ENCYCLOPEDIA OF COGNITIVE SCIENCE

Vagueness
Achille C Varzi
Columbia University, New York

(Final version published in Lynn Nadel (editor in chief), Encyclopedia of Cognitive Science, London: Macmillan and Nature Publishing Group, 2003, pp. 459–464)

What is vagueness?
Standardly, one says that vagueness arises whenever a concept or linguistic expression admits of borderline cases of application. A predicate such as ‘bald’, for example, is vague because there can be situations in which it is indeterminate whether or
not it applies to (a name of) a certain object. Some people are clearly bald (Picasso),
some are clearly not bald (the count of Montecristo), and some are borderline
cases—our concept of baldness and our linguistic practices do not specify any exact
number of hairs that marks the boundary between the bald and the non-bald. Similarly, a singular term such as ‘Mount Everest’ is vague because there is no determinate way of tracing the geographical limits of its referent. Some rocks are clearlypart of Everest and some are clearly not, but some rocks enjoy a borderline status.
There is, however, dispute concerning this way of characterizing vagueness. For
the statement that a term t admits of borderline cases of application—that it is indeterminate whether or not such-and-such objects fall within the boundaries of the entity designated by t—can be given a de re reading, as in (1), or a de dicto reading, asin (2):(1)

The term t designates an entity x such that it is indeterminate whether suchand-such objects fall within the boundaries of x.

(2)

It is indeterminate whether the term t designates an entity x such that suchand-such objects fall within the boundaries of x.

On the first reading the indeterminacy is ontological. The predicate ‘bald’ isvague, on this reading, because it stands for a vague set: there is no objective, determinate fact of the matter about whether the borderline cases are included in that set(or about whether they enjoy the corresponding property). Likewise, on this reading‘Mount Everest’ is vague because it stands for a genuinely vague denizen of reality:1there is no objective, determinate fact of the matter about whether the borderline
rocks are part of the mountain. There may also be no determinate fact of the matter
about when the mountain itself came into being, for the temporal boundaries of an
object may be vague too.
By contrast, the de dicto reading corresponds to a purely linguistic (or conceptual) notion of vagueness. On this view the set of bald people is not vague at all.
There are exactly 2n sets of people (where n is the number of all people at the present time), each with its perfectly precise membership function; yet our linguistic
stipulations do not fully specify which of those sets can do duty for the extension of
the predicate ‘bald’. There is, similarly, no vague mountain on this view: instead
there are plenty of aggregates of matter, each with its precise spatio-temporal boundary, and when we say ‘Mount Everest’ we are just being vague as to which such aggregate we are referring to.
The two views are not strictly incompatible, at least insofar as one may be willing to treat some vagueness as ontological and some as linguistic. One could also
construe some terms as involving both sorts of vagueness: it would be indeterminate
which particular sets or objects those terms designate, and the relevant candidates would include vague specimens along with sharp ones. However, these waysof combining ontological and linguistic vagueness have attracted little attention andcurrent views on vagueness divide rather clearly between one approach and theother. (See Tye 1990 and Lewis 1993, respectively, for two representative position
statements, and Evans 1978 for a much debated way of setting up the issue.)
Problems and paradoxes
Insofar as vagueness involves borderline cases, whether de re or de dicto, it manifests itself semantically in the generation of truth-value gaps. If Jones is a baldish
person, then the statement that he is bald appears to lack a definite truth-value; if it isindeterminate whether this rock is part of Everest, then the statement that it is part ofEverest is likewise neither true nor false. This has been a natural source of concern
for philosophers and logicians since Frege, for the admission of truth-value gaps
amounts to a failure of the classical principle of bivalence.
The main source of concern, however, is that vagueness precipitates a deep puzzle. For not only do vague terms involve borderline cases. They also seem to involve
borderline borderline cases, or borderline borderline borderline cases. For example,
just as there is no sharp line between the bald and the non-bald there does not seemto be any sharp line between the bald and the baldish (or the baldish-ish). In neithercase can a single hair make a difference. Intuitively, this means that our notion ofbaldness satisfies the following principle:

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(3)

For every n: if a man with n hairs on his head are bald, then a man with n +1 hairs on his head is also bald.

(Let us suppose that baldness supervenes exclusively on the number of hairs.) However, it is enough to combine this principle with
(4)

A man with no hairs on his head is bald.

to reach the paradoxical conclusion that(5)

A man with 500,000 hairs on his head is bald.

