=Paper= {{Paper |id=Vol-136/paper-8 |storemode=property |title=Contextual Effects on Vagueness and the Sorites Paradox: A Preliminary Study |pdfUrl=https://ceur-ws.org/Vol-136/85.pdf |volume=Vol-136 }} ==Contextual Effects on Vagueness and the Sorites Paradox: A Preliminary Study== https://ceur-ws.org/Vol-136/85.pdf

Contextual Effects on Vagueness and the Sorites
Paradox:
A Preliminary Study

Richmond H. Thomason

Philosophy Department
University of Michigan
Ann Arbor, MI 48109-2110, USA

Abstract. Context is a prominent theme in accounts of the semantics of
vagueness. Contextual theories of vagueness have been motivated in part
by a desire to disarm skeptical arguments that make use of vagueness;
another motive is the hope that they might help to resolve the Sorites
Paradox. This paper is intended as part of a larger project, which aims at
describing and comparing the leading theories that have been presented,
to bring out the ways in which contextual theories have informed the se-
mantic phenomenon of vagueness, and to make strategic assesments and
suggestions concerning the crucial case of adjectives. It concentrates on
Hans Kamp’s 1975 theory, and two recent contributions by Haim Gaif-
man and Joseph Halpern. The specific goal of this paper is to examine the
role of context in the semantics of certain adjectives, and assess to what
extent context can help to solve what I take to be the most challenging
version of the Sorites Paradox.

1 Introduction
Since it began to attract the attention of philosophical logicians in the late 1960s,
the Sorites paradox has puzzled many authors, and has inspired a wide spectrum
of proposed solutions. After nearly fifty years of work on the topic, it still seems
that we lack a satisfactory methodology for attacking this problem. The main
positive contribution of this paper is to propose such a methodology; at the same
time I provide a survey of some of the proposals for bringing context into the
semantics of vagueness and the solution of the Sorites.

2 Kamp’s semantic theory of adjectives
Contextual theories of vagueness, formulated using the apparatus of formal se-
mantics, go back at least to (Goguen, Jr., 1969), but perhaps the most influen-
tial (and, in my opinion, the most rewarding) early work of this sort is (Kamp,
1975).1 Kamp’s theory is confined to adjectives: adjectives in predicative posi-
tion, like ‘tall’ in ‘That man is tall’, are a notorious source of vagueness. Kamp’s
1
(Fine, 1975) appeared in the same year as Kamp’s paper. Fine’s paper is important
for the theory of vagueness, but has little or nothing to say about context.
chief goal in the paper seems to have been to give an account of the semantics
of adjectives according to which the meaning of the comparative would be de-
termined from the meaning of the absolute form. But in the course of providing
such an account he formulates a detailed model theory for vague predicates,
a theory which also provides for contextual sharpenings of (reductions in the
vagueness of) predicates.
Kamp’s theory treats contexts as unanalyzed primitives. A contextual “de-
vaguification”2 is a function from contexts to (partial) models.3 Kamp does not
have a lot to say about the informational side of contexts—about what factors
in practice help us to devaguify adjectives. But he does illustrate how the infor-
mational context can influence the interpretation of various sorts of adjectives.
He notes that if the adjective modifies a noun, the noun can play an impor-
tant part in these contextual effects.4 And he indicates that the interests of the
speakers—what issues they are trying to resolve—can also be important.
(Kamp, 1975), then, provides two important ideas: the use of partial models
(with a supervaluational interpretation) for interpreting vagueness, and the idea
that the appropriate partial model can be a function of context.

