## 1. The Sorites in History



The Megarian philosopher Eubulides (4th century BC) is
usually credited with the first formulation of the puzzle. (The name
'sorites' derives from the Greek word *soros*,
meaning 'heap'.) Although we don't know his
motivations for introducing it (along with several other legendary
puzzles), the paradox was later used by Greek philosophers as a
dialectical weapon, most notably by the Sceptics against the Stoics'
claims to knowledge.



Curiously, the paradox attracted little subsequent interest until the
late 19th century. Marxist philosophers in the neo-Hegelian
tradition, like Plekhanov (1908 [1937: 114]), cited the paradox as
evidence of the failure of "customary" logic and the
utility of the 'logic of contradiction'. In this way some
Marxists sought to establish the triumph of the dialectic. Meanwhile,
in Anglo-American philosophy, formal logic regained its central place,
and its classical formalisation left no room for the vagueness of
natural language. Vagueness and the associated paradox were seen as
lying beyond the scope of logic and so posing no challenge to it.
However, since the demise of the ideal language doctrines of the
latter half of the twentieth century (see
 SS3.1),
 interest in the idiosyncrasies of natural language, including its
vagueness, has greatly increased. 


## 2. Different Formulations of the Paradox



At least three conditions must be met for an argument to be an
instance of the sorites paradox. (1) It must be possible to construct
a sorites series for the predicate in question, *viz.*, a
finitely-membered ordering of values on a dimension decisive of the
predicate's application. A sorites series for 'tall'
is an ordering on the dimension of height (an ordering of heights),
for 'old' an ordering on the dimension of age (an ordering
of ages), and so forth. (2) Neighboring values in the series must be
only incrementally different, i.e., either indiscriminable or just
slightly different. An incremental difference is supposed to guarantee
that if a vague predicate applies to one of a pair of neighbors, it
applies equally to the other. (Following Wright [e.g., 1975], the
property of applying across incremental differences on a decisive
dimension is often called the *tolerance* of a vague term.) (3)
The predicate must be true of the first value in the series and false
of the last.



The paradox is often presented in the conditional form discussed
above. More formally: let '\(\Phi\)' be a soritical
predicate and let '\(\alpha\_{n}\)' (where *n* is a
natural number) represent a value in a sorites series for
'\(\Phi\)'. Then the paradox can be represented most
simply this way, using Modus Ponens:



Conditional Sorites 

 

\[
\begin{align}
& \Phi\alpha\_{1}\\
&\textrm{If } \Phi\alpha\_{1} \textrm{ then } \Phi\alpha\_{2}\\
&\textrm{If } \Phi\alpha\_{2} \textrm{ then } \Phi\alpha\_{3}\\
&\textrm{Etc.}\\
&\textrm{If } \Phi\alpha\_{n-1} \textrm{ then } \Phi\alpha\_{n}\\\hline
&\Phi\alpha\_{n} \textrm{ (where \(n\) can be arbitrarily large)}
\end{align}
\]


A different formulation of the paradox replaces the set of conditional
premises with a universal generalization and proceeds by mathematical
induction. Let '*n*' be a variable ranging over the
natural numbers and let '\(\forall n(\ldots n\ldots)\)'
assert that every number *n* satisfies the condition
...*n*.... Further, let us represent the claim 'For any
\(n\), if \(\alpha\_n\) is \(\Phi\) then \(\alpha\_{n+1}\) is
\(\Phi\)' as '\(\forall n (\Phi\alpha\_n \rightarrow
\Phi\alpha\_{n+1})\)'. 



Mathematical Induction Sorites

 

\[\begin{align}
&\Phi\alpha\_1 \\
&\forall n (\Phi\alpha\_n \rightarrow \Phi\alpha\_{n+1}) \\\hline
&\forall n(\Phi \alpha\_n)
\end{align}\]


For example, since a man with 1 hair on his head is bald, and since,
for any number of hairs *n*, if a man with *n* hairs is bald
then so is a man with *n*+1 hairs, every number *n* is such
that a man with *n* hairs on his head is bald.



Another version of the puzzle is a variant of the inductive form. We
know that a sorites series for 'bald' contains some
numbers of hairs such that men having those numbers of hairs are not
bald. By the least number principle (equivalent to the principle of
mathematical induction) there must be a least number, say *i*+1,
such that a man with *i*+1 hairs on his head is not bald. Since a
man with 1 hair on his head is bald, it follows that *i*+1 must
be greater than 1. Hence the series contains a number of hairs
*n* (=*i*) such that a man with *n* hairs is bald
whilst a man with *n*+1 hairs is not. Let '\(\exists
n(\ldots n\ldots)\)' assert that some number *n* satisfies
the condition ...*n*.... Then we can schematize the latter
reasoning this way:



Line-drawing Sorites

 

\[\begin{align}
&\Phi\alpha\_1 \\
&{\sim}\forall n(\Phi \alpha\_n) \\\hline
&\exists n \ge 1 (\Phi\alpha\_{n} \amp {\sim}\Phi\alpha\_{n+1})
\end{align}\]


The line-drawing and inductive forms of the puzzle illustrate well the
soritical predicament; apparently, competent users of
'bald' both must, and must not, draw a line in the series.
For convenience in what follows, most of the examples are framed in
terms of the conditional or inductive forms of the paradox. Of course,
an adequate resolution of the sorites will presumably need to disarm
all versions of it.



We should mention also an informal version of the paradox, known as
the "forced march sorites" (Horgan 1994a; Soames 1999).
Here it is framed in terms of the hypothetical classifications that
would be made by a competent speaker proceeding step by step along a
sorites series. A competent speaker must say that a single grain of
wheat does not make a heap; but if that's right, then she must
also say that two grains do not make a heap; and that three grains
don't; and so on until she must say that, e.g., one million
grains don't make a heap. As will emerge, the forced march
sorites plays an important role in several treatments of the
paradox.



It is worth noting that the popular definition of vagueness in terms
of soriticality (e.g., Wright 1976; Bueno & Colyvan 2012) may well
be incorrect. If the sorites is a resolvable fallacy, as most
theorists of vagueness believe, then vagueness is not after all a
source of paradox. Maybe someone will say that even after the correct
diagnosis of the puzzle has been discovered, the argument will remain
a paradox because it will still *appear* to consist in
unimpeachable reasoning from true premises to a false conclusion. But
such a view makes vagueness far too contingent a property; for all we
know, once we have discovered the proper solution to the puzzle, the
major premise will no longer appear true. It may appear true to the
uninitiated, but this too would be a dubious way to define
vagueness--*viz.*, as the property of generating an
argument that previously appeared, or appears to the uninitiated,
paradoxical. Bueno and Colyvan say that "a predicate is vague
just in case it can be employed to generate a sorites argument"
(2012). But what does 'can be employed' mean here? If a
sorites argument is a fallacy, a vague predicate cannot be
*correctly* employed in it. Is the criterion supposed to be
that a vague predicate is a term which, when employed incorrectly,
appears temporarily (to the uninitiated?) to generate a sorites
paradox? (Raffman 2014: 18-19)



In all likelihood, soriticality is an illusory feature of words like
'old' and 'rich'; their vagueness is real. If
that's right, then apparent soriticality may be best viewed as a
temporary *symptom* of vagueness, or perhaps as an element of
its "surface characterization" (Smith 2008, e.g., 132; see
Smith's third chapter for insight as to what vagueness is).


