## 1. Inquiry Resistance



If you cut one head off of a two-headed man, have you decapitated him?
What is the maximum height of a short man? When does a fertilized egg
develop into a person?



These questions are impossible to answer because they involve absolute
borderline cases. In the vast majority of cases, the unknowability of
a borderline statement is only relative to a given means of settling
the issue (Sorensen 2001, chapter 1). For instance, a boy may count as
a borderline case of 'obese' because people cannot tell
whether he is obese just by looking at him. His curious mother could
try to settle the matter by calculating her son's body mass
index. The formula is to divide his weight (in kilograms) by the
square of his height (in meters). If the value exceeds 30, this test
counts him as obese. The calculation will itself leave some borderline
cases. The mother could then use a weight-for-height chart. These
charts are not entirely decisive because they do not reflect the ratio
of fat to muscle, whether the child has large bones, and so on. The
boy will only count as an absolute borderline case of
'obese' if no possible method of inquiry could settle
whether he is obese. When we reach this stage, we start to suspect
that our uncertainty is due to the concept of obesity rather than to
our limited means of testing for obesity.



Absolute borderline cases are first officially targeted by Charles
Sanders Peirce's entry for 'vague' in the 1902
*Dictionary of Philosophy and Psychology*:



> 
> A proposition is vague when there are possible states of things
> concerning which it is *intrinsically uncertain* whether, had
> they been contemplated by the speaker, he would have regarded them as
> excluded or allowed by the proposition. By intrinsically uncertain we
> mean not uncertain in consequence of any ignorance of the interpreter,
> but because the speaker's habits of language were indeterminate.
> (Peirce 1902, 748)
> 



Peirce connects intrinsic uncertainty with the sorites paradox:
"vagueness is an indeterminacy in the applications of an idea,
as to how many grains of sand are required to make a heap, and the
like" (1892, 167). In the case of relative borderline cases, the
question is determinate but our means for answering it are incomplete.
For absolute borderline cases, there is incompleteness in the question
itself. When a term is applied to one of its absolute borderline cases
the result is a statement that resists all attempts to settle whether
it is true or false (Hu 2021). For instance, no amount of conceptual
analysis or empirical inquiry can establish the minimum number of
grains needed for a heap. Equally futile is inquiry into the
qualitative vagueness illustrated by the decapitation riddle. An
inventive speaker could give the appearance of settling the matter by
stipulating that 'decapitate' means 'remove a
head' (as opposed to 'make headless' or
'remove the head'). But that semantic revision would
change the topic to an issue that merely sounds the same as
decapitation. The switch in meaning might not be detected if the
inventive speaker was misperceived as merely applying antecedent
knowledge of decapitation.



We should expect absolute borderline cases to arise for unfamiliar
cases. Where there is no perceived need for a decision, criteria are
left undeveloped. Since philosophical thought experiments often
involve unfamiliar cases, there is apt to be intrinsic uncertainty. W.
V. O. Quine was especially suspicious of the far-out hypotheticals
popular in the literature on personal identity. He globalizes his
doubts about determinate answers in his thesis of radical translation:
for any language, there are indefinitely many, equally adequate rival
translation manuals.



The indeterminacy of translation applies to one's own language.
Introspection will not settle whether '\(x\) is the same person
as \(y\)' means '\(x\) has the same body as \(y\)'
or '\(x\) has same memories as \(y\)'. You have never
encountered a case in which these two possible rules of usage diverge.
John Locke imagines a prince and cobbler who awake with each
other's memories. Is the prince now in the cobbler's bed
or is the prince in his royal bed with errant memories? Quine sees a
deflating resemblance with the decapitation riddle. Either of the
incompatible answers fits past usage. Yet Locke answers that the
prince and cobbler have switched bodies. Materialists counter that the
prince and cobbler have instead switched psychologies. Locke and his
adversary should have instead shrugged their shoulders.



The Cambridge University philosophers Bertrand Russell (1923) and
Frank Ramsey were heavily influenced by American pragmatism (Misak
2016). Their reading of Peirce made semantic indeterminacy grow into
an international pre-occupation among analytic philosophers. Although
vagueness seems holistic, Peirce's definition is reductive.
'Tall' is vague by virtue of there being cases in which
there is no fact of the matter. A man who is 1.8 meters in height is
neither clearly tall nor clearly non-tall. No amount of conceptual
analysis or empirical investigation can settle whether a 1.8 meter man
is tall. Borderline cases are inquiry resistant.