3 The Sorites Paradox
The Sorites Paradox (or the Paradox of the Heap), deserves to be counted among
the more profound logical paradoxes, and has been known since ancient times.
The paradox is well known, and I will present it in a brief and abstract way
here—the chief purpose of this section is merely to establish some terminology.
It takes the form either of an induction or of a finite chain of applications
of modus ponens. Given a vague predicate P , you set up the paradox by finding
a series P (a1 ), . . . , P (an ) of applications of the predicate meeting the following
conditions.
(1) For all i, 1 ≤ i < n, P (ai ) and P (ai+1 ) are indistinguishable for
the purposes at hand, so that (apparently) the conditionals P (ai ) →
P (ai+1 ) are all true, for 1 ≤ i < n.
(2) P (a1 ) is definitely true.
(3) P (an ) is definitely false.
In this form of the paradox (which depends on n−1 modus ponens applications),
I’ll call (1) the major premises of the paradoxical argument, (2) the base premise,
and (3) the absurd conclusion.
2
Awkward as it is, I prefer this term to Kamp’s ‘disambiguation’. Ambiguity is not
the same as vagueness and it is good policy to keep the two distinct.
3
Kamp’s paper is rich in technical detail, and actually develops a sophisticated for-
malism for formalizing vagueness and natural language adjectives. Since what I say
will not rely on these technical details, I am suppressing them here.
4
For instance, ‘That author is famous’ is (apparently), less vague than ‘that is a
famous author’, since it rules out cases in which the author is famous, but not
famous as an author.
In the inductive form, you define a numerical predicate Q with analogous
properties. (Indeed, if we express the dependence of ai on i in the above example
as an explicit numerical function, so that ai = f (i), we can let Q = λnP (f (n)).)
In particular, Q is chosen so that the following universal generalization appears
be plausible.

(10 ) ∀n[Q(n) → Q(n + 1)].

I’ll call condition (10 ) the inductive premise of this form of the paradox.

4 Kamp on the Sorites

Kamp’s 1975 paper doesn’t mention the Sorites Paradox, but it is impossible
to develop a theory of vagueness to the point the paper reaches without being
aware of the threat of this paradox.5 Six years later, in (Kamp, 1981), Kamp
addresses the paradox directly—and the results are discouraging.
Kamp had adopted and improved van Fraassen’s “supervaluational” theory
of satisfaction in partial models,6 and had originally thought (as I myself had)
that this approach provided a plausible or at least defensible account of the
Sorites paradox. However, the fact that on this theory
W
(4) {P (ai ) ∧ ¬P (ai+1 ) / 1 ≤ i < n} must be definitely true

(in the conditional version of the paradox), and that

(5) ∃n[Q(n) ∧ ¬Q(n + 1)] must be definitely true

(in the inductive version of the paradox), came to seem more and more worrisome
to Kamp.
Although you might be able to convince yourself that these consequences of
the supervaluational approach (which are closely connected to the conservatism
of this approach with respect to classical logic) are harmless in the case of ad-
jectives like ‘tall’, it really seems to be impossible to reconcile them with the
behavior of what Kamp (following Dummett) calls “observational predicates.”
These are predicates with semantic connections to the outcomes of acts of per-
ception or observation.
The problematic status of observational predicates was pointed out in (Dum-
mett, 1975) and (Wright, 1975). (Kamp indicates that the point was originally
due to Dummett.) In this connection, the source of the difficulty consists in the
fact that an observational adjective or adjective phrase P (‘red’, or if you prefer,
‘apparently red’) will satisfy a condition to the following effect:
5
Since Kamp’s theory is presented model-theoreticaly, the threat of the paradox
doesn’t come in the form of a worry that the reasoning endorsed by the theory
will in itself lead to a contradiction. The worry is that adding axioms that seem,
according to common sense, to be plainly true, will induce a contradiction within
the theory.
6
See, for instance, (Van Fraassen, 1966, van Fraassen, 1969).
(6) If a and b are observationally indistinguishable, then P (a) ↔ P (b)
is definitely true.

