vague one (e.g., when talking
about the rain). In other cases, vagueness might theoretically be avoidable, but
there is no sufficient practical reason to actually do it (as in the argument from
cognitive costs). The costs of vagueness may not be high enough. In many sit-
uations, the vagueness of vague terms does not become a problem because
there are no borderline cases and no risk of running into Sorites series. In
other words, the vague terms are only potentially vague (there are possible
borderline cases) but not actually vague (there is no borderline case in the
given situation). If this is (likely to be) the case, there is no reason not to use
vague terms (even though there is no positive reason to use them for their
vagueness either).
There is yet another more serious and pervasive problem that inheres in most
arguments for the value of vagueness. Vague terms always have several properties
besides their vagueness. The problem is that in virtually all cases these properties
clearly are better candidates for suitable means to desirable ends than vagueness,
whereas there is no evidence that vagueness (in any viable sense) is suitable to
produce the desired effects (as in the argument from stability, the argument from
metaphysics, the argument from learning, the argument from memory, the argu-
ment from cognitive costs, the argument from measurement, and the argument
from adaptability). Thus, even when vagueness is detrimental, there can still be
sufficient reason to use vague terms if the expected benefits of some other prop-
erty of the vague terms outweigh the negative expected utility of encountering
borderline cases and Sorites series.
The ingenious experiments by Green and van Deemter indicate a number of
these properties such as the possibility to avoid mathematical terms, which are gen-
erally more difficult to process. Also, other forms of indeterminacy have a number
of effects that are falsely attributed to vagueness in these arguments. Consider the
term “many”again. As pointed out above, it is not only vague but also more gen-
eral than the original “324,542.”Moreover, it is relative to a contextually valued
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
24 DAVID LANIUS

standard and can thus vary in its extension in a way “324,542”or “more than
1,000”cannot. Nor do these precise expressions lean themselves toward figurative
or other pragmatic uses as do terms such as “many.”Furthermore, if a speaker says
that the number of perished people in an earthquake is “many,”she might not only
say something about the earthquake but also give a value judgement. She provides
information about her values and assessment of the state of affairs.
27
As Frank
Veltman put it, the speaker adds an opinion.
28
She can pragmatically imply that the
number is high relative to other contexts that she considers comparable. The audi-
ence can learn something about the speaker herself from her uttering “many”—
something they could not learn from her uttering “more than 1,000”or “324,542.”
We can thus identify three underlying problems with the nine arguments for the
value of vagueness discussed here. First, some of them equivocate two distinct
forms of fuzziness. Second, some of them use an impoverished understanding of
value, assuming vagueness to be valuable if it is unavoidable or not detrimental.
Third, some of them conflate vagueness with other properties of vague terms such
as generality or context sensitivity that are valuable in a more robust sense. It seems
to me that none of the nine arguments ultimately survives these problems.
29
Do we
have to conclude that there simply is no value of vagueness?
5. The Argument from Strategic Fallacy
If we do allow for something to have value if it is a suitable means to an end that
is desirable for someone, then another argument can be made. There may be
somewhat sinister functions for vagueness that have so far been overlooked in the
literature entirely. It is possible that someone exploits the vagueness of vague
terms in slippery slope arguments to make other people believe or do something
that she desires. The argument is the following:
(1) Someone desires to make other people φ.
(2) Making a slippery slope argument by exploiting the Sorites susceptibility
of vague terms is a suitable causal means to make other people φ.
27 Jucker et al. (2003) show, for instance, that imprecise speech acts better convey the speaker’s atti-
tude and inform also about her assumptions about the audience’s beliefs in addition to what is said.
28 This is discussed by van Deemter (2010, pp. 266–267) and van Rooij (2011, pp. 129–130). Both
refer to an inaugural lecture by Veltman (2002).
29 Also formal models have been developed by, for example, de Jaegher (2003), de Jaegher and van
Rooij (2011), Franke et al. (2011), Douven (2019), and Correia and Franke (2019), in an attempt to show
why vagueness is so ubiquitous in language despite its being apparently suboptimal. The results have
been mixed. Some models employ a notion of vagueness that is closer to generality or context sensitivity.
Those models that do not are much less conclusive in showing that there is value in using what the
authors call “vagueness”.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
25WHAT IS THE VALUE OF VAGUENESS?

(3) If making a slippery slope argument by exploiting the Sorites susceptibil-
ity of vague terms is a suitable causal means to bring about what some-
one desires, then vagueness has instrumental value.
(4) Thus: Vagueness has instrumental value.