Inquiry resistance recurs. For in addition to the unclarity of the
borderline case, there is normally unclarity as to where the unclarity
begins. Twilight governs times that are borderline between day and
night. But we are equally uncertain as to when twilight begins. This
suggests there are borderline cases of borderline cases of
'day'. Consequently, 'borderline case' has
borderline cases. This higher-order vagueness seems to show that
'vague' is vague (Hu 2017). We are slow to recognize
higher-order vagueness because we are under a norm to assert only what
we know. When we discuss cases of twilight, we focus on definite cases
of this borderline between day and night, not the marginal cases
between definite day and definite twilight.



The vagueness of 'vague' would have two destructive
consequences. First, Gottlob Frege could no longer coherently
characterize vague predicates as incoherent. For this attribution of
incoherence uses 'vague'. Frege's ideal of precision
is itself vague because 'precise' is the complement of
'vague'. Second, the vagueness of 'vague'
dooms efforts to avoid a sharp line between true and false with a
buffer zone that is neither true nor false. If the line is not drawn
between the true and the false, then it will be between the true and
the intermediate state. Any finite number of intermediates just delays
the inevitable.



Several philosophers repudiate higher-order vagueness as an illusion
(Wright 2021, chapter 5). They deny that there is an open-ended
iteration of borderline status. They find it telling that speakers do
not go around talking about borderline borderline cases and borderline
borderline borderline cases and so forth (Raffman 2005, 23). Defenders
of higher-order vagueness reply that ordinary speakers avoid iterating
'borderline' for the same reason they avoid iterating
'million' or 'know'. The iterations are
confusing but perfectly meaningful. 'Borderline' behaves
just like a vague predicate. For instance, 'borderline'
can be embedded in a sorites argument. Defenders of higher-order
vagueness have also tried to clinch the case with particular-specimens
such as borderline hermaphrodites (reasoning that these individuals
are borderline borderline males) (Sorensen 2010).



Second thoughts about higher-order vagueness are difficult to sustain
unless one also rethinks first-order vagueness. Perhaps Peirce
overreacted to the threat of futile inquiry! Where does the tail of a
snake begin? When posed as a rhetorical question, the speaker
intimates that there is no definite answer. But the tail can be
located by tracing down from the snake's rib cage. A false
attribution of indeterminacy will lead to the premature abandonment of
inquiry. Is heat the absence of cold or cold the absence of heat? Many
physicists ceased to inquire after Ernst Mach ridiculed the ancient
riddle as indeterminate. Israel Scheffler (1979, 77-78) draws a
moral from such hasty desertions. Never give up! In an internal
criticism of his colleague Quine, Scheffler condemns any attribution
of semantic indeterminacy as a defeatist relic of the distinction
between analytic and synthetic statements. Appeals to meaning never
conclude inquiry or preclude inquiry.



No one pursues the many lines of inquiry Scheffler refuses to close.
Yet some agree that the attribution of borderline status is never
mandatory. Crispin Wright (2021) concludes there are no
*definite* borderline cases. Diana Raffman (2014, 153) offers
some empirical support for borderline status always being optional. A
minority of fluent speakers will draw a sharp line between yellow and
orange.



Mandatory status is defended by listing paradigm cases of
'borderline'. Just as a language teacher tests for
competence by having students distinguish clear positives from clear
negatives, she also tests by their recognition of borderline cases.
Any pupil who uses '-ish' without recognizing borderline
cases of 'noonish' has not yet mastered the suffix.
'Noonish' is not a synonym of 'within ten minutes of
noon' or any other term with a precisely delimited interval.


## 2. Comparison with Ambiguity and Generality



'Tall' is relative. A 1.8 meter pygmy is tall for a pygmy
but a 1.8 meter Masai is not tall for a Masai. Although relativization
disambiguates, it does not eliminate borderline cases. There are
shorter pygmies who are borderline tall for a pygmy and taller Masai
who are borderline tall for a Masai. The direct bearers of vagueness
are a word's full disambiguations such as 'tall for an
eighteenth-century French man'. Words are only vague indirectly,
by virtue of having a sense that is vague. In contrast, an ambiguous
word bears its ambiguity directly--simply in virtue of having
multiple meanings.



This contrast between vagueness and ambiguity is obscured by the fact
that most words are both vague and ambiguous. 'Child' is
ambiguous between 'offspring' and 'immature
offspring'. The latter reading of 'child' is vague
because there are borderline cases of immature offspring. The contrast
is further complicated by the fact that most words are also general.
For instance, 'child' covers both boys and girls.