In Sorites examples where adjacent instances are observationally indistin-
guishable, (6) is incompatible with (4) (in the conditional versions of the para-
dox).7 So it seems that a supervaluational theory is incompatible with the se-
mantic properties of some adjectives. If difficulties of this sort were restricted
to supervaluational approaches, it would simply be evidence against these the-
ories. However, the objections to other logical theories of vagueness are equally
powerful.
In (Kamp, 1981), Kamp formalizes a theory of the context for adjectives ac-
cording to which contexts are sets of sentences that may or may not be coherent;
the idea is that Dummett-like color spectrum examples may gradually force an
agent to occupy an incoherent context.
I won’t go into the details of this theory in this paper (which is intended as a
brief version of a longer study). I’ll merely indicate why I think that an approach
of this sort is unsatisfactory.8
Contexts consisting of sets of sentences (or of sets of propositions) are ap-
propriate for representing presuppositions. But, since (Stalnaker, 1975), the idea
that such contexts have direct effects on the semantic content of expressions has
lost all or most of its attractiveness. Linguistic evidence and considerations such
as theoretical simplicity provide a great deal of support for this view.9
Presuppositional contexts can have indirect effects on content (for instance,
by indicating the appropriateness of a context that does affect content). And, in
dynamic theories of semantics, the distinction between presuppositional effects
on appropriateness and contextual effects on content can become blurred.
Now, formulations of the Sorites along the lines I sketched in Section 3 do
not seem to involve dynamic operators in any essential way. (While they can
be formulated, for instance, using dynamic conjunction, the paradoxical aspects
seem to remain if they are formulated using only static operators.) Because
of this, I assume that it is best to seek for a solution within a static logical
framework—dynamic logic will only complicate the issues with which we have
to deal.
Secondly, although, as I said, presuppositional contexts can indirectly affect
content, they can do so only by interacting with independent contextual effects
on content. Since the Sorites (like any paradox that results in an explicit contra-
diction) is plainly a problem of content, the core of any contextual solution has
to lie in the identification of contextual effects on content that could resolve the
7
The example that is usually used for this purpose is a colored line, extending a great
distance, graduated so that any local area appears to be uniform in color, and with
distinctly different colors at the extremes.
8
From conversations with Kamp at the time that the paper was written, I have the
impression that Kamp himself wasn’t entirely happy with it, either.
9
See (Beaver, 1996) for a survey of work on presupposition up to 1996, and an as-
sessment of the status of semantic theories of presupposition as of that time.
paradox. This is my reason for abandoning solutions along the lines that Kamp
explored in his 1975 paper.

5 Fast forward

This section will draw attention to the leap I now take over many years and
pages. It marks a large gap in my account of the literature.
In the 1980s, vagueness and the Sorites paradox came to be perceived as
a central problem in metaphysics, and received growing attention in the philo-
sophical literature. See (Keefe and Smith, 1997) for a survey and collection from
the late 1990s, and (Keefe, 2000) for more recent material relating to these de-
velopments. Delia Graff’s work, beginning with her 1997 dissertation, (Graff,
1997), and continued in (Graff, 2000, Graff, 2001, Graff, 2003), deserves partic-
ular mention in this regard.
In this paper, I will settle for a discontinuous history, and will pass over
this period to two more recent contributions by non-metaphysicians. Actually,
although this leaves out interesting and important discussion of the philosophical
issues, it omits much less if the purpose is to focus on the incorporation of context
into a logical formalization that addresses the Sorites Paradox.

6 Gaifman’s proposal

In (Gaifman, 2002), available online as of the date of this paper but as yet (as
far as I know) unpublished, Haim Gaifman presents a fully formalized theory
of vagueness that presents the role of context in a different light. The paper
also presents an extended discussion of the phenomena of vagueness, containing
several new insights.
Gaifman proposes that a class of predicates, the “tolerant predicates,” in-
cluding Dummett’s observational predicates, satisfy a version of (6), which he
formalizes as follows.