This, now, is a function that clearly relies on the Sorites susceptibility of vague
terms. Premise (1) is evidently unproblematic. Premise (2) may require an exam-
ple to be warranted. But it is easy to produce one: Just consider the argumenta-
tion by many opponents to what they call “political correctness.”They argue that
if we were to restrict the freedom of speech, there will soon be a tyranny of politi-
cal correctness, which will prohibit even the most innocent expressions of opin-
ion. By doing so, they try to make other people stop criticizing them by
exploiting the Sorites susceptibility of the vague term “restriction to freedom of
speech.”Does a criticism restrict the freedom of speech? Does a taboo? A moral
rule with strict social sanctions? A legal provision? Indeed, a Sorites series with a
corresponding slippery slope argument can easily be constructed —and often in
public debate such argumentations do have the desired effect.
Finally, whereas premise (3) requires little from the concept of instrumental
value, it is not at all absurd —in contrast to the senses of “value”that were
required to make some of the other arguments work. Here, we need not assume
that vagueness has instrumental value simply due to its being an unavoidable side
effect of something desirable. The vagueness of the term has instrumental value
for the speaker because it helps to further her goal of manipulating other people.
This may be morally reprehensible, and certainly it is not a value in any objective
sense. Thus, although this might not be the function people have in mind when
claiming that vagueness is valuable, I take this to be the only “value”of vague-
ness that has (so far) been clearly shown.
30
6. Summary
Unfortunately, the discussion of the arguments for the view that vagueness is
valuable has not yielded very promising results. Most arguments fail because
either they only show that vagueness is not necessarily bad, that it is simply
unavoidable, or that some feature of vague language other than its vagueness has
30 There are other (more or less similar) arguments for the strategic use of vagueness, which fail, how-
ever, for the same reasons as the nine arguments of the value of vagueness discussed here; they conflate
vagueness with generality or standard relativity or in some other way fail to show that it is the vagueness
of the vague terms that can be strategically used. See Lanius (2019) for a discussion of some of these
arguments with respect to the law as well as for a general argument why vagueness is unlikely to play
any strategic role in communication at all.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
26 DAVID LANIUS

a value. Only the last argument for the value of vagueness as exploiting the Sori-
tes susceptibility in slippery slope arguments appears conclusive. Although there
seems to be some value of vagueness in the sense that it can strategically be
exploited, it proved to be much harder to show a more objective value of vague-
ness. If we are interested in value in this stronger sense, we should better refrain
from talking about “the value of vagueness”given that, at the moment, we do not
have a sufficiently good grasp of what such a value of vagueness could be.
Acknowledgements
I am very grateful to the participants of the research seminar at Karlsruhe Institute
of Technology for their constructive feedback to an earlier draft of this paper (in
particular, Gregor Betz, Inga Bones, and Christian Seidel), as well as to my two
anonymous reviewers for their incredibly useful comments. Open Access funding
enabled and organized by Projekt DEAL.
References
ÅKERMAN, J. and GREENOUGH, P. (2010) “Hold the Context Fixed, Vagueness Still Remains.”
In R. Dietz and S. Moruzzi (eds), Cuts and Clouds: Vaguenesss, its Nature, and its
Logic, pp. 275–288. Oxford; New York: Oxford University Press.
ASGEIRSSON, H. (2015) “On the Instrumental Value of Vagueness in the Law.”Ethics 125(2):
425–448.
ASGEIRSSON, H. (2020) The Nature and Value of Vagueness in the Law. Oxford: Hart
Publishing.
BACH, K. (1994) “Conversational Impliciture.”Mind & Language 9(2): 124–162. https://doi.
org/10.1111/j.1468-0017.1994.tb00220.x.
BADDELEY, A. (2007) Working Memory, Thought, and Action. Oxford: Oxford University
Press.
BARTSCH, R. (1984) “Norms, Tolerance, Lexical Change, and Context-Dependence of Mean-
ing.”Journal of Pragmatics 8(3): 367–393.
BARWISE, J. and PERRY, J. (1983) Situations and Attitudes. Cambridge, MA: MIT Press.
BIX, B. (1991) “H. L. A. Hart and the “Open Texture”of Language.”Law and Philosophy
10: 51–72.
BLACK, M. (1937) “Vagueness: An Exercise in Logical Analysis.”Philosophy of Science 4
(4): 427–455.
BONES, I. (2020) How to Swim in Sinking Sands: The Sorites Paradox and the Nature and
Logic of Vague Language. Paderborn: Mentis.
CHOI, A. H. and TRIANTIS, G. G. (2010) “Strategic Vagueness in Contract Design: The Case
of Corporate Acquisitions.”Yale Law Journal 119: 848–924.