## 3. Responses to the Paradox



As with any paradox, four broad types of response appear to be
available. One might:


1. deny that logic applies to soritical expressions.



Alternatively, one might accept that the paradox is a legitimate
argument to which logic applies, but then deny its soundness by
either


2. rejecting some premise(s), or
3. denying that it's valid.



The most drastic response would be to


4. embrace the paradox and conclude that vague terms are either
incoherent or vacuous.



In what follows, we consider the major philosophical treatments of the
sorites and the ways in which they have employed these strategies to
dissolve the puzzle.


### 3.1 Ideal Language Approaches



Committed as Frege and Russell (see entries on
 Gottlob Frege
 and
 Bertrand Russell)
 were to ideal language doctrines, it is not surprising to find them
pursuing a type (1) response (e.g., Frege 1903[1960], Russell 1923). A
key attribute of the ideal language is said to be its precision; hence
the vagueness of natural language, including all soritical terms, is a
defect to be eliminated. If that's right, then contrary to what
many theorists believe, soritical terms cannot be marshalled to
challenge classical logic. Logic simply does not apply to them.
Echoing this response, Quine contends that although eliminating vague
terms may incur some cost to ordinary ways of talking, it's a
cost worth paying insofar as it allows us to preserve the "sweet
simplicity" of classical logic (1981: 31-37).



However, with the demise of ideal language doctrines and subsequent
revival of interest in ordinary language, vagueness was no longer
regarded as a superficial or easily dispensable feature. If logic was
to have teeth, it had to apply to natural language as it stands;
soritical expressions are unavoidable and the paradox must be faced
head on. Responses of type (2) do just this and are the most common
family of responses. Logic is seen as applicable to natural language,
in particular to the paradoxical argument, and the latter is diagnosed
as resting on a faulty premise. 


### 3.2 The Epistemic Theory



Most theorists of vagueness conceive of vagueness as a semantic
phenomenon, as somehow rooted in the meanings of words like
'tall' and 'old'. As we shall see, semantic
theories typically introduce special nonclassical logics and/or
semantics in order to resolve the paradox (and accommodate the
phenomenon of borderline cases). In contrast, epistemicists think that
vagueness is just a form of ignorance: vague terms have sharp
boundaries whose locations are hidden from us. In fact, heaps are
sharply divided from non-heaps, and tall heights are sharply divided
from average ones, but we cannot discover where those divisions lie
(e.g., Sorensen 1988, 2001; Williamson 1994a,b, 2000; Graff 2000, Fara 2008;
Rescher 2009). On this view, the sorites paradox is dispatched
immediately: the major premise, or one of the conditional premises, is
simply false. And bivalence is preserved: any application of a vague
term either is true or is false, though we cannot always know
which.



What facts about the world or natural language or competent speakers
could serve to fix sharp boundaries for vague words? According to
Williamson (e.g., 1994b: 184), meaning supervenes on use; in other
words, the locations of the sharp boundaries of a vague term are a
function of speakers' dispositions to use it as they do.
(Insofar as the use of a vague term varies across time, its boundaries
may be unstable.) Of course, we cannot know the totality of those
dispositions, and we do not know the relevant function; and our
ignorance of these factors blocks one path to knowledge of the
locations of the term's boundaries. 



Another possible route to knowledge of the boundary locations is
blocked by the fact that our knowledge of the application of a vague
term is *inexact*. Inexact knowledge is governed by *margin
for error* principles, *viz.*, principles of the form
'If *x* and *y* differ incrementally on a decisive
dimension and *x* is known to be \(\Phi\) (old, blue, etc.), then
*y* is
 \(\Phi\)'.[2]
 For example, where knowledge is inexact, we can know of a blue object
that it is blue only if objects whose colors are incrementally
different are blue as well--hence, only in clear cases. In
contrast, in the borderline or "penumbral" region of a
sorites series for 'blue', where the boundary lives, some
shade of blue is only incrementally different from, indeed may look
the same as, a shade that is not blue; and we cannot know where this
difference lies. Consequently, if we classify the former shade as
blue, that classification is correct by luck, and so does not
constitute knowledge. (On the plausible assumption that *seeing
that something x is blue* is sufficient for *knowing that x is
blue*, it follows that some blue things are such that we cannot
*see* that they are blue, even under ideal viewing
conditions.)



The virtues and the appeal of the epistemic theory are significant,
and it has earned its share of supporters. At the same time, the view
may be hard to accept. Even its proponents grant that epistemicism is
intuitively implausible; and it seems to multiply mysteries. As a
first approximation, the epistemicist says that



> 
> 
> vague terms have unknowable sharp boundaries that are fixed by an
> unknown function of their unknowable (i.e., not fully knowable)
> patterns of use. 
> 
> 
> 



However, it seems that the function too must be unknowable, not just
unknown; for how could we recognize it if we came across it? How could
we tell whether we'd gotten hold of the right function but by
determining whether it yields the correct boundaries as its values? If
that's right, then the epistemicist's thesis must actually
be that



> 
> 
> vague terms have unknowable sharp boundaries that are fixed by an
> unknowable function of their unknowable patterns of use.
> 
> 
> 



(Raffman 2014: 10) To be sure, explanations are provided for our
irremediable ignorance in these cases: for example, we can't
know where the sharp boundaries lie because our knowledge is inexact,
and we can't know the total pattern of the term's use
because "the data are infinite" (Williamson 1994b:
184-185), and so on. Nevertheless, epistemicism may have the
feel of a "just so"
 story.[3]
 (See SS5.1 for further discussion of Williamson.)



Graff Fara defends a different strain of epistemicism (Graff 2000, Fara 2008). Akerman
and Greenough (2010) observe that her account



> 
> 
> is a form of epistemicism in that vague predicates draw sharp,
> bivalent,
>  boundaries.[4]
>  Unlike the epistemicism of Sorensen (1988) and Williamson (1994a,b),
> however, it is constitutive of vagueness that the boundary can shift
> as a function of changes in [speakers'
>  interests].[5]
>  (2010: 277)
> 
> 
> 



Such a sharp boundary is unknowable because (among other things) it is
constantly moving around in a sorites series, changing location with
the speaker's interests in such a way that he never encounters it
(Fara 2008: 328). As Stanley puts it, 



> 
> 
> when we look for [a] boundary of the extension of [a vague term] in
> its penumbra, our very looking has the effect of changing the
> [extension] of the vague expression so that the boundary is not where
> we are looking. (2003: 269) 
> 
> 
> 



Thus we can never discover where the boundary lies, and each
conditional premise seems true so long as we are considering it. (The
role of interest-relativity in Graff Fara's account is discussed
further in
 SS3.3.4.)
 



Retention of classical logic and bivalence is supposed to be a chief
advantage of the epistemic approach over other views (e.g., Williamson
1992: 162). Indeed, because bivalence is widely supposed to entail
sharp boundaries, many theorists of vagueness believe that, for all
intents and purposes, epistemicism is the only theory that can employ
a bivalent semantics (e.g., Rosenkranz 2003, Keefe
 2000).[6]
 In particular, they believe that no semantic theory of vagueness can
be classical. Subsequent developments cast doubt on this view, however;
see
 SS3.3.5.
 