(This can be shown by 500,000 repeated applications of the rules of universal instantiation and modus ponens.) In other words, the intuition that the applicability of‘bald’ cannot be a matter of a single hair seems to force us to reason from the true
premise that Picasso is bald to the false conclusion that the count of Montecristo isalso bald. And a corresponding point can be made about the intuition that the applicability of ‘Everest’ is not a matter of millimeters. In both cases it’s hard to come up
with a diagnosis, but the clash between logic and intuition is deep.
In its oldest form, this problem is known as the phalakros puzzle (from theGreek word for ‘bald’) and is attributed to Eubulides of Miletus, a contemporary ofAristotle. Eubulides is also standardly credited with the formulation of the sorites
puzzle, which builds in a similar way on the vagueness of ‘heap’ (Greek: soros).
There are also versions of the puzzle that rely on a different way of expressing theinductive principle (3). For example, already the Stoics considered replacing the embedded conditional ‘if ... then’ with a negated conjunction (it is not the case that a
man with n hairs on his head is bald and a man with n + 1 hairs is not bald). This
makes the paradox even harder, since one cannot just blame the material conditional
for the trouble. Another common variant involves replacing (3) with a long chain ofconditionals (or negated conjunctions), one for each relevant n. Again this makes the
puzzle more robust, since one cannot blame the universal quantifier for the trouble.
As it turns out, today one does not make a big deal of these differences and all such
puzzles are collectively referred to as sorites puzzles. The problem with vagueness isthat it yields such puzzles, in some form or other. And theories of vagueness arenaturally compared on the basis of how successful they are in providing a systematicway out.
Theories of vaguenessBroadly speaking, there are two strategies for dealing with the sorites paradox. Focusing on the version exemplified by (3)–(5), one can either (i) reject the argument3

as invalid, or (ii) reject it as valid but unsound. (One can also bite the bullet and accept the conclusion, but few would be willing to go that far.)
Strategy (i) comes in two main varieties. On the one hand, one can just insist
that logically valid reasoning can only be formulated in a precise language. This was,for example, the response advocated by Russell (1923) in the first full-length paper
devoted to the topic of vagueness. Today this is not a popular position because it
enforces an intolerable restriction on the scope of logic: since many of the words
that we use in ordinary discourse (as well as in much scientific discourse) are vague,
logic would be of very little practical use. On the other hand, one can question thevalidity of the sorites argument by questioning the adequacy of classical logic. Herea popular stance is to adopt some kind of many-valued logic in which statements areallowed to take intermediate truth-values and in which the validity of the inference of(5) from (4) decreases as the number of application of modus ponens increases.
In fact, because the notion of a borderline case is itself vague, a natural implementation of this strategy allows for a continuum of intermediate truth-values andthe result is a fuzzy logic in which sentential connectives, for example, are represented by operations on the real numbers in the interval [0,1] rather than on the twovalued truth set {0,1}. (See e.g. Machina 1976.) If vagueness is thought of as an
ontological phenomenon, this account is naturally combined with a fuzzy semantics
in which a predicate, for example, is assigned an extension whose membership function is itself continuum-valued. The closer to 1 the value is, the more the argument isa member of the set (Zadeh 1965). This way of proceeding allows one to resolve theparadox as follows. First, the connectives are characterized so that a conditional of
the form(6)

If a man with n hairs on his head is bald, then a man with n + 1 hairs on hishead is also bald

is sure to come out true or nearly true. For example, on a popular account the truthvalue of a conditional is set equal to 1 minus the surplus of the antecedent over theconsequent (if any). Second, there will be values of n such that the truth-value of theantecedent of (6) is slightly higher than that of the consequent. The underlying intuition is that one hair does make some difference after all, albeit a very small andnegligible difference. Thus, as long as validity is defined so that the conclusion of avalid argument must be at least as true as each of the premises, the relevant instancesof modus ponens will be invalid whenever the truth-value of the antecedent is lessthan or equal to the truth-value of the conditional but (slightly) greater than the truthvalue of the consequent. The paradox arises because the error is so small as to beundetectable, and yet it compounds each time a new application of modus ponens isinvoked.