(7) NP (x, y) → [P (x) ↔ P (y)]

Here, NP is an appropriate “sufficiently near” relation for the predicate P : if P
is the predicate of being red, for instance, NP relates two things if and only if
they are observationally indistinguishable with respect to redness.
Citing (Kofka, 1922), who in turn refers to experiments performed by others
late in the 19th century, as well as (Raffman, 1994), Gaifman suggests that
applications of some vague predicates, at least, are relative to a comparison class.
In the psychological experiments that Gaifman cites, a subject is presented with
several stimuli, and asked to compare some of them. The judgments depend not
only on the items compared, but on the comparison class.
Here, it is easy to see what the context is: it will consist of the examples
that are presented (and, presumably, attended to) as the comparison is made. In
other cases (for instance, in evaluating sentences like ‘If the car is red, we’ll have
to repaint it’ or the sentence ‘Tomatoes are red’), it may be harder to reconstruct
a comparison class.
Gaifman formalizes the context as a list of compared items (he feels that in
some cases the order in which the items are presented can be important) and
proposes a contextual logic in which context is formalized using modal operators
[ C ], where C is a context. Thus, the formula [ C ]A means that A holds in
context C.10
Gaifman reformalizes the Tolerance Condition, (6), in the following way to
take context into account.
(8) NP (x, y) → [ C ][P (x) ↔ P (y)]
This idea is well motivated by the comparison experiments, and by the in-
terpretation that psychologists like Kofka made of these results. And it makes
good logical sense to say that whether or not two instances are distinguishable
depends on the comparison class. Dummett’s account of indistinguishability may
overlook a hidden variable—whether two presented objects are perceptually dis-
tinguishable does not in general depend only on the two objects. Even though,
compared just with each other, a and b may appear equally red, introducing a
third object c that (compared just with b) appears to be just as red as b but
that perceptably differs from a in redness seems to create a context in which a
is now (indirectly) perceptually different from c in redness.11
As Gaifman points out, this idea disarms the color-spectrum versions of the
Sorites that Kamp and others found so disturbing. The disarmament strategy
depends on formalizing the conditional version of paradoxical situation so that
the major premise conditionals P (x) → P (y) give way to [ x, y ][P (x) → P (y)],
whereas a conjunction like P (a1 ) ∧ P (an ) gives way to [ C ][P (a1 ) ∧ P (an )], where
C is a list containing at least a1 and an . It’s plausible enough that in this example
all these formulas are true.
Gaifman’s idea raises a number of issues, some of them calling for extended
reflection. (1) There is the question of motivating this use of context, and of how
in general to motivate a hypothesized contextual effect on content. Since in this
case there is some independent plausibility to the psychological motivation, I
pass this over here. (2) We may ask whether Gaifman’s idea actually solves the
Sorites paradox. Of course, Gaifman, like Kamp, is presenting his theory model-
theoretically, so consistency is not in doubt. But the question is whether Sorites-
like examples could be presented that force you, using Gaifman’s formalization
policies, to represent sentences that intuitively are true in a way that makes
them false. I will return to this question later in this paper; the main issue is
how to make a plausible case that a logic and formalization strategy does in fact
10
Gaifman seems to be unaware of the AI literature on the logic of context; he thinks
that this formal proposal is new.
11
However, if one takes this point seriously, it seems as if Gaifman’s (8) should now
give way to

(80 ) [ C ][NP (x, y) → [P (x) ↔ P (y)]].
solve a paradox. (3) Kamp began his investigation with a project having to do
with the semantics of adjectives. That project tends to be forgotten in the later
work on vagueness, but it is worthwhile to restore it to a proper place in the
inquiry. For instance, it provides us with additional constraints that may help us
to evaluate proposed solution: a solution that fails to produce a way of defining
the comparative form in terms of the absolute form is not as successful, all other
things equal, as one that does provide such a definition.