### 3.3 Semantic Approaches



As indicated above, vagueness is usually taken to be a semantic
feature of language. And if it is a semantic feature, its logic and/or
semantics cannot be classical, or so the standard thinking goes.
Starting in the later part of the 20th century, a number of
non-classical logics and semantics have been developed for vague
terms, each advancing its proprietary resolution of the sorites
paradox. The extent of the proposed logical innovation varies.



Most semantic theories of vagueness and treatments of the sorites
conceive of the application of a vague term as *indeterminate*
in a certain range of cases. Specifically, in a sorites series for
vague predicate '\(\Phi\)', it is said to be
indeterminate--there is "no fact of the
matter"--which value is the last \(\Phi\) value. The
indeterminacy is usually thought to be manifested in the
predicate's possession of (possible) borderline cases.
Borderlines are variously conceived as neither definitely (or
determinately) \(\Phi\) nor definitely not \(\Phi\), or as such that
the sentence '*x* is \(\Phi\)' is neither true nor
false, or neither super-true nor super-false, or neither true to
degree 1 nor false to degree 1, for
 example.[7]
 The shared idea seems to be that the regions of borderline cases in a
sorites series for '\(\Phi\)' constitute the
predicate's blurred boundaries; and that because the series
contains these indeterminate values, the major premise (or one or more
conditional premises) of the paradox is either less than true or
flat-out false. In what follows we review some of the major semantic
treatments of the paradox.


#### 3.3.1 Supervaluationism



In accord with a principle of least mutilation, one approach adapts
Van Fraassen's supervaluation semantics (1966) to the sorites paradox
and vagueness more generally (e.g., Fine 1975; Keefe 2000). As a
result, it endorses a non-bivalent logic that, at least on the face of
it, retains the classical consequence relation and classical laws
while admitting truth-value gaps. On this view, the challenge posed by
the sorites paradox can be met by logical revision in the metatheory
alone, and a type (2) response is advocated.



Unlike the epistemic conception of vagueness, a semantic conception
will treat the apparent semantic indeterminacy of vague predicates as
real. Borderline cases are values to which the predicate neither
definitely applies nor definitely does not apply, where
'definitely' gets a semantic analysis as opposed to an
epistemic one. The positive extension of a predicate is given by those
values to which the predicate definitely applies, the negative
extension by those values to which the predicate definitely does not
apply, and the remaining (penumbral) cases are values to which the
predicate neither definitely does, nor definitely does not, apply.
Consistently with a view of vagueness as a semantic deficiency (e.g.,
Fine 1975) or as semantic indecision (e.g., Lewis 1986),
supervaluationists define a notion of "super-truth"
("super-falsity") as the status of being true (false)
irrespective of how the semantic deficiency or indecision is resolved
or precisified, i.e., true (false) on every precisification of the
predicate. Applying the predicate to something in its positive
extension results in a super-true sentence, while applying it to
something in its negative extension results in a super-false sentence.
Equating super-truth with truth simpliciter and super-falsity with
falsity simpliciter then results in a non-bivalent logic with
borderline cases giving rise to truth-value gaps.



With validity then defined in the usual way as preservation of truth
(simpliciter), the supervaluationist account of validity coincides
with classical validity. In particular, treating laws as zero-premise
arguments, supervaluationism preserves all classical laws. Thus,
despite its abandonment of bivalence, supervaluationism validates the
law of excluded middle. For example, irrespective of the vagueness of
'heap' it is logically true of any number of grains of
wheat that it either does or does not make a heap. As a consequence,
supervaluation semantics is not truth-functional. It countenances
instances of true disjunctions neither of whose disjuncts is (super)
true. Conjunction and the conditional exhibit analogous non-classical
features.



Since all of the forms taken by the sorites paradox are classically
valid, they are also supervaluationally valid. The conclusion of the
conditional form using Modus Ponens is resisted by noticing that some
conditional premise fails to be true; though, admittedly, none is
false. The conditional sorites is valid but unsound. More revealing is
the diagnosis of the version employing a universal major premise. This
version is also deemed unsound due to the failure of one of the
premises--the universal premise. The universally quantified
conditional is not true; in fact it is false. While there is no one
conditional premise that is false, it is nonetheless true according to
supervaluation theory that some conditional is. That is to say, it is
true that some n is such that it's not the case that if
\(\Phi\alpha\_{n}\) then \(\Phi\alpha\_{n+1}\) (where
'\(\Phi\)' is soritical relative to the subjects of the
form \(\alpha\_{n}\)).



Since supervaluation semantics admits that the falsity of
'\([?]n(\Phi\alpha\_{n} \rightarrow
\Phi\alpha\_{n+1})\)' is logically equivalent to the truth of
'\(\exists n(\Phi\alpha\_n \amp {\sim}\Phi\alpha\_{n+1})\)',
the line-drawing form of the sorites is sound: it is
supervaluationally valid since classically valid and its premises are
uncontestably true. What supervaluation semantics claims to provide is
a formal account of how, contrary to appearances, such a conclusion
could be true; it is true since true no matter how one resolves the
indeterminacy of the vague term involved (i.e., the soritical
predicate).



In this way the sorites paradoxes are said to be defused. With
vagueness viewed as a semantic phenomenon, classical semantics is no
longer appropriate as a semantics of vague language and supervaluation
semantics is proposed in its place. One immediate concern facing this
solution is the fact that it ultimately treats the mathematical
induction and line-drawing forms of the sorites in the same manner as
the logically conservative epistemic theory does. We are forced to
accept the avowedly counterintuitive truth of '\(\exists
n(\Phi\alpha\_n \amp {\sim}\Phi\alpha\_{n+1})\)' which seems to
postulate the existence of a sharp boundary, yet the existence of such
a boundary is just what the semantic theory of vagueness is meant to
deny. Supervaluationists respond by denying that the conclusion of the
line-drawing sorites expresses the existence of a sharp boundary.
Though committed to the claim expressed by

 

\[\tag{a}
 \mathrm{T} '\exists n(\Phi\alpha\_n \amp {\sim}\Phi\alpha\_{n+1})\apos,
\]


semantic precision is properly captured only by the expression



\[\tag{b} 
\exists n \mathrm{T} '(\Phi\alpha\_n \amp {\sim}\Phi\alpha\_{n+1})\apos
\]


and this is clearly denied by supervaluation theory. Whilst it is true
that there is some cut-off point, there is no particular point of
which it is true that it is the cut-off point. Since only the latter
sort of cut-off point is taken to be a sharp boundary, no commitment
is made to such a boundary of which we are ignorant (contra the
epistemic theorist).



With this explanation, however, doubts arise as to the adequacy of the
logic. Not only must (b) be properly taken to represent the semantic
precision of '\(\Phi\)' but we must also be prepared to
admit that some existential statements can be true without having any
true instance, thus blocking any inference from (a) to (b). Just as
the failure of the metatheoretic principle of bivalence in conjunction
with the retention of the law of excluded middle commits the
supervaluationist to the presence of true disjunctions lacking true
disjuncts, so too must we countenance analogous non-standard behavior
in the logic's quantification theory. In effect, the counterintuitive
aspects of the epistemic theory are avoided only at a cost to other
intuitions.