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This account has a certain prima facie appeal but it is open to a number of objections. For one thing, the fuzzy-theoretic machinery appears to replace vaguenesswith extremely refined precision. To what degree, exactly, is it true that Jones isbald? To degree 0.6? Perhaps to degree 0.59? Or maybe 0.5999? Second, the assumption of a totally ordered set of truth-values is itself problematic. How does thedegree to which Jones is bald compare to the degree to which Smith is tall? How
does it compare to the degree to which a certain borderline rock is part of Everest?
Third, there is the embarrassing presupposition that a point must still exist where
one goes from fully fledged truth to partial truth, or from partial truth to fully
fledged falsehood. What is the maximum value of n such that a person with n hairsis truly bald, i.e., bald to degree 1? What is the last rock, along a continuous pathdescending from the peak of Everest, which is definitely part of the mountain? All ofthese are questions which may not have practical relevance but which appear to undermine the theoretical force of the account.
Turning to strategy (ii) for dealing with the sorites paradox—to accept the validity of the argument but to reject one of its premises as false—one can again distinguish two main approaches. One can either reject the “base step” expressed by(4), or one can reject the “inductive premise” expressed by (3). (In the version of
the paradox where (3) is replaced by a chain of conditionals, this amounts to rejecting one of the conditionals. Here we shall not elaborate on this variant.) A rejectionof (4) amounts to a radical response to the effect that a vague term such as ‘bald’ isultimately incoherent. (See e.g. Unger 1979.) Given the pervasiveness of such terms
in natural language, this line of response seems to have little advantages over Russell’s version of the first strategy. A rejection of (3), on the other hand, amounts toasserting the existence of a precise number n of hairs separating the bald from thenon-bald. This appears to contradict the intuition that ‘bald’ is vague and, by generalization, that there are any vague words at all. Indeed, this conflict is real and ineliminable if vagueness is understood entirely in ontological terms, for then the existence of a relevant cut-off value of n amounts to the existence of a sharp boundaryaround the relevant set or object. However, if vagueness is understood in linguistic
terms there is one popular way of resolving this intuitive conflict. This is what hascome to be known as supervaluationism (Fine 1975).
The basic idea underlying supervaluationism is that a vague term is one that
admits of various alternative “precisifications”. A vague predicate such as ‘bald’,for instance, could be made precise by deciding that a man is bald if and only if he
has at most 10,000 hairs. Or it could be made precise by deciding that a man is bald
if and only if he has at most 9,999 hairs. And so on. The predicate is vague precisely
because there is indeterminacy between these various ways of picking out a precise
cut-off value. Likewise, a vague singular term such as ‘Everest’ could be made pre-5

cise by drawing a precise boundary around its referent, but there are many ways of
doing this and all of them are compatible with the way we use the name. Given thisunderstanding of vagueness, supervaluationism says that the truth-value of a statement involving vague terms is a function of its truth-values under the various admissible precisifications of those terms. If the statement is true under all such precisifications, then it is true simpliciter: the unmade linguistic stipulations don’t matter. In
other words, it makes no difference to suppose that the meaning of those expressions could be defined more precisely: what the statement says is true regardless (or“super-true”). Likewise, if the statement comes out false under every precisification
then we may regard it as false (or super-false) in spite of its vagueness. This explains, for example, why we can confidently assert (4) and deny (5). On the other
hand, when a statement comes out true under some precisifications and false underothers the unmade linguistic stipulations become relevant. In such cases, the statement suffers a truth-value gap. This is why, for example, we must suspend judgment
when it comes to statements of the form
(7)

A man with n hairs on his head is bald

for various intermediate values of n: the truth-value of such statements depends crucially on how we imagine the extension of ‘bald’ to be precisified.
As it turns out, this account preserves all theorems of classical logic eventhough it violates some of its fundamental semantic presuppositions, such as bivalence and truth-functionality. For example, an instance of the law of the excluded
middle such as(8)Either a man with n hairs on his head is bald, or he is not bald.

is sure to come out true even when both disjuncts suffer a truth-value gap. It is precisely this sort of non-standardness that allows supervaluationism to explain away
the sorites paradox. Supervaluationally the inductive premise (3) is false because itcomes out false on every precisification. However, contrary to the standard semantics for the quantifiers, the falsity of (3) does not imply the existence of a specific n
for which the corresponding conditional (6) is false, and this is what allows a supervaluationist to save the intuition that ‘bald’ is vague. Supervaluationally it is true that
there is a number n of hairs that marks the boundary between bald and non-bald, butthere is no number n of hairs such that it is true of it that it marks the boundary.
(The same account applies, mutatis mutands, to a sorites for a singular term such as‘Everest’.)This account is attractive because it reflects a deep, preanalytical intuition concerning vagueness as it arises in ordinary language: we speak vaguely because inordinary circumstances the vagueness of our words does not matter. Still, various