CORREIA, J. P. and FRANKE, M. (2019) “Towards an Ecology of Vagueness.”In R. Dietz
(ed.), Vagueness and Rationality in Language Use and Cognition, pp. 87–113.
New York; Berlin: Springer.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
27WHAT IS THE VALUE OF VAGUENESS?

DE JAEGHER, K. (2003) “A Game-Theoretic Rationale for Vagueness.”Linguistics & Philoso-
phy 26: 637–659.
DE JAEGHER, K. and VAN ROOIJ, R. (2011) “Strategic Vagueness, and Appropriate Contexts.”In
A. Benz, E. Christian, G. Jäger and R. van Rooij (eds), Language, Games, and Evolution,
pp. 40–59. Trends in Current Research on Language and Game Theory. Berlin: Springer.
DOUVEN, I. (2019) “The Rationality of Vagueness.”In R. Dietz (ed.), Vagueness and Ratio-
nality in Language Use and Cognition, pp. 115–134. New York; Berlin: Springer.
ENDICOTT, T. A. O. (2000) Vagueness in Law. Oxford: Oxford University Press.
ENDICOTT, T. A. O. (2005) “The Value of Vagueness.”In V. K. Bhatia, J. Engberg, M. Gotti
and D. Heller (eds), Linguistic Insights:Vo l .yes. Vagueness in Normative Texts, pp. 27–
48. Bern: Lang.
FRANKE, M., JÄGER, G., and VAN ROOIJ, R. (2011) “Vagueness, Signaling and Bounded Ratio-
nality.”Logic and Engineering of Natural Language Semantics 7:41–55.
FREGE, G. (1882) “Über die wissenschaftliche Berechtigung einer Begriffsschrift.”Zeitschrift
für Philosophie und philosophische Kritik 81: 48–56.
GREEN, M. J. and VAN DEEMTER, K. (2011) “Vagueness as Cost Reduction: An Empirical
Test.”Proceedings of the ‘Production of Referring Expressions’Workshop at the Annual
Meeting of the Cognitive Science Society, Vol. 33, pp. 1–6. https://pre2011.uvt.nl/pdf/
green_van_deemter_7_8.pdf.
GREEN,M.J.,andvanDEEMTER, K. (2013). “The Utility of Vagueness: Does It Lie Elsewhere?”Pro-
ceedings of the ‘Production of Referring Expressions’Workshop at the Annual Meeting of the
Cognitive Science Society, Vol. 35, pp. 1–5. https://pre2013.uvt.nl/pdf/green-vandeemter.pdf.
GREEN, M. J. and VAN DEEMTER, K. (2019) “The Elusive Benefits of Vagueness: Evidence
from Experiments.”In R. Dietz (ed.), Vagueness and Rationality in Language Use and
Cognition, pp. 63–86. New York; Berlin: Springer.
GRICE, P. (1989) Studies in the Way of Words. Cambridge, MA: Harvard University Press.
HADFIELD, G. K. (1994) “Weighing the Value of Vagueness: An Economic Perspective on
Precision in the Law.”California Law Review 82(3), 541–554.
HEMPEL, C. G. (1939) “Vagueness and Logic.”Philosophy of Science 6(2): 163–180.
JUCKER, A. H., SMITH, S. W., and LÜDGE, T. (2003) “Interactive Aspects of Vagueness in
Conversation.”Journal of Pragmatics 35: 1737–1769.
KEEFE, R. (2000) Theories of Vagueness. Cambridge: Cambridge University Press.
KENNEDY, C. (2011) “Vagueness and Comparison.”In P. Égré and N. Klinedinst (eds),
Vagueness and Language Use, pp. 73–97. Houndmills; Basingstoke; Hampshire;
New York: Palgrave Macmillan.
KLUCK, N. (2010) “Some Notes on the Value of Vagueness in Everyday Communication.”In
E. Hüllermeier, R. Kruse and F. Hoffmann (eds), Communication in Computer and
Information Science. Information Processing and Management of Uncertainty in
Knowledge-Based Systems. Applications, pp. 344–349. Berlin, Heidelberg: Springer.
KLUCK, N. (2014) Der Wert der Vagheit [The Value of Vagueness]. Berlin; Boston: de
Gruyter. https://doi.org/10.1515/9783110347791.
KRIFKA, M. (2009) “Approximate Interpretations of Number Words: Case for Strategic Com-
munication.”In E. W. Hinrichs and J. Nerbonne (eds), Theory and Evidence in Seman-
tics, pp. 109–132. Stanford: CSLI Publications.
LANIUS, D. (2019) Strategic Indeterminacy in the Law. New York: Oxford University Press.