At this point the supervaluationist might seek to explain these
semantic anomalies by showing how they are mandated by a proper
understanding of the underlying phenomenon of vagueness. More exactly,
the suggestion is that a view of vagueness as merely semantic, not
reflecting any underlying phenomenon of metaphysical vagueness (i.e.,
a view of vagueness as merely representational) might underpin a
supervaluationist approach. Fine (1975) appears to promote this
representational view when defending the law of excluded middle, for
example, and Varzi (2001) amongst others also defends
supervaluationism in this way. (If successful, such a defense would
also provide a principled justification of the common *de
facto* linkage of supervaluation theory and a representational
view of vagueness.) If this explanation is to be pursued, then the
formal machinery of supervaluationism solves the paradox only in
conjunction with a denial of metaphysical vagueness. The metaphysical
debate is ongoing. Keefe (2000), on the other hand, opts for a risky
pragmatic defense: though counterintuitive, the semantic anomalies
that afflict supervaluationism should be accepted because they are
part of a theory that fares better overall than any other; no
additional defense is necessary.



Williamson (1994a) points to two further problems apparently
afflicting the supervaluationist account. First, classical inferences
like conditional proof, constructive dilemma, and *reductio ad
absurdum* no longer hold in a language extended to express
vagueness by the addition of a determinately operator 'D'
or similar. The logic of the extended language is decidedly
non-classical. (Dummett [1975] offers an alternative definition of
validity that does not encounter this problem, but Williamson raises
other objections to it. However, Graff Fara [2003] shows that if we
strengthen the notion of consequence to penumbral consequence, we get
failures of these principles even in the absence of a determinately
operator.) Second, problems arise also with regard to the phenomenon
of higher order vagueness. In accommodating higher order vagueness,
the supervaluationist must admit that his proffered concept of truth,
*viz*., super-truth, lacks properties that truth is standardly
thought to possess. Contrary to claims by supervaluationists, then,
truth is not super-truth (see Keefe 2000 for a rebuttal).


#### 3.3.2 Relatives of Supervaluationism



Some criticisms of supervaluationism are mounted from positions closer
to the supervaluationist's own perspective, sharing some of its
central insights while abandoning others.



While agreeing to the type (2) response advocated by
supervaluationists, Burgess and Humberstone (1987) take issue with the
theory's much-discussed retention of the law of excluded middle,
adopting instead a variant of supervaluationist logic that abandons
the classical law in the face of apparent counter-examples presented
by vagueness. (For discussion and criticism from a supervaluationist
perspective see Keefe 2000: ch.7.)



Another variant of supervaluationism is Jaskowski's
paraconsistent (see the entry on paraconsistent logic)
"discussive logic" which underwrites a type (3) response
to the conditional sorites. A decade before Mehlberg (1958) first
proposed what was, in effect, a supervaluationist treatment of
vagueness, a student of
 Lukasiewicz (see entry),
 Stanislaw Jaskowski, published an account of a logic that
he proposed as a logic of vague concepts. It was, in fact, the first
formal system of paraconsistent logic. (Interestingly, both Mehlberg
and Jaskowski were students of the
 Lvov-Warsaw School of philosophy (see entry)
 where Lukasiewicz was a professor.) Paraconsistent approaches to
the sorites paradox had been advocated by Marxists for some time, with
borderline case predications providing paradigm examples of
dialectical situations. The paradox was commonly cited as evidence of
the inadequacy of classical logic; but it was not until
Jaskowski's pioneering work that the proposal received formal
explication. This logic, sometimes now referred to as
"subvaluationism" to emphasise its duality with the more
familiar supervaluationism, represents the postulated semantic
indeterminacy as semantic overdetermination, rather than the
underdetermination typical of truth-value gap responses to the
phenomenon of vagueness. While arguing for a supervaluationist
semantics for vagueness, Fine (1975) noted that the (subvaluationist)
truth-value glut approach can be arrived at by a simple
reinterpretation of the truth-value gap approach advocated therein.
(For more on this system and a limited defense thereof see Hyde 1997.
For criticism see Keefe 2000: ch.7 and Beall & Colyvan 2001.)


#### 3.3.3 Degree and Many-Valued Theories



In contrast to the non-truth-functional logics outlined above, several
truth-functional non-classical logics have been proposed, and in
particular, many-valued logics (see the entry on many-valued logic).
Again vagueness is seen as a properly semantic phenomenon, with the
attendant indeterminacies providing cases of semantic
underdetermination or overdetermination, but truth-functionality is
preserved. The approaches vary as regards the number of non-classical
truth-values deemed appropriate to model vagueness and defuse the
sorites paradox.



An initial proposal, first developed in Hallden 1949 and
Korner 1960 and revamped in Tye 1994, uses a three-valued logic.
The motivation for such a logic is similar to the supervaluationist's.
Just as a vague predicate divides objects into the positive extension,
negative extension and penumbra, vague sentences can be divided into
the true, the false and the indeterminate. Unlike supervaluation
semantics, however, the connectives are all defined
truth-functionally. Though Hallden proposed Kleene's weak
three-valued tables, Kleene's strong three-valued tables have
dominated as the preferred choice. (For the relevant tables see Haack
1974: Appendix.) A recent variation on this theme is Field 2003, which
supplements Kleene's strong tables with an improved,
non-truth-functional conditional and distinguishes the three-valued
semantics from the common truth-value gap approach.



The particular response to the sorites paradox then further depends on
the definition of validity adopted. A common generalisation of the
concept of validity to many-valued logic involves the designation of
certain values. A sentence holds (or is assertible) in a many-valued
interpretation just if it takes a designated value. Validity may then
be defined as the necessary preservation of designated value. (In
classical logic, of course, only truth is designated and thus the
generalised concept reduces to the classical concept of necessary
truth-preservation.) There are then two non-trivial choices: let the
set of designated values be {true} or {true, indeterminate}. The
former proposal, advocated by Korner and by Tye, results in a
type (2) response to the paradox. The latter proposal results in a
paraconsistent logic and yields a type (3) response (see the section
on many-valued systems in the entry on paraconsistent logic). When
coupled with the Kleene strong tables, it results in the
paraconsistent system LP, elsewhere proposed to deal with the liar
paradox and offered as a logic of vagueness in Weber (2010).



While some are motivated to adopt the foregoing three-valued
approaches for their truth-functionality, others find the consequences
unacceptable. Those who, for example, find supervaluationist arguments
for classical laws plausible will balk at excluded middle claims
sometimes being other than wholly true or contradictions sometimes
being other than wholly false, as may be the case in such systems. A
further concern with such approaches, also applicable to
supervaluationism, is that the invoked tripartite division of
sentences seems to face objections similar to those that led to the
abandonment of the bipartite division effected by two-valued classical
logic. Due to the phenomenon of higher order vagueness (in particular
second order vagueness) there would seem to be no more grounds for
supposing there to exist a sharp boundary between the true sentences
and indeterminate ones or the indeterminate sentences and false
sentences than there was for supposing a sharp boundary to exist
between the true sentences and the false ones. The phenomenon of
vagueness that drives the sorites paradox no more suggests two sharp
boundaries than it did one. Vague concepts appear to be concepts
without boundaries at all. No finite number of divisions seems
adequate. Tye (1994) seeks to avoid such difficulties by employing a
vague metalanguage; Sainsbury (1990) proposes that vague terms are
"boundaryless", and that belonging to the extension of a
vague predicate is more like being attracted to a magnetic pole than
like fitting into a pigeon hole (as conventional wisdom might have
it).