Ambiguity and vagueness also contrast with respect to the
speaker's discretion. If a word is ambiguous, the speaker can
resolve the ambiguity without departing from literal usage. For
instance, he can declare that he meant 'child' to express
the concept of an immature offspring. If a word is vague, the speaker
cannot resolve the borderline case. For instance, the speaker cannot
make 'child' literally mean anyone under eighteen just by
intending it. That concept is not, as it were, on the menu
corresponding to 'child'. He would be understood as taking
a special liberty with the term to suit a special purpose.
Acknowledging departure from ordinary usage would relieve him of the
obligation to defend the sharp cut-off.



When the movie director Alfred Hitchcock mused 'All actors are
children' he was taking liberties with clear negative cases of
'child' rather than its borderline cases. The aptness of
his generalization is not judged by its literal truth-value (because
it is obviously untrue). Likewise, we do not judge precisifications of
borderline cases by their truth-values (because they are obviously not
ascertainable as true or false). We instead judge precisifications by
their simplicity, conservativeness, and fruitfulness. A
precisification that draws the line across the borderline cases
conserves more paradigm usage than one that draws the line across
clear cases. But conservatism is just one desideratum among many.
Sometimes the best balance is achieved at the cost of turning former
positive cases into negative cases.



Once we shift from literal to figurative usage, we gain fictive
control over our entire vocabulary--not just vague words. When a
travel agent says 'France is a hexagon', we do not infer
that she has committed the geometrical error of classifying France as
a six-sided polygon. We instead interpret the travel agent as speaking
figuratively, as meaning that France is shaped like a hexagon.
Similarly, when the travel agent says 'Reno is the biggest
little city', we do not interpret her as overlooking the
vagueness of 'little city'. Just as she uses the obvious
falsehood of 'France is a hexagon' to signal a metaphor,
she uses the obvious indeterminacy of 'Reno is the biggest
little city' to signal hyperbole.



Given that speakers lack any literal discretion over vague terms, we
ought not to chide them for indecisiveness. Where there is no decision
to be made, there is no scope for vice. Speakers would have literal
discretion if statements applying a predicate to its borderline cases
were just permissible variations in linguistic usage. For the sake of
comparison, consider discretion between alternative spellings.
Professor Letterman uses 'judgment' instead of
'judgement' because he wants to promote the principle that
a silent E signals a long vowel. He still has fond memories of Tom
Lehrer's 1971 children's song "Silent E":



> 
> Who can turn a can into a cane?
>   
> 
> Who can turn a pan into a pane?
>   
> 
> It's not too hard to see,
>   
> 
> It's Silent E.
>   
> 
>   
> 
> Who can turn a cub into a cube?
>   
> 
> Who can turn a tub into a tube?
>   
> 
> It's elementary
>   
> 
> For Silent E.
> 



Professor Letterman disapproves of those who add the misleading E but
concedes that 'judgement' is a permissible spelling; he
does not penalize his students for misspelling when they make their
hard-hearted choice of 'judgement'. Indeed, Professor
Letterman scolds students if they fail to stick with the same spelling
throughout the composition. Choose but stick to your choice!



Professor Letterman's assertion 'The word for my favorite
mental act is spelled j-u-d-g-m-e-n-t' is robust with respect to
the news that it is also spelled j-u-d-g-e-m-e-n-t. He would continue
to assert it. He can conjoin the original assertion with information
about the alternative: 'The word for my favorite mental act is
spelled j-u-d-g-m-e-n-t and is also spelled j-u-d-g-e-m-e-n-t'.
In contrast, Professor Letterman's assertion that 'Martha
is a woman' is not robust with respect to the news that Martha
is a borderline case of 'woman' (say, Letterman learns
Martha is younger than she looks). The new information would lead
Letterman to retract his assertion in favor of a hedged remark such as
'Martha might be a woman and Martha might not be a woman'.
Professor Letterman's loss of confidence is hard to explain if
the information about her borderline status were simply news of a
different but permissible way of describing her. Discoveries of
notational variants do not warrant changes in former beliefs.



News of borderline status has an evidential character. Loss of clarity
brings loss of warrant. If you do not lower your confidence, you are
open to the charge of dogmatism. To concede that your ferry is a
borderline case of 'seaworthy' is to concede that you do
not know that your ship is seaworthy. That is why debates can be
dissolved by showing that the dispute is over an absolute borderline
case. The debaters should both become agnostic. After all, they do not
have a license to form beliefs beyond their evidence. If your ferry is
borderline seaworthy, you are not free to believe it is seaworthy.