LIPMAN, B. L. (2000) In A. Rubinstein (ed.), Comments: Economics and Language. Econom-
ics and Language, pp. 114–123. New York: Cambridge University Press.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
28 DAVID LANIUS

LIPMAN, B.L. (2009). Why Is Language Vague? Working Paper. Retrieved from Boston Uni-
versity website: http://people.bu.edu/blipman/Papers/vague5.pdf
MCNEILL, D. and FREIBERGER, P. (1993) Fuzzy Logic. New York: Simon & Schuster.
PARIKH, R. (1994) “Vagueness and Utility: The Semantics of Common Nouns.”Linguistics &
Philosophy 17(6): 521–535.
PEIRCE, C. S. (1960) “Pragmatism and Pragmaticism.”In C. Hartshorne, P. Weiss and A. W.
Burks (eds), Collected Papers. Cambridge, MA: Harvard University Press.
PINKAL, M. (1995) Logic and Lexicon: The Semantics of the Indefinite. Dordrecht: Springer.
QUINE, W. V. O. (1960) Word and Object. 24th ed. Cambridge, MA: MIT Press.
ROSCH, E. H. (1973) “Natural Categories.”Cognitive Psychology 4(3): 328–350.
ROSCH, E. H., MERVIS, C. B., GRAY, W. D., JOHNSON, D. M., and BOYES-BRAEM, P. (1976)
“Basic Objects in Natural Categories.”Cognitive Psychology 8(3): 382–439.
RUSSELL, B. (1923) “Vagueness. The Australasian Journal of Psychology and.”Philosophy 1
(2): 84–92.
SAINSBURY, R. M. (1997) “Concepts without Boundaries.”In R. Keefe and P. Smith (eds),
Vagueness: A Reader, pp. 251–264. Cambridge, MA: MIT Press.
SORENSEN, R.A. (1997). Vagueness. Retrieved from http://plato.stanford.edu/entries/vagueness/
SORENSEN, R. A. (2001) “Vagueness Has no Function in Law.”Legal Theory 7(04): 387–417.
SPENCE, A. M. (1974) Market Signaling: Information Transfer in Hiring and Related Pro-
cesses. Cambridge, MA: Harvard University Press.
STATON, J. K. and VANBERG, G. (2008) “The Value of Vagueness: Delegation, Defiance, and
Judicial Opinions.”American Journal of Political Science 52(3): 504–519.
SWINBURNE, R. G. (1969) “Vagueness, Inexactness, and Imprecision.”The British Journal for
the Philosophy of Science 19(4): 281–299.
VAN DEEMTER, K. (2009) “Utility and Language Generation: The Case of Vagueness.”Jour-
nal of Philosophical Logic 38(6): 607–632.
VAN DEEMTER, K. (2010) “Not Exactly.”Praise of Vagueness. Oxford: Oxford University
Press.
VAN ROOIJ, R. (2011) “Vagueness and Linguistics.”In G. Ronzitti (ed.), Vagueness: A Guide,
pp. 123–170. Dordrecht: Springer.
VELTMAN, F. (2002) Het Verschil Tussen “Vaag”en “Niet Precies”(the Difference between
“Vague”and “Not Precise”). Amsterdam: Vossiuspers.
WAISMANN, F. (1945) “Open Texture.”Proceedings of the Aristotelian Society, Supplemen-
tary Volumes 19: 119–150.
WAISMANN, F. (1965) The Principles of Linguistic Philosophy. London: Macmillan.
WAISMANN, F. (1978) “Language Strata.”In A. Flew (ed.), First Series. Logic and Language.
Oxford: Basil Blackwell (Original work published 1953).
WILLIAMSON, T. (1992) “Inexact Knowledge.”Mind 101(402): 217–242.
WILLIAMSON, T. (1994) Vagueness. London: Routledge.
WITTGENSTEIN, L. (2009) Philosophical Investigations. 4th ed. Chichester: Wiley-Blackwell
(Original work published 1953).
WRIGHT, C. (1975) “On the Coherence of Vague Predicates.”Synthese 30(3/4): 325–365.
WRIGHT, C. (1997) “Language-Mastery and the Sorites Paradox.”In R. Keefe and P. Smith
(eds), Vagueness: A Reader, pp. 151–173. Cambridge, MA: MIT Press (Original work
published 1976).
ZIPF, G. K. (1949) Human Behavior and the Principle of Least Effort: An Introduction to
Human Ecology. Cambridge, MA: Addison-Wesley Press.
© 2021 The Authors. Theoria published by John Wiley & Sons Ltd on behalf of Stiftelsen Theoria.
29WHAT IS THE VALUE OF VAGUENESS?