Goguen (1969) and Zadeh (1975), on the other hand, suggest replacing
classical two-valued logic with an infinite-valued one. Hyde (2008)
also adopts this approach, though the infinite-valued semantics is
considered a purely formal device and not a commitment to degrees of
truth (see Cook 2002 in this connection). Infinite-valued or fuzzy
logics (see the entry on fuzzy logic) have, however, also been
promoted precisely for their recognition of degrees of truth. Just as
baldness comes in degrees so too, it is argued, does the truth of
sentences predicating baldness of things. The fact that John is more
bald than Jo is reflected in the sentence 'John is bald'
having a higher degree of truth than 'Jo is bald'. Smith
(2008) advocates a fuzzy logic for just this reason.



Infinite-valued logics are then developed to solve the sorites paradox
in a variety of ways. As with all many-valued logics, the connectives
and validity can be defined variously, giving rise to a number of
distinct logics. A standard proposal proceeds by way of the
continuum-valued, truth-functional semantics of Lukasiewicz (see
Haack 1974: Appendix). As with the three-valued case, the type of
response offered to the paradox also crucially depends on the
definition of validity. Where validity is defined as preservation of
designated value and only the maximum value is designated, the
conditional sorites admits of a type (2) response, as in Hyde 2008.
However, reinstating the validity of classical laws on this general
approach would require designating more than the maximum value, and a
type (3) response results. In contrast, Machina (1976) suggests
defining validity as preservation of the lowest degree of truth
possessed by any of the argument's premises. On this approach, the
conditional sorites is invalid and thus a type (3) response again
ensues. Edgington (1996) expounds a distinctly different
non-truth-functional degree theory that preserves the principle of
bivalence and classical logic. On this approach the conditional form
of the sorites is valid and a type (2) response is advocated. Smith
(2008) combines a non-bivalent, truth-functional degree theory with
classical logic through a distinctive definition of validity. Smith's
unique approach provides another type (2) response to the paradox.



As with three-valued approaches, a number of problems beset
infinite-valued approaches to vagueness. Firstly, where the infinitude
of semantic values are taken to model degrees of truth, the very idea
of a degree of truth needs explanation. Secondly, if numerical
truth-values are used some justification seems required for the
particular truth-value assignments. Thirdly, the full implications of
abandoning the well-understood classical theory in favour of degree
theory need spelling out before a proper evaluation of its worth can
be made. (On these points see Sainsbury 1995: ch.2; Keefe 2000: ch.4.
For an extended defense see Smith 2008: ch.5.) Furthermore, it is far
from clear whether such an approach successfully avoids problems of
higher order vagueness. And the assumption of a totally ordered
truth-set is overly simple. Not all natural language sentences are
comparable as regards their truth. Due to the multi-dimensional nature
of a concept such as redness, we may be unable to say of two reddish
patches differing in hue or brightness or colour-saturation, whether
one is redder than the other. (On the latter points see Williamson
1994a: ch.4; Keefe 2000: ch.4. Smith [2008: ch.6] argues that the
so-called problem of higher order vagueness is in fact a distinct
phenomenon and proposes a distinct response.)



Smith defends the view he calls *fuzzy plurivaluationism*,
blending together elements of degree theories and supervaluationism.
The plurivaluationist semantics departs from the supervaluationist in
assigning to each vague predicate multiple precise classical
extensions ("acceptable interpretations") and abandoning
the semantic notion of super-truth. He replaces super-truth with
"just a level of talk" governed by the instruction
'Say that a sentence is simply true if it is true on every
acceptable interpretation' (2008: 109-110). Smith
writes:



> 
> 
> The plurivaluationist will tell us that 'This leaf is red'
> and 'This leaf is not red' can be said neither to be
> simply true nor [to be] simply false, while 'This leaf is red or
> not red' can be said to be simply true....[W]e have no
> violation of truth-functionality [because t]here is no level of
> semantic fact at which...a disjunction is assigned the value
> True, while neither of its disjuncts is. For the only
> *semantic* facts are the facts about what is happening in each
> acceptable interpretation--and these are entirely classical
> (hence truth-functional). What we have is just a *level of
> talk* laid on top of these semantic facts. The talk
> *sounds* non-truth-functional, but it is in fact
> epiphenomenal...[I]t does not literally describe a
> non-truth-functional semantic reality. (2008: 110)
> 
> 
> 



While "simple truth" may differ significantly from
super-truth, the plurivaluationist endorses the notion that properties
defined across multiple valuations (interpretations) play a
significant role in the verbal behavior of competent speakers.


#### 3.3.4 Contextualism and Its Relatives



Dissatisfied with many-valued and supervaluationist approaches, Kamp
(1981) introduced a *contextualist* solution to the paradox.
Focussing on the inductive form of the sorites, Kamp maintained that
every instantiation of the major premise is true in its individual
context, where a context consists of the sentences (containing the
given predicate) previously accepted as true. (For Kamp, a context
just is a set of sentences.) Each instance is true because its
antecedent must be added to the operative context before its
consequent is evaluated, and the neighboring values referred to in the
antecedent and consequent are only incrementally different. In a
classical semantics the universal major premise would then be true as
well; but Kamp adopts a non-classical definition dictating that the
universal premise is true in contexts (i) where its instances are true
*and* (ii) that remain coherent when the universal premise
itself is added. The trouble is that adding that premise produces an
incoherent context that "assign[s] opposite truth values to one
and the same sentence" (Kamp 1981: 252). Hence the premise is
false, despite all of its instances being true. The contextual
relativity of this view is intuitively appealing, and it is free of
the need to explain why each instance of the universal premise seems
true when at least one must be false. At the same time, the
nonstandard semantics for the universal quantifier may seem
unintuitive.



Inspired by Kamp, a subsequent contextualist approach (Raffman 1994)
has it that the major premise of the paradox is false but seems true
for at least two reasons. First, we confuse it with the true claim
that if \(\alpha\_{i}\) is \(\Phi\) then \(\alpha\_{i+1}\) is \(\Phi\)
when the two values are considered together, pairwise. The pairwise
claim, though true, does not license the paradoxical conclusion, which
makes reference only to a value considered individually. The second
reason is a hypothesis, *viz.*, that the major premise can be
false, while seeming true, because the speaker performing a forced
march undergoes a characteristic change in her verbal dispositions at
the moment of switching from '\(\Phi\)' to
'not-\(\Phi\)'. This dispositional change constitutes a
change of context (akin to a Gestalt-shift) that allows the
co-ordinated extensions of '\(\Phi\)' and
'not-\(\Phi\)' to shift so that the values \(\alpha\_{i}\)
and \(\alpha\_{i+1}\) flanking the switching place are now both
classified as not-\(\Phi\); in particular, \(\alpha\_{i}\) is
classified as \(\Phi\) before the switch and as not-\(\Phi\)
afterward. Thus neither predicate is ever applied in such a way that
it distinguishes between the two values relative to the same context,
and so the speaker is able to switch from '\(\Phi\)' to
'not-\(\Phi\)' without crossing a boundary. The major
premise seems true because we fail to realize that truth can be
secured for all of its instances together only by equivocating on
context.