According to W. K. Clifford, belief beyond the evidence is always
morally forbidden, not just irrational. William James countered that
belief beyond the evidence is permissible when there is a forced,
momentous, live option. Can I survive the destruction of my body,
simply by virtue of a strong psychological resemblance with a future
person? The answer affects my prospects for immortality. James says
that my belief that this is borderline survival does not forbid me
from believing that it is survival. Whether I exercise this
prerogative depends on my temperament. Since momentous borderline
cases are so prevalent in speculative matters, much philosophical
disagreement is a matter of personality differences.



The Clifford drop into agnosticism is especially quick for those who
believe that evidence supports a unique judgment. A detective who
thinks the evidence shows Gray is a murderer cannot concede that the
evidence equally permits the judgment that Gray is not a murderer. If
the detective concedes that Gray's act is borderline murder,
then he cannot shed the hedge and believe the act is murder.



In the case of relative borderline cases, the hedge can be shed by the
discovery of a novel species of evidence. Murder inquiries were
sometimes inconclusive because the suspect had an identical twin.
After detectives learned that identical twins have distinct
fingerprints, some of these borderline cases became decidable.
Presently, there are some borderline cases arising from the extreme
similarity in DNA between identical twins. Soon, geneticists will be
able to reliably discern the slight differences between twin DNA that
arise from mutations.



News of an alternative sense is like news of an alternative spelling;
there is no evidential impact (except for meta-linguistic beliefs
about the nature of words). Your assertion that 'All bachelors
are men' is robust with respect to the news that
'bachelor' has an alternative sense in which it means a
male seal. Assertions are not robust with respect to news of hidden
generality. If a South African girl says 'No elephant can be
domesticated' but learns there is another species of elephant
indigenous to Asia, she will lose some confidence; maybe Asian
elephants can be domesticated. News of hidden generality has
evidential impact. When it comes to robustness, vagueness resembles
generality more than vagueness resembles ambiguity.



Mathematical terms such as 'prime number' show that a term
can be general without being vague. A term can also be vague without
being general. Borderline cases of analytically empty predicates
illustrate this possibility.



Generality is obviously useful. Often, lessons about a particular
\(F\) can be projected to other \(F\)s in virtue of their common
\(F\)-ness. When a girl learns that *her* cat has a nictitating
membrane that protects its eyes, she rightly expects that her
neighbor's cat also has a nictitating membrane. Generality saves
labor. When the girl says that she wants a toy rather than clothes,
she narrows the range of acceptable gifts without going through the
trouble of specifying a particular gift. The girl also balances
values: a gift should be intrinsically desired and yet also be a
surprise. If uncertain about which channel is the weather channel, she
can hedge by describing the channel as 'forty-something'.
There is an inverse relationship between the contentfulness of a
proposition and its probability: the more specific a claim, the less
likely it is to be true. By gauging generality, we can make sensible
trade-offs between truth and detail.



'Vague' has a sense that is synonymous with abnormal
generality. This precipitates many equivocal explanations of
vagueness. For instance, many commentators say that vagueness exists
because broad categories ease the task of classification. If I can
describe your sweater as red, then I do not need to ascertain whether
it is scarlet. This freedom to use wide intervals obviously helps us
to learn, teach, communicate, and remember. But so what? The problem
is to explain the existence of borderline cases. Are they present
because vagueness serves a function? Or are borderline cases
side-effects of ordinary conversation--like the echoes indoor
listeners learn to ignore?



Every natural language is both vague and ambiguous. However, both
features seem eliminable. Indeed, both are eliminated in miniature
languages such as checkers notation, computer programming languages,
and mathematical descriptions. Moreover, it seems that both vagueness
and ambiguity ought to be minimized. 'Vague' and
'ambiguous' are pejorative terms. And they deserve their
bad reputations. Think of all the automotive misery that has been
prefaced by



> 
> Driver: Do I turn left?
>   
> 
> Passenger: Right.
> 



English can be lethal. Philosophers have long motivated appeals for an
ideal language by pointing out how ambiguity creates the menace of
equivocation:



> 
> No child should work.
>   
> 
> Every person is a child of someone.
>   
> 
> Therefore, no one should work.
> 



Happily, we know how to criticize and correct all equivocations.
Indeed, every natural language is self-disambiguating in the sense
that each has all the resources needed to uniquely specify any reading
one desires.


## 3. The Philosophical Challenge Posed by Vagueness



Vagueness, in contrast, precipitates a profound problem: the sorites
paradox. For instance,


* Base step: A one-day-old human being is a child.
* Induction step: If an \(n\)-day-old human being is a child, then
that human being is also a child when it is \(n + 1\) days old.
* Conclusion: Therefore, a 36,500-day-old human being is a
child.



The conclusion is false because a 100-year-old man is clearly a
non-child. Since the base step of the argument is also plainly true
and the argument is valid by mathematical induction, we seem to have
no choice but to reject the second premise.