7 Halpern’s proposal

Joseph Halpern proposes a formalization that develops and formalizes the intu-
ition that subjectivity plays an important role in the semantics of vagueness. The
formalization proposed in (Halpern, 2004) uses a multimodal logic of the sort de-
scribed in (Fagin et al., 1995). The semantics of multimodal logic treats possible
worlds as global states, which are combinations of the (local) states of agents and
another state, representing the objective facts in the world: w = ho, s1 , . . . , sn , i,
where si (1 ≤ i ≤ n) is the state of agent i and so is the objective component. In
multimodal epistemic logic, the truth of a modal-free sentence in w will depend
only on so , and an agent state is determined by a two-place relation over worlds.
The truth of a sentence involving a modal operator [ a ] (tracking, say, what
agent a knows) may depend on a’s local state, which is modeled using a 2-place
relation over worlds. To understand the application to vagueness, we can resrict
attention to the case where n = 1; here a world w is a pair ho, si.
Suppose that we include all the subjective aspects affecting vagueness (the
agent’s ability to perceptually discriminate objective states, the choice of a refer-
ence class, the agent’s possibly idiosyncratic interpretation of an adjective, and
any other such factors) in the subjective state s. And suppose that all the ob-
jective factors affecting vagueness (if there are any) are included in the objective
state o, and that these things are built into a system of possible worlds. This
will yield a model for a logic of vagueness.
Halpern’s logic has two modal operators:12 [ R ] (tracking what the agent
reports) and [ D ] (tracking what is objectively definite). Both are interpreted
as usual, using relations ∼s and ∼o over worlds. ho, si ∼o ho0 , s0 i iff o = o0 and
ho, si ∼s ho0 , s0 i iff s = s0 . The operators are interpreted in the standard way;
hM, wi |= [ D ]φ iff hM, w0 i |= φ for all w0 such that w ∼s w0 , and hM, w0 i |=
[ R ]φ iff hM, w0 i |= φ for all w 0 such that w ∼o w 0 and w ∈ P , where P is a
designated set of “initially plausible worlds.”
Take the color spectrum sorites, with 3 propositions. Pi says that the i th
sample is red. Say that the objective state records the physical light-reflecting
properties of the samples. The samples differ in objective ways that make higher-
numbered ones closer to orange and further from red than the lower-numbered
ones. The interpretation of ‘red’ depends on both the subjective and subjective
components of a world. Suppose that the agent can perceive that the first sample
12
Two operators for each agent, but I am considering the single-agent case here.
is red and that the third is not; in all w0 such that w ∼s w0 , P1 is true and P3 is
false. But w ∼s w1 and P2 is true in w1 ; w ∼s w2 and P2 is false in w2 . In these
models, the conditional [ R ]P1 → [ R ]P2 is simply false at the world w. So on
this way of formalizing the Sorites (which is close to the one Halpern uses), the
major premisses of the argument are false.
However, Halpern’s logic is very flexible and context can be absorbed into
his models in various ways. For instance, Gaifman’s idea could be incorporated
into the formalization of the color spectrum sorites (say, by making the samples
to which the agent is attending part of the objective state). Attempts of this
kind show that Halpern’s logical language may need to be extended to provide
an adequate formalization—to represent Gaifman’s effect, the modal operators
need to contain a representation of the attention list—but Halpern seems to be
willing to entertain extensions of this kind.
For this reason, Halpern’s proposal is probably best viewed as a general
framework for formalizing solutions to the Sorites that are conservative with
respect to classical logic, and the main contribution is the distinction between
the subjective and objective aspects of possible worlds. I am not sure myself
how useful this distinction is. Sometimes it seems to be helpful, in other cases it
seems to be arbitrary and perhaps even unhelpful.