The latter view has been criticized for (among other things) applying
only to the forced march paradox as opposed to the sorites proper; the
sorites concerns a series of values (properties like colors, heights,
ages, etc.) in the abstract, independently of anything to do with
speakers' verbal dispositions or behaviors. To put the criticism
another way, Raffman's account may explain why the major premise of
the forced march sorites seems true, but it does not touch the paradox
proper. Inasmuch as their solutions often involve a dynamical element,
other contextualist treatments of the paradox may be vulnerable to
this objection as well.



Soames (1999, 2002) maintains that vague terms are context-sensitive in the manner
of indexical expressions. Stanley (2003) objects that if Soames is right,
then a diagnosis of the paradox as equivocating on an implicit
contextual parameter is precluded because indexicals do not admit of
variable interpretation in verb phrase ellipsis. Consider the
statement 'Jack is tired now and Jill is too'. Both the
first and second (implicit) occurrences of the indexical
'now' must receive the same interpretation: Jack and Jill
are tired at the same time. As a result of this fixity of
interpretation, versions of the sorites paradox that employ such
ellipses are not open to contextualist resolution, even in the
presence of the relevant sort of contextual variation. Stanley
provides the following example:



> 
> 
> If that\(\_{1}\) is a heap then that\(\_{2}\) is too, and if
> that\(\_{2}\) is, then that\(\_{3}\) is, and if that\(\_{3}\) is, then
> that\(\_{4}\) is, ... and then that\(\_{n}\) is...(2003: 272)
> 
> 
> 
> 



where 'that\(\_{n}\)' refers to the *n*th element of a
sorites series for 'heap'. If 'heap' is
indexical, as Soames proposes, there is no room to suppose that its
extension shifts from conjunct to conjunct in Stanley's formulation.
Defending the contextualist, Raffman (2005) responds by denying that
vague terms are indexicals. She contends that in verb phrase ellipsis,
vague terms should be understood on the model of 'That elephant
is big, and so is that flea'. Here the extension of
'big' varies between the two conjuncts despite the
ellipsis (Ludlow 1989). 



Although Graff Fara defends an epistemic solution to the
paradox, she proposes a dynamical contextualist explanation of the
intuitive appeal of the conditional premise(s). On her view, vague
predicates express properties that are interest-relative in the sense
that their extensions are determined by what counts as significant for
a speaker at a time. The premises of the paradox seem true because a
speaker performing a forced march has a "standing interest in
efficiency that causes [him] to avoid making discriminations that are
too costly" (Fara 2008: 327-8). For example, for any pair of
neighboring, incrementally different heights in a sorites series for
'tall': when the speaker is actively considering the pair,
so that the similarity between the two heights is *salient*, the
cost of discriminating between them outweighs the benefits: 



> 
> 
> [S]uppose my primary purpose is to choose a cherry tree for the yard.
> A discrimination between two cherry trees that are very similar in
> height will be very costly given my interest in efficiency. But the
> discrimination will be costlier still when I am actively considering
> the two trees as live options for my purpose. (Fara 2008: 328) 
> 
> 
> 



Owing to the high cost, the two heights in question will be treated as
"the same for present purposes", and if one of the trees
is tall, so is the other. In effect, the application of
'tall' is governed by an interest-relative form of
tolerance. (Presumably the speaker's interest must invariably be
in cost whenever he considers saliently similar values; in particular,
the latter interest cannot be superseded by a different one, like,
say, an interest in the location of the sharp boundary.)




Some contextualizers, like Burns (1991), make use of the idea that a
predicate's sharp boundaries never lie where one is looking, to
defend a purely pragmatic analysis of the sorites paradox that leaves
classical semantics and logic intact; others see consequences for
logic and semantics and advocate a non-classical approach. Shapiro's
(2006) book develops a dynamical contextualist theory employing a
distinctive variant of supervaluationist logic and semantics to
provide a type (2) solution to the paradox. Soames (1999) appeals to
context-sensitivity to defend a three-valued logic of vague
predicates, postulating boundaries between the determinate exemplars,
the determinate non-exemplars, and the borderline cases. Coupled with
Kleene's strong, three-valued semantics, this non-classical
contextualism denies the truth of the universally quantified major
premise of the paradox while nonetheless also denying its falsity.
(Tappenden [1993] suggests a similar three-valued approach that
appeals to context to explain the apparent truth of the universally
quantified premise, but his use of the notion of context here differs
subtly from that of Kamp and Soames.) The conditional sorites also
admits of solution. Accepting the standard three-valued
truth-conditions for the universal quantifier, Soames (1999) takes the
conditional sorites to have some non-true conditional premise.



For criticisms and a helpful taxonomy of different varieties of
contextualism (with particular attention to the distinction between
"extension-shifting" and "boundary-shifting"
forms of the view), see Akerman 2009 and Akerman &
Greenough 2010.


#### 3.3.5 The Multiple Range Theory



The multiple range ("multi-range") theory is
a semantic theory of vagueness that purports to retain classical logic
and 
bivalence.[8]
Here, the vagueness of an expression consists in its
having multiple equally permissible, arbitrarily different ways of
being applied, relative to a given context (Raffman
2014: ch.4). In a sorites series, the vagueness of a term is reflected in its
possession of multiple equally permissible, arbitrarily different
places to stop applying it. Any adequate theory of vagueness must
acknowledge the existence of permissible stopping places in a sorites
series, since competent users of a vague term are required to stop
applying it before the end. For example, in a sorites series of ages
proceeding from a clearly old age of 90 to a clearly middle-aged
(hence not old) age of 50, say for Americans in 2018--make the
context as fine-grained as you like--speakers can permissibly
stop applying 'old' at 70, or at 67, or 65, or 63.5,
 etc.[9]
 Different speakers will stop at different ages, and the same speaker
will stop at different ages on different occasions. Any particular
stopping place in the series is arbitrary, hence without legislative
force; speakers cannot justifiably charge each other with error when
they stop at different places. In contrast, a boundary would be
legislative; speakers who failed to stop applying 'old' at
its boundaries would be making mistakes. The distinction between
boundaries and permissible stopping places is a cornerstone of the
multi-range approach.



This multiplicity of application is said to be reflected in the
predicate's semantics in the form of multiple *ranges of
application.* A range of application is just an abstract
representation, in the semantics, of a permissible way of applying the
predicate. More formally: a range is a set of values (e.g., ages) to
whose instantiations the predicate can competently be applied. In a
series from 90 to 50, one range of application of 'old'
might contain the ages 90-70, another 90-65, another
90-63.5, and so forth; and the ages (e.g., 90-70) in those
various ranges will be instantiated by different people at different
worlds. 



According to the multi-range view, a sentence applying a vague
term to a given value is true relative to *each* of its ranges
that contain that value, and false relative to each of the others.
Some ranges of 'old', 'borderline', and
'middle-aged' are pictured in the figure below. 



![](sorites-fig.svg)

Figure 1: Some ranges of application of
'old', 'middle-aged', and
'old[middle-aged] borderline' 




Note that each predicate has some ranges that overlap with some ranges
of the other two. The figure indicates that for a
63-year-old, the sentence '*x* is old' is
true relative to the 3rd range of 'old' and
relative to the 4th range and relative to the
5th, and false relative to the 1st and to the
2nd. The sentence '*x* is middle-aged' is
true relative to the 1st range of 'middle-aged'
and to the 2nd, and false relative to each of the
others. The sentence '*x* is borderline old' is true
relative to each range of 'borderline' except the
4th, and false relative to the latter.