George Boolos (1991) observes that we have an autonomous case against
the induction step. In addition to implying plausible conditionals
such as "If a 1-day-old human being is a child, then that human
being is also a child when it is 2 days old", the induction step
also implies ludicrous conditionals such as "If a 1-day-old
human being is a child, then that human being is also a child when
36,500 days old".



Boolos is puzzled why we overlook these clear counterexamples. One
explanation is that we tend to treat the induction step as a
*generic* generalization such as "People have ten
toes" (Sorensen 2012) rather than a generalization with a
quantifier such as 'all' or 'most'. Whereas
the universal generalization "All people have ten toes" is
refuted by a single person with eleven toes, the generic
generalization tolerates exceptions.



This hypothesis is plausible for newcomers to the sorites paradox. But
it is less plausible for those being tutored by Professor Boolos. He
guides logic students to the correct, universal reading of the
induction step. When students drift to a generic reading, Boolos
reminds them that the induction step is a generalization that
tolerates no exceptions.



Guided by Boolos' firm hand, logic students drive a
*second* stake into the heart of the induction step (the
entailment of preposterous conditionals). Yet the paradox seems far
from dead. The hitch is that death for the induction step is life for
its negation. The negation of the second premise classically implies a
sharp threshold for childhood. For the falsehood of the induction step
implies the existential generalization that there is a number \(n\)
such that an \(n\)-day-old human being is a child but is no longer a
child one day later.



Epistemicists accept this astonishing consequence. They stand by
classical logic and conclude that vagueness is a form of ignorance.
For any day in the borderline region of 'child', there is
a small probability that it is the day one stopped being a child.
Normally, the probability is negligible. But if we round down to zero,
we fall into into inconsistency.



This diagnosis is reminiscent of Carl Hempel's solution to the
Raven Paradox. At first blush, we deny that a white handkerchief could
confer any degree of confirmation on 'All ravens are
black'. But this is incompatible with us regarding a white
handkerchief as providing some confirmation of 'All non-black
things are non-ravens'. Since this contrapositive is logically
equivalent to 'All ravens are black', whatever confirms
one hypothesis confirms the other. Hempel advises us to regain
consistency by conceding that a white handkerchief slightly confirms
'All ravens are black'. Similarly, Hempel might advise us
to slightly raise the probability of a borderline child being a child
upon learning that she is a day younger than previously believed.



Hempel's only reservation about indoor ornithology is
inefficiency. Given how the actual world is arranged, more news about
'All ravens are black' can be gained outdoors. We do not
bother to look indoors because the return on inquiry is too low.
Bottom-up Bayesians envisage extraordinary scenarios in which the
return on inquiry could be high. However, even their ample imagination
fails to yield a possible world in which a single day has a more
discernible opportunity to make a difference to whether someone is a
child. All pressure to shift credence is exerted top down from
probability theory.



Timothy Williamson (1994) traces our ignorance of the threshold for
childhood to "margin for error" principles. If one knows
that an \(n\)-day-old human being is a child, then that human being
must also be a child when \(n + 1\) days old. Otherwise, one is right
by luck. Given that there is a threshold, we would be ignorant of its
location. Under a use theory of meaning, the threshold depends on many
speakers. Williamson characterizes this threshold as sharing the
unpredictability of long-term weather forecasts. According to the
meteorologist Edward Lorenz, a tornado in Kansas could be precipitated
by the flapping of a butterfly's wings in Tokyo.



The use theory of meaning does not apply to asocial languages. For
instance, psycholinguists postulate a language of thought that is not
used for communication. This innate language is vague.



Natural languages lexicalize only a small share of our concepts. The
primary bearers of vagueness are concepts rather than words (Bacon
2018). Aphasia does not end vague thinking. Insofar as we attribute
thinking to non-linguistic animals, we attribute vague thinking.
Peirce's emphasis on a community of inquirers encourages a
sociology of language. Ludwig Wittgenstein's attack on the
possibility of a private language was intended to remove any rival to
a community-based approach to language. Yet asocial vagueness remains
viable. This motivates some epistemicists to explain our ignorance
more metaphysically, say, as an effect of there being no truth-maker
to control the truth-value of a borderline statement (Sorensen 2001,
chapter 11).