6objections have been raised. For example, some critics just consider the supervaluationary account of the logical operators in (8) and (3) unacceptable. Moreover, supervaluationists have been pressed to provide an account of the phenomenon ofhigher-order vagueness. This difficulty manifests itself not only in the supposition
that there is a clear demarcation between the clear cases and the borderline cases (asin fuzzy logic) but also in the supposition that each vague term comes with a precise
set of precisifications. Presumably, if ‘bald’ could be made precise by deciding thata man is bald if and only if he has at most n hairs, then it could be made precise by
deciding that a man is bald if and only if he has at most n+1 hairs—and this yields
immediately a sorites paradox for the semantic predicate ‘could be made precise’.
For a supervaluationist this only shows that the metalanguage within which the semantics is formulated is itself vague, but some critics find this line of response unsatisfactory. Lastly, the very assumption that every vague expression can in principle
be precisified, or that any number of vague expressions can in principle be simultaneously precisified, has sometimes been regarded with suspicion.
Vagueness and cognitive science
To the extent that vagueness is not entirely a matter of ontology, it falls naturally
within the range of interest of the cognitive sciences. Supervaluationism, for example, may be viewed as implementing a certain view about how ordinary speakers
manage to communicate and reason even in the absence of a precise language.
We speak vaguely because in normal circumstances the vagueness of our words
does not matter. In normal circumstances what we say is true under all the admissible interpretations of our words, hence we don’t bother being more precise (Lewis
1993).
More generally, the linguistic conception of vagueness has often been associated with the idea that language is but one of many different representation systems.
Thoughts and mental images, for some authors, can likewise suffer from the phenomenon of vagueness, and so can every private or publicly accessible representation. Russell himself combined his conservative views on logic with the view that allvagueness is analogous to the vagueness that may exist in a photograph, let alone thefigures of an impressionist painting.
It is not clear, however, whether one and the same account can indeed be madeto fit all these different cases (Dummett 1975). Compare the vagueness of ‘bald’with that of ‘looks bald’. If Jones is a borderline case of the latter predicate, a linguistic account would have to say that on certain precisifications Jones will look bald
(to me) even though his identical twin, who has just one more hair on his head, will
not look bald. Since the two men look exactly alike to me, this seems to contradict7

the idea that the predicate ‘looks bald’ is entirely observational, i.e., that it applies
only in virtue of appearances. A similar point can be made for observational predicates such as ‘looks square’ (where ‘square’ is non-vague), or for any other predicate expressing properties whose reality, as some like to say, is their appearance—e.g., color predicates.
For another example, if there is such a thing as the language of thought, or
Mentalese, then it would seem to suffer from a different sort of vagueness than public languages, at least to the extent that the meaning of Mentalese expressions does
not depend on their use. A supervaluationary account would therefore seem unjustified in this case. A fuzzy-theoretic semantics would also be inadequate because of
the psychologically unrealistic rich of nuances in the underlying space of truthvalues. In the case of public languages one may try to ground a fuzzy truth-value
assignment on statistical measurements, but Mentalese would defy this way of proceeding. (There is a tradition of psychological studies aimed at measuring the degree
to which people are inclined to classify a penguin as a bird, say, but this is irrelevant
here: something may fail to be a typical P without being a borderline case of P, justas a perfectly clear case of prime number may fall short of typicality. See Armstrong
et al, 1983.)
Sorensen (1991) has suggested that cases such as these favor an epistemic account whereby vagueness is a kind of ignorance. On this account, the indeterminacy
associated with a vague expression stems primarily from our inability to determine
its exact reference (extension). More generally, the epistemic account has been proposed as an alternative to all the theories mentioned in the previous section, also because it provides a straightforward way out of all sorts of sorites paradoxes. If thevagueness of ‘bald’ is a matter of ignorance, then a critical cut-off value of n does
exist which separates the bald from the non-bald, except that it is unknown to us.
Moreover, the relevant value cannot be known by us and this would explain our inclination to regard a statement such as (3) as true when it is, in fact, false. In thissense, epistemicism can be viewed as an alternative to supervaluationism in providing
an implementation of strategy (ii) above. Both validate all theorems of classical logic;
unlike supervaluationism, however, epistemicism also validates all classical semantic
presuppositions, including the principle of bivalence.
The epistemic account of vagueness is generally met with astonishment. How
can there be a sharp boundary demarcating the extension of ‘bald’ if nobody everbothered making the necessary semantic stipulations? What could possibly be theexplanation of the relevant ignorance? One response, articulated in some detail by
Williamson (1992), is that the boundaries associated with vague terms are unknowable because they violate a general principle that characterizes reliable knowledge.
Briefly, this is a principle to the effect that our beliefs are reliable only if we leave a8

margin for error. For example, the belief that a general condition obtains in a particular case can be reliably true only if that condition obtains in every similar case(the relevant notion of similarity depending on context and cognitive capacities). In
the case at issue this would mean that we cannot know that a certain person is bald if
people with just one more hair on their head are not bald. The vagueness of a predicate such as ‘bald’ would then be captured, intuitively, not by a principle such as (3)but rather by a margin-of-error principle such as (9):(9)

For every n: if a man with n hairs on his head is known to be bald, then aman with n + 1 hairs on his head is bald.