Raffman warns against two potential confusions. (1) Ranges of
application are not precisifications (2014: 102-3). To see why, note
that on the multi-range view, the predicate
'borderline' has ranges of application like any other
vague term; ranges of application of 'borderline' contain
borderline values. In contrast, by their nature, precisifications
contain no borderline values. Second, a range of (e.g.)
'old' contains only old ages, whereas a precisification of
'old' contains old ages and not-old (e.g., middle-aged)
ages. Therefore a range contains only a permissible stopping place,
whereas a precisification contains a sharp boundary. Consequently,
third, whereas a range specifies a way in which competent speakers can
actually apply the term, a speaker who applied 'old'
according to a precisification would be (mis)applying it as if it had
sharp boundaries. Fourth, the multi-range view contains no
analogue of super-truth; ordinary truth is truth relative to a single
range. (2) Ranges of application are not (aspects of) contexts. Among
other things, whereas speakers typically are (or can be) aware of the
context they are relativizing to, and can choose a given context for a
certain reason, they do not (cannot) *choose* the ranges to which
they will relativize their applications of a vague term. Rather,
speakers simply choose how they will classify a given value, and that
classification is relativized--automatically, as it were, in
virtue of the semantics of the term--to each of its ranges that
contains the value in question. Relativization to ranges is not
something *speakers do*. (In this connection it is worth noting
that whereas contextualist treatments of the sorites are typically
coupled with a distinct type of semantics for vague terms, e.g., an
epistemicist or supervaluationist or three-valued semantics, the
multi-range solution employs a proprietary multi-range semantics.)



 On the multi-range view, the sorites is said to dissolve
because, on pain of equivocation on ranges, every line in the paradox
must have its truth-value relative to the same ranges of application
of 'old'. And since each range contains a last age--a
permissible stopping place--the major premise of the paradox is
false relative to each range of the predicate for any context.


 The multi-range theorist hypothesizes that the major premise of
the paradox seems true because we confuse it with two pragmatic rules
for the use of vague words (2014: 172-5):


* (I)  For
any vague term '\(\Phi\)': If \(\alpha\_{n}\) and
\(\alpha\_{n+1}\) are only incrementally different on a decisive
dimension, then any differential application of the predicate as
between them, i.e., any application of '\(\Phi\)' to one
but not to the other, must be *arbitrary*. (That is: arbitrary
as opposed to *impermissible*).
* (II) For
any vague term '\(\Phi\)': If \(\alpha\_{n}\) and
\(\alpha\_{n+1}\) are only incrementally different on a decisive
dimension, then if \(\Phi\alpha\_{n}\) then \(\Phi\alpha\_{n+1}\),
*insofar as \(\alpha\_{n}\) and \(\alpha\_{n+1}\) are considered
 pairwise*.[10]



Of its aspects discussed here, the multi-range approach has been
criticized most prominently for its commitment to an extreme
relativism about truth. Opponents object that it is one thing to
relativize truth to possible worlds, and to such contextual factors as
speakers, times, spatial locations, comparison classes, speaker
interests and purposes, stakes, and standards of assessment; and quite
another to relativize truth to factors that vary even after all of
those contextual parameters have been fixed. The extremely
fine-grained relativity proposed by the multi-range theorist
seems to stretch the notion of truth to the breaking point. Also,
questions arise about the (higher order) vagueness of the predicate
'range of application' itself; and it's not clear that
speakers ever follow a rule like (I) above. See Akerman 2014,
Egre 2015, Sainsbury 2015, Scharp 2015, and Caie 2015
for these and other criticisms; and Raffman 2015 for some replies.



While there isn't space to review them here, we should note
that theorists of vagueness have done a variety of revealing empirical
studies investigating ordinary speakers' use of vague terms.
Just for example, see Egre 2009, Ripley 2011, Alxatib
& Pelletier 2010, Serchuk *et al.* 2011, Huang 2012, 2013,
Egre *et al.* 2013. 


### 3.4 Embracing the Paradox



Several philosophers have endorsed a type (4) response, drawing the
radical conclusion that the paradox is unsolvable; we are just stuck
with it. The question then is what the paradox shows. Dummett (1975),
for example, maintains that vague observational predicates whose
application is supposed to be governed by a nontransitive
indiscriminability relation are incoherent. Such a view appears fatal
to the familiar notion of a determinate shade of color (see, e.g.,
Jackson 1975; Wright 1975; Peacocke 1992; Graff
2001; Mills 2002; Hellie 2005; Chuard 2007 for discussion).



A different type (4) response holds that, contrary to appearances,
conditional sorites paradoxes are sound. For example, it is true,
after all, that no number of grains of wheat make a heap. However,
such a view immediately runs into trouble because the paradoxes come
in pairs. As observed above, there are negative and positive versions
of the puzzle depending on whether the soritical predicate is negated.
To accept all sorites arguments as sound requires assent to the
additional claim that, since one grain of wheat makes a heap, any
number does. A radical incoherence follows since there is a commitment
to all and any number both making a heap and not making a heap.
Similarly, everyone is bald and no one is; everyone is rich and no one
is, and so on.



The problem is that the soundness of any positive conditional sorites
undercuts the truth of the unconditional premise of the corresponding
negative version, and vice versa. Unless one is prepared to accept a
pandemic of contradictions in natural language, not all sorites can be
sound. Unger (1979) and Wheeler (1979) propose a more restricted
embrace. Dissatisfied with responses of types (1) and (3), one accepts
the applicability and validity of classical norms of reasoning.
Nonetheless, dissatisfaction with responses of type (2) considered so
far--rejecting some conditional premise--leaves open the
option of either rejecting the minor (unconditional) premise or
accepting it and, with it, the soundness of the paradox. What is
advocated is the soundness of those sorites which deny heapness,
baldness, hirsuteness, richness, poverty, etc. of everything--a
type (4) response--and the corresponding falsity of the
unconditional premise of all respective positive variants of the
argument--a type (2) response. Terms like 'heap',
'bald', 'hirsute', 'rich' and
'poor' apply to nothing. (For criticisms, see Williamson
1994a: Ch. 6.)


## 4. Unification with the Liar Paradox



The sorites paradox has traditionally been seen as unrelated in any
substantially interesting way to the semantic and set-theoretic
paradoxes of self-reference. However, McGee (1991) and Tappenden
(1993) proposed a unified treatment of the liar and sorites paradoxes
based on similarities between vague predicates and the truth
predicate. More recently, Field (2003: 262) speaks of 



> 
> 
> some temptation to connect up the [semantic paradoxes and the
> paradoxes of vagueness] by viewing the semantic paradoxes as due to
> something akin to vagueness or indeterminacy in semantic concepts like
> 'true'. 
> 
> 
> 



Field 2008 further develops this theme, though it is devoted primarily
to a resolution of the liar, Curry's and other paradoxes. Field's
approach is by way of a logic that abandons the law of excluded
middle.



Some see unification as much more clearly indicated by the supposed
fact that the semantic and sorites paradoxes themselves are "of
a kind". Thus Colyvan (2009) points to a number of ways in which
paradoxes might be thought to be of a kind *per se*, concluding
that the liar and the sorites are examples and thus deserving of a
similar solution. Priest (2010) adds weight to this claim, arguing
that both the paradoxes of self-reference and the sorites paradox have
a common underlying structure, satisfaction of what Priest calls
"the inclosure schema". On the assumption that this common
structure is sufficient to warrant a similar treatment, Priest
advocates a paraconsistent response to the sorites having elsewhere
defended a paraconsistent response to the paradoxes of self-reference.
In fact, as with paradoxical sentences, some vague sentences involving
borderline cases will furnish examples of true contradictions,
dialethias.