Debate over Williamson's margin for error principle draws us
deep into epistemology and modality (Yli-Vakkuri 2016). Preferring to
explore the shallows, some commentators survey attitudes weaker than
knowledge. According to Nicholas Smith (2008, 182) we cannot even
guess that the threshold for baldness is the 400th hair. Hartry Field
(2010, 203) denies that a rational man can fear that he has just
passed the threshold into being old. Hope, speculation, and wonder do
not require evidence but they do require understanding. So it is
revealing that these attitudes have trouble getting a purchase on the
threshold of oldness (or any other vague predicate). A simple
explanation is that bare linguistic competence gives us knowledge that
there are no such thresholds. This accounts for the comical air of the
epistemicist. Just as there is no conceptual room to worry that there
is a natural number between sixty and sixty one, there is no
conceptual room to worry that one has passed the threshold of oldness
between one's sixtieth and sixty-first birthday.



An old epistemicist might reply: My piecemeal confidence that a given
number is not the threshold for oldness does not agglomerate into
collective confidence that there is no such number. If I wager against
each number being the threshold, then I must have placed a losing bet
somewhere. For if I won each bet then there was no opportunity for me
to make the transition to oldness. My bookie could have made a
"Dutch book" against me. He would have been entitled to
payment without having to identify which bet I lost. Since
probabilities may be extracted from hypothetical betting behavior, I
must actually assign some small (normally negligible) probability to
hypotheses identifying particular thresholds. So must you.



Stephen Schiffer (2003, 204) denies that classical probability
calculations apply in vague contexts. Suppose Donald is borderline old
and borderline bald. According to Schiffer we should be just as
confident in the conjunction 'Donald is old and bald' as
in either conjunct. Adding conjuncts does not reduce confidence
because we have a "vague partial belief" rather than the
standard belief assumed by mathematicians developing probability
theory. Schiffer offers a calculus for this vagueness-related
propositional attitude. He crafts the rules for vague partial belief
to provide a *psychological* solution to the sorites
paradox.



The project is complicated by the fact that vague partial beliefs
interact with precise beliefs (MacFarlane 2010). Consider a statement
that has a mixture of vague and precise conjuncts: 'Donald is
old and bald and has an even number of hairs'. Adding the extra
precise conjunct should diminish confidence. Schiffer also needs to
accommodate the fact that some speakers are undecided about whether
the nature of the uncertainty involves vagueness. Even an idealized
speaker may be unsure because there is vagueness about the borders
between vagueness-related uncertainty and other sorts of
uncertainty.



Attitudes toward determinacy vary over time. Bernard Williams (2010,
162) says Thucydides (460-400 BC) invented historical time.
Thucydides insists that any two events occurred simultaneously or one
occurred before the other. Thucydides does not countenance a vague
"olden days"--a period in which events fail to be
chronologically ordered. Every event is situated on a single timeline.
There can be ignorance of when an event took place but there is always
a fact of the matter. With similar focus, Archimedes (287-212
BC) insists that every quantity corresponds to a specific number. No
quantity, however large, is innumerable. In "The Sand
Reckoner" Archimedes devises a special notation to express the
total number of grains of sand. The reductionism of Enlightenment
thinkers led them to further analysis of shades of gray into black and
white dots of precision.



In a holistic reaction, the Romantics portray the new boundaries as no
more real than the grid lines of a microscope. The specimen under the
slide lacks boundaries. Understanding does not require boundaries.
Organizing items along a spectrum is a natural form of classification.
We comprehend this analog grouping more readily and smoothly than we
grasp the digital groupings instilled by Descartes' algebraic
representation of space. If our metaphysics of classification is
constricted by a logic designed for mathematics, the sorites paradox
will make us see disorder where there is really order of a more fluid
sort. Spontaneous arrangements will bring what order is permitted by
the subject matter. Romantic biologists take to the field and refuse
to pen species within the axes of the Cartesian coordinate system.
Confronted with novel phenomena such as evolution, magnetism and
electricity, Romantic scientists bypassed boundaries.



A geologist views a pair of aerial photographs cross-eyed. His brain
recognizes the two images as representations of the same landscape.
The flat images fuse into a unifying stereoscopic perspective. The
Romantics portray vague language as perceptively de-focussed.
Vagueness is not the cloudy vision of an old man with cataracts.



The epistemicist fits into the Thucydidean tradition of replacing
indeterminacy with determinacy. Crispin Wright (2021, 393) compares
the epistemicist's postulation of hidden boundaries to the
mathematical realist's postulation of a hidden Platonic realm.
Both the epistemicist and the mathematical realist add metaphysical
infrastructure to ensure truth will transcend our means of proof. This
infrastructure validates proofs of generalizations that are not
constructed from examples. The impossibility of finding a witness for
an existential generalization, such as 'There is a youngest old
man', is compatible with there being a non-constructive
proof.



Despite rearguard defenses by the Romantics, Newtonian physics
appeared to show that empirical reality fits classical logic. The hard
sciences illustrated the crystalline growth of determinacy. But in a
shocking reversal, quantum mechanics suggests the possibility of
reversing from determinacy to indeterminacy. What had been formerly
regarded as determinate, the position and velocity of an object, was
regarded as indeterminate by Werner Heisenberg. Quantum logicians,
such as Hilary Putnam (1976), abandoned the distributive laws of
propositional logic.



Putnam (1983) went on propose a solution that seems to apply L. E. J.
Brouwer's intuitionism to the sorites paradox. (In response to a
rebuttal, Putnam [1985] surprisingly denied that intuitionistic
semantics was part of the proposal.) Brouwer had achieved fame with
his fixed point theorem. But reflection on Kant's philosophy of
mathematics led Brouwer to retract his lovely proof. *Any*
non-constructive proof had to be replaced! Brouwer's recall of
established theorems yielded more specific proofs that had higher
explanatory power. Theorems that could not be retrofitted became
objects of agnosticism. This was sad news for Brouwer's fixed
point theorem. But Putnam welcomes the recall of the
epistemicist's theorem that there is a smallest natural number.
The classical logician relies on double negation to deduce a hidden
boundary from the negation of the generalization "If \(n\) is
small, then \(n + 1\) is small". Since the intuitionist does not
accept this classical rule, his refusal to acquiesce to the induction
step does not saddle him with hidden thresholds. Like the
epistemicist, the intuitionist treats vagueness as a cognitive problem
that does not require rejecting the law of bivalence. Unlike the
epistemicist, the intuitionist must not claim to know the law of
bivalence. To sustain this agnosticism about bivalence, the
intuitionist must parry the sorites monger's reformulations
(Wright 2021, chapter 14). For instance, the intuitionist's
least number principle threatens to re-ignite the sorites.



Many commentators concede to the epistemicist that it is logically
possible for vague predicates to have thresholds. They just think it
would be a miracle:



> 
> It is logically possible that the words on this page will come to life
> and sort my socks. But I know enough about words to dismiss this as a
> serious possibility. So I am right to boggle at the possibility that
> our rough and ready terms such as 'red' could so
> sensitively classify objects.
> 



Epistemicists counter that this bafflement rests on an over-estimate
of stipulation's role in meaning. Epistemicists say much meaning
is acquired passively by default rather than actively by decision. If
some boundaries are more eligible for reference than others, then the
environment does the work. If nothing steps in to make a proposition
true, then it is false. Or so opines Timothy Williamson.



Most doubt whether precise analytical tools fit vague arguments. The
scientific Romantic H. G. Wells was amongst the first to suggest that
we must moderate the *application* of logic:



> 
> Every species is vague, every term goes cloudy at its edges, and so in
> my way of thinking, relentless logic is only another name for
> stupidity--for a sort of intellectual pigheadedness. If you push
> a philosophical or metaphysical enquiry through a series of valid
> syllogisms--never committing any generally recognised
> fallacy--you nevertheless leave behind you at each step a certain
> rubbing and marginal loss of objective truth and you get deflections
> that are difficult to trace at each phase in the process. Every
> species waggles about in its definition, every tool is a little loose
> in its handle, every scale has its individual error. (1908, 18)
> 



Many more believe that the problem is with logic itself rather than
the manner in which it is applied. They favor solving the sorites
paradox by replacing standard logic with an earthier deviant
logic.



There is a desperately wide range of opinions as to how the revision
of logic should be executed. Every form of deviant logic has been
applied in the hope of resolving the sorites paradox.


## 4. Many-valued Logic



An early favorite was many-valued logic. On this approach, borderline
statements are assigned truth-values that lie between full truth and
full falsehood. Some logicians favor three truth-values, others prefer
four or five. The most popular approach is to use an infinite number
of truth-values represented by the real numbers between 0 (for full
falsehood) and 1 (for full truth). This infinite spectrum of
truth-values might be of service for a continuous sorites argument
involving 'small real number' (Weber and Colyvan 2010).
Classical logic does not enforce a sharp boundary for infinite
domains. But it still enforces unlimited sensitivity for vague
predicates. If two quantities \(A\) and \(B\) are growing but \(A\) is
given a headstart, then \(A\) will cease to equal a small real number
before \(B\). Since this transitional difference holds regardless of
the lead's magnitude, "\(A\) equals a small real
number" passes from true to false within an arbitrarily narrow
interval. If the number of truth-values is instead as numerous as the
real numbers, there is no longer any mismatch between the tiny size of
the transition and the large difference in truth-value.



There is a new mismatch, however. The number of truth-values now
dwarfs the number of truth-value bearers. Since a natural language can
express any proposition, the number of propositions equals the number
of sentences. The number of sentences is \(\aleph\_0\) while (assuming
the continuum hypothesis) the number of real numbers is at least
\(\aleph\_1\). Consequently, there are infinitely many truth-values
without bearers.



This new mismatch seems more bearable than the old mismatch it is
designed to avoid. Yet some find even a lower infinity of truth-values
over-bearing. Instead of having just one artificially sharp line
between the true and the false, the many-valued logician has
infinitely many sharp lines such as that between statements with a
truth of .323483925 and those with a higher truth-value. Mark
Sainsbury grimaces, '... you do not improve a bad idea by
iterating it' (1996, 255).



A proponent of an infinite-valued logic might cheer up Sainsbury with
an analogy. It is a bad idea to model a circle with a straight line.
Using two lines is not much better, nor is there is much improvement
using a three-sided polygon (a triangle). But as we add more straight
lines to the polygon (square, pentagon, hexagon, and so on) we make
progress--by iterating the bad idea of modeling a circle with
straight lines.



Indeed, it would be tempting to triumphantly conclude "The
circle has been modeled as an infinite-sided polygon". But has
the circle been revealed to be an infinite-sided polygon? Have curved
lines been replaced by straight lines? Have curved lines (and hence
circles) been proven to not exist? A model can succeed without being
clear what has been achieved.



But it is premature to dwell on the analogy "Precision is to
vagueness as straightness is to curvature". The many-valued
logician must first vindicate the comparison by providing details
about how to calculate the truth-values of vague statements from the
truth-values of their component statements.



Proponents of many-valued logic approach this obligation with great
industry. Precise new rules are introduced to calculate the
truth-values of compound statements that contain statements with
intermediate truth-values. For instance, the revised rule for
conjunctions assigns the conjunction the same truth-value as the
conjunct with the lowest truth-value. These rules are designed to
yield all standard theorems when all the truth-values are 1 and 0. In
this sense, classical logic is a limiting case of many-valued logic.
Classical logic is agreed to work fine in the area for which it was
designed--mathematics.



Most theorems of standard logic break down when intermediate
truth-values are involved. (An irregular minority, such as 'If
\(P\), then \(P\)', survive.) Even the classical contradiction
'Donald is bald and it is not the case that he is bald'
receives a truth-value of .5 when 'Donald is bald' has a
truth-value of .5. Many-valued logicians note that the error they are
imputing to classical logic is often so small that classical logic can
still be fruitfully applied. But they insist that the sorites paradox
illustrates how tiny errors accumulate into a big error.



Critics of the many-valued approach complain that it botches phenomena
such as hedging. If I regard you as a borderline case of 'tall
man', I cannot sincerely assert that you are tall and I cannot
sincerely assert that you are of average height. But I can assert the
hedged claim 'Either you are tall or of average height'.
The many-valued rule for disjunction is to assign the whole statement
the truth-value of its highest disjunct. Normally, the added disjunct
in a hedged claim is not more plausible than the other disjuncts. Thus
it cannot increase the degree of truth. Disappointingly, the proponent
of many-valued logic cannot trace the increase of assertibility to an
increase in the degree of truth.



Epistemicists ascribe the increase in assertibility to the increasing
probability of truth. Since the addition of disjuncts can raise
probability indefinitely, the epistemicists correctly predict that we
can hedge our way to full assertibility. However, epistemicists do not
have a monopoly on this prediction.


## 5. Supervaluationism



According to supervaluationists, borderline statements lack a
truth-value. This neatly explains why it is universally impossible to
know the truth-value of a borderline statement. Supervaluationists
offer details about the nature of absolute borderline cases. Simple
sentences about borderline cases lack a truth-value. Compounds of
these statements can have a truth-value if they come out true
regardless of how the statement is admissibly precisified. For
instance, 'Either Mr. Stoop is tall or it is not the case that
Mr. Stoop is tall' is true because it comes out true under all
ways of sharpening 'tall'. Thus the method of
supervaluations allows one to retain all the theorems of standard
logic while admitting "truth-value gaps".



One may wonder whether this striking result is a genuine convergence
with standard logic. Is the supervaluationist characterizing vague
statements as propositions? Or is he merely pointing out that certain
non-propositions have a structure isomorphic to logical theorems?
(Some electrical circuits are isomorphic to tautologies but this does
not make the circuits tautologies.) Kit Fine (1975, 282), and
especially David Lewis (1982), characterize vagueness as
hyper-ambiguity. Instead of there being one vague concept, there are
many precise concepts that closely