And this principle does not combine with (4) and (5) to generate a paradox even ifclassical logic is retained holus bolus.
Some support for the epistemic conception of vagueness seems to derive fromrecent experimental data. Notably, Bonini et al (1999) have found that ordinary
speakers react to questions about vague predicates as if they were not sure about
their boundaries, which leads to the hypothesis that vague predicates are mentally
represented like sharp predicates with crisp true/false boundaries of whose location
one is uncertain. On the other hand, such findings seem compatible also with theview that vagueness is at bottom a phenomenon that reflects the fluid judgmental
spreadings involved in human categorization. According to Raffman (1994), ordinary subjects are always likely to break the slippery slope of a sorites series precisely because a sharp category shift is likely to occur at some point on each run of
judgments. The point of shift varies with the judgments of different speakers andthose judgments in turn vary with the contexts in which they are made. Rather than
explaining this phenomenon in epistemic terms, however, Raffman conjectures thatthe point of shift is determined by a constellation of psychological factors, such as
the strength of the judgmental inertia induced by the anchoring heuristics employedby the subjects as they proceed along the series. (One will categorize a greater number of people as bald if one begins from the hairless side of a corresponding sorites
series than if one begins from the hairy side: see Tversky and Kahneman 1974.) In
other words, the relevant category shifts are not to be viewed as boundary crossings
but as Gestalt-like changes of perspective. If this is right, then it is also plausible tosuppose that a subject’s judgments may vary depending on whether the items in asorites series are considered individually or pairwise. The basic premise and theconclusion of a sorites argument derive their plausibility from individual judgments.
But only the second, pairwise type of judgment satisfies the inductive premise of thesorites paradox. This means that (3) would have to be rewritten as(10) For every n: if a man with n hairs on his head is bald then a man with n + 1
hairs on his head is also bald, insofar as the two men are judged pairwise.

9And with (3) replaced by (10) the paradoxicality of the argument would dissolve intoa fallacy of equivocation.
It is indeed regretful that the available experimental data are still too scarce tothrow light on these conjectures. A psychologically plausible account can hardly failto include some hypothesis about the mental representations that underlie our usageof vague words. Still, few theorists seem inclined to believe that the paradox can succumb to empirical considerations, just as few theorists are willing to accept a purely
epistemic account. For a vast majority, the paradox is a genuine one. And for manyvagueness remains a deep and bewildering philosophical conundrum.
References
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Be. Cognition 17: 263–308.
Bonini N, Osherson D, Viale R and Williamson T (1999) On the Psychology of Vague
Predicates. Mind & Language 14: 377–393.
Dummett M (1975) Wang’s Paradox. Synthese 30: 301–324.
Evans G (1978) Can There Be Vague Objects? Analysis 38: 208.
Fine K (1975) Vagueness, Truth and Logic. Synthese 30: 265–300.
Lewis DK (1993) Many, but Almost One. In: Bacon J, Campbell K and Reinhardt L(eds) Ontology, Causality, and Mind, pp. 23–38. Cambridge: Cambridge University Press.
Machina K (1976) Truth, Belief, and Vagueness. Journal of Philosophical Logic 5:
47–78.
Raffman D (1994) Vagueness Without Paradox. Philosophical Review 103: 41–74.
Russell B (1923) Vagueness. Australasian Journal of Psychology and Philosophy 1:
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Sorensen RA (1991) Vagueness Within the Language of Thought. Philosophical
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Tversky A and Kahneman D (1974) Judgement Under Uncertainty: Heuristics andBiases. Science 185: 1124–1131.
Tye M (1990) Vague Objects. Mind 99: 535–557.
Unger P (1978) There Are No Ordinary Things. Synthese 41: 117–154.
Williamson T (1992) Vagueness as Ignorance. Proceedings of the Aristotelian Society,Suppl. Vol. 66: 145–162.
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