8 A possible resolution of the color-spectrum Sorites

As I indicated, Dummett’s version of the Sorites has struck many people as
especially troublesome, because of the very strong intuition in this case (in the
conditional version) that the major premisses are fully true. So a solution to this
version of the paradox might be a good place to begin attacking the Sorites. We
can think of this case as a problem concerning the semantics of a color adjective,
say, ‘red’.
Even though it will not deal with the generality problem, a plausible solution
to the color spectrum version would certainly be encouraging, since this is gener-
ally thought to be the hardest known case. In this section, I’ll (briefly) examine
a solution that incorporates Gaifman’s contextual resolution.
There are many ways in which adjectives can depend on context. The most
obvious of these, and probably the most frequently cited, is dependence on a
standard of comparison, perhaps given by a comparison class. When something
is said to be tall, for instance, or heavy, the content of what is said will vary
according to what sorts of things it is compared with. A 10-story building could
be truly said to be tall if (implicitly) it is compared with buildings in Missoula,
but not at all tall if (implicitly) compared with buildings in New York.
Color adjectives may depend on this factor as well, though it seems to be
relatively unimportant for them. However, there does seem to a dependence of
color adjectives on what Graff calls interest. In (Graff, 2000) she illustrates this
with an example in which the purpose is to identify a book on that is gray (but
grayish blue) that is on a shelf with other books that are grayish red. In this
case, it seems appropriate to say “bring me the blue book.” If this is a matter of
truth and not appropriateness, we have another sort of contextual dependence.
Suppose that this sort of contextual effect is determined by a set of alterna-
tives (in this case, the alternatives would be a partition of the set of books on the
shelf into two sets, one of which is a unit set). Representations of this sort are
also useful in the interpretation of focus; see, for instance, (Rooth, 1996), and
I think they may provide a better way of coming to grips with the phenomena
than ascribing them to purposes or interests.
In that case (discounting other contextual effects on color adjectives) we could
treat a context as a pair consisting (as Gaifman proposes) of a comparison list
and a set of alternatives. Rather than formalizing this, with a logical language
and a model theory, I will discuss the idea briefly. The role of the alternative set
is to locate the boundary of a color adjective in a region where it can achieve
one of the partitions (if there is such a place for the adjective in question). In
general, this will shift the interpretation of the adjective (in much the same way
that an intensifier like ‘very’ does) without, however, fixing it precisely. We can
assume for the purposes of this example that the effects of the attention lists
are as Gaifman describes them. In that case (with, of course, an appropriate
formalized language and model theory) we can inherit Gaifman’s solution to the
Sorites in this version of the Sorites.

9 Other adjectives
Other adjectives, however, may not exhibit dependence on a context list: ‘intel-
ligent’ and ‘expensive’ may be cases of this sort. Here, we will need to look for
other resolutions of the Sorites. If, for instance, we can show in these cases that
it is independently plausible to deny the major premisses, we do not even need
to look for appropriate contextual effects.
The point is that, with an approach that treats the context appropriate for
interpreting an adjective as a complex structure that can contain many factors
affecting the boundary of the adjective, we can develop a general approach that
can examine various adjectives on a case-by-case basis, while at the same time a
general logical framework is maintained. Any formalized solution in the literature
that respects classical logic could, I think, be incorporated into this framework.
This is methodologically useful, for the reasons I mention in the next section.

10 Responding to paradoxical challenges
A paradox confronts us with a plausible argument that has plausible premisses
and an implausible conclusion. The profound paradoxes (including at least the
semantic paradoxes and the Sorites) are generalizable, appearing in a variety of
forms. This feature makes them difficult to solve. A resolution of one form of
a paradox may not generalize to other forms. For instance, a solution to the
version of the Liar Paradox involving ‘This sentence is false’ that simply invokes
truth-value gaps will not solve versions that invoke ‘This sentence is not true’.
And formalisms that seem to avoid the Russell paradox by eliminating negation
can run afoul of the Curry paradox.
A formally adequate solution to a class of paradoxes will involve a formalized
language in which the paradoxes can be formalized in a plausible way. It must be
semantically sensible, and it must be consistent. The best way of showing this is
to produce models of the formalism, and to show that these models are plausible
and match the formalism’s informal motivation. This part of the project is more
or less mathematical. As such, it can be carried through to completion and
evaluated using the criteria generally applicable to logical formalizations.
But in the case of generalizable paradoxes, there has to be a more open-ended
part of the solution project, in which various generalizations of the paradox are
considered, and solved. The main components of the solution—assuming that
the formalism doesn’t have to be extended to deal with generalizations of the
paradox—will consist in exhibiting plausible formalizations of the generalized
paradox and showing that the solution strategy extends to these cases. What
makes this part of the solution open-ended is the lack of a complete list of all
the conceivable variations of a generalizable paradox. In the absense of such a
list, one has to fall back on the methods that are used to support claims like
Church’s thesis and the adequacy of first-order logic as a vehicle for formalizing
mathematical logic: (1) an extended suite of case studies illustrating the gener-
ality of the formalism, (2) the absence of any plausible counterexamples, once
the claim has achieved a certain amount of publicity.

11 Generality and the Sorites

The known Sorites paradoxes do not vary much in the actual reasoning. As we
said, they come in two forms; and any solution that deals with the conditional
form is pretty certain to deal with the inductive form. But they do occur with
predicates that differ widely in their linguistic realizations and semantic prop-
erties. Generality poses a threat to solutions of the Sorites—and especially to
contextual solutions—mainly through the variety of vague predicates.
A solution that—like Gaifman’s, for instance—depends for its plausibility on
the way in which an adjective like ‘red’ depends on context, may be unable to
deal with a Sorites paradox involving a different adjective that does not depend
on context in this particular way. Not all adjectives are related to perception in
a way that makes contextual dependence on a comparison list plausible: ‘honest’
doesn’t seem to enter into any such contextual dependence, for instance. Analysis
of other examples, and case-by-case consideration of whether there is a plausible
way of disarming the paradox, will be needed to make the generality of the
solution plausible. Fortunately, adjectives, like other items in the lexicon, group
together into semantic classes, so that the number of cases to examine will be
much smaller than the number of adjectives in the lexicon.
Beyond vague adjectives, there are vague adjective phrases. This is uncharted
territory; but hopefully, an extended examination of adjectives might yield gen-
eral techniques for dealing with conjunctions and modifications of adjective
phrases, and other complex cases.
And, of course, vagueness is not confined to adjectives and adjective phrases.
After all, ‘sorites’ (‘heap’) is a noun; and a complete account will have to sys-
tematically consider the whole list of syntactic categories.
The program I have proposed here attempts to resolve the Sorites by cases.
The list of cases may prove to be so open-ended that a completed argument may
not emerge from the program; but, with the help of some linguistics, there is
some hope that this might happen.
An alternative way to structure the cases would look at the meanings of
adjectives and other words; such an attempt, for instance might proceed by
distinguishing different types of vague predicates, and showing that each type
is tractable. This is more or less what Delia Graff undertakes in her solution
strategy. Setting aside the arguments she gives in specific cases, I think that
this semantic way of structuring the argument is unsuccessful. Graff has one
sort of solution for “phenomenal continua” Sorites arguments and another sort
for the rest. The phenomenal continua arguments involve observational predi-
cates, which are those predicates whose “applicability to an object (given a fixed
context of evaluation) depends only on the way that object appears” (Graff,
2001)[p. 907]. Approaching the problem in this way, by dividing the paradoxical
cases into a positively defined class (observational predicates) and a residue class
(the other predicates) simply transfers the generality problem to the residue
class. Graff deals with some arguments in the residue class elsewhere (Graff,
2000), but even so the generality of her solution strategy (which deploys only
two solution methods) is left in doubt. There is no reason to think that the strat-
egy covers all versions of the paradoxical arguments. In fact, “semiobservational”
predicates such as redness, which (apparently) depend partly on appearances,
may pose a problem.
To address the generality problem, it seems that a semantical case structure
would have to provide an exhaustive partition of the predicates without appeal-
ing to a residue category—and this strikes me as difficult, if not impossible.

12 Conclusion

The purpose of this paper has been to bring out some useful ideas in the recent
literature on the logic of vagueness and to develop a methodology that does
justice to the very deep problems raised by the Sorites, rather than to propose
a solution. The hope is that, using this framework, incremental progress can be
made. If that turns out to be the case (and I can’t guarantee at all that it would),
this would show that the Sorites paradox, despite its apparent intractibility, is
different from the semantic paradoxes. There, I see no hope at present for an
incremental program of this kind.
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