## 5. Philosophical Lessons



Having considered several major families of responses to the logical
and semantic challenges posed by the sorites, it is worth reflecting
upon some of the broader philosophical issues that the problem raises.
Since the deeply puzzling phenomenon of vagueness manifests itself
first and foremost as a linguistic phenomenon, it is unsurprising that
the responses variously intersect with problems concerning meaning,
truth and reference.


### 5.1 Meaning as Use



A challenge posed for the epistemic theorist's response is that on
such a view the commonly supposed connection between meaning and use
appears to be strained if not severed altogether (see again
 SS3.2).
 While the margin-for-error principle discussed in Williamson 1994a
might serve to explain how we could be ignorant of the postulated
sharp boundaries, it might be thought that since our use of vague
terms does not draw sharp boundaries, it could not contain them given
the generally accepted connection between meaning and use. As
Williamson reports this concern others might have, "the
epistemic view of vagueness sets truth-conditions floating
unacceptably free of our dispositions to assent and dissent"
(1994a: 205). It seems that such a view must abandon the idea that our
use determines meaning.



One obvious response is that the connection between meaning and use is
not as strong as might be supposed. Nature might also sometimes play a
role in determining meaning, e.g., in the case of natural kind terms;
but for a predicate like, say, 'thin', it is unlikely that
nature provides what our use does not. Williamson further responds by
pointing out that the determination thesis at issue is really a
supervenience thesis--meaning supervenes on use--and this
thesis can be agreed to by epistemicists. Granted, the epistemicist
cannot say exactly how meaning supervenes on use, and so cannot
calculate the meaning or truth-conditions of an application of a vague
term from facts about use. However, the response continues, this
inability is something which all theorists ought to accept. To suppose
that the epistemic theory must make good on this count is to place
unreasonable expectations on the theory (see Williamson 1996 and
Burgess 2001 for further discussion). 



The supervenience thesis is also challenged by symmetry
considerations. When confronted with a borderline case of
'thin', the argument goes, a language user will neither
assent to the application of the term nor to the application of its
negation. Patterns of dissent are similarly symmetrical with respect to the two 
claims.[11] 
And yet despite this symmetry at the level of use
it must be broken at the level of truth and falsity where one of the
terms or its negation truly applies according to the theory; one or
the other claim is true and the other false. If our patterns of use
leave the matter equally unsettled either way then how can the truth
of the matter be settled without arbitrariness and a severing of the
connection between meaning and use? The answer, Williamson suggests,
lies in the fact that truth and falsity are not symmetrical notions.
Falsity obtains in the absence of truth, so where there is symmetry at
the level of use, falsity reigns. Whether this response succeeds is
debated in Burgess 2001 and Weatherson 2003.


### 5.2 Truth and the T-schema



As already noted in connection with supervaluationism, theories that
abandon bivalence have been charged with having to reject the required
Tarskian constraint on truth encapsulated in his T-schema:
'p' is true if and only if p. The rejection of bivalence
in the context of the T-schema is said to lead to absurdity
(Williamson 1994a: ch. 7; see Wright 1994 for criticism). This charge
applies more generally to any non-bivalent theory of vagueness coupled
with the T-schema. If validated, the pressure to abandon bivalence in
the presence of vagueness would then cast doubt on a deflationary
account of truth. Many will find this consequence untoward. Field
(2008), for example, is devoted to saving this account of truth from a
range of paradoxes and rejects a truth-value gap approach.



Supervaluationists have responded by noting that though the T-schema
is not true, a corresponding mutual entailment thesis is not
threatened: "'p' is true" entails and is
entailed by 'p'. However, the latter claim is strictly
weaker than the corresponding claim involving the conditional
according to supervaluationism, and one may wonder whether the weaker
commitment is sufficient to capture what matters about truth (see
Keefe 2000: ch. 8). Others have taken issue with the Williamson
argument by pointing out that in the context of nonbivalent approaches
to vagueness, negation can be variously defined and that the argument
supposes a rejection of bivalence invoking a particularly strong
reading of negation. In response Williamson contends that while an
appropriately weak account of negation can be offered sufficient to
undermine the argument for a general acceptance of bivalence, in the
special case of vagueness the phenomenon of higher-order vagueness
provides the materials for similarly reducing this weaker rejection to
absurdity. (See Williamson 1994a: 193f. and Pelletier & Stainton
2003 for further discussion.)



The claim then is that even if there were a sense in which truth was
non-bivalent and nonetheless satisfied the T-schema, thus making
available a deflationary account, the particular nature of the problem
posed by vagueness precludes such a synthesis. The depth of the
problem, as evidenced by the phenomenon of higher-order vagueness,
shows that it cannot be accounted for by a rejection of bivalence
alone.


### 5.3 The Inscrutability of Reference



Attempts to solve the sorites paradox also throw issues of reference
into sharp relief. Unlike epistemic responses to the sorites which
postulate inscrutable boundaries, supervaluationism is frequently
associated with a semantic approach to vagueness seemingly committed
to the inscrutability of reference.



Consider a sorites paradox using the predicate 'is on
Everest' using the series of millimetre discriminations along a
line from its peak to the valley floor below. The first point (the
summit) is clearly on Everest. The last (in the valley) clearly is
not. And there is no clear point in between where we would draw the
sharp boundary separating the mountain from its surrounds. The
vagueness or indeterminacy that underwrites this sorites paradox is,
on this approach, not a result of epistemic limitations, nor a result
of indeterminacy in Everest itself but, rather, arises as a result of
indeterminacy surrounding what to count as the referent of the term.
According to the supervaluationist, vagueness is a matter of semantic
indecision, as it is frequently put. In the case to hand, there is
simply no fact of the matter as to exactly what portion of earth is
referred to. There is a range of admissible candidates, all with equal
claim to be Everest, amongst which we have simply not decided, nor (to
paraphrase Lewis) is anyone stupid enough to try. In such a case,
overlapping the problem of the many (see the entry on the problem of
the many), the theory commits to a singular term
'Everest', though apparently a denoting phrase, lacking
any unique determinate referent. This accords with Russell's much
earlier analysis of vagueness as "one-manyness in
denotation".



As Keefe (2000: ch. 7.1) points out, supervaluationism thus understood
nonetheless makes true the claim that there is but one (sharply
bounded) Mt Everest (thus claiming a solution to the problem of the
many, and to the foregoing sorites paradox since it is true that there
is a sharp cut-off point to being on Everest) even if there is no one
(sharply bounded) mountain of which it is true that it is the thing
referred to by 'Everest' (and thus no point on the
mountain of which we can say that it is truly the cut-off point).
There is but one Everest but there is no fact of the matter what it
is.



As with earlier problems concerning the role of existential
quantification in supervaluationism, one can debate whether this is a
consequence to be embraced or an untoward consequence undermining the
theory being advanced. It is certainly surprising that reference is
inscrutable in this way. Moreover, such cases are not the exception;
given the ubiquity of vague singular terms such cases appear to be the
rule (see Lewis 1993; McGee & McLaughlin 2000; Morreau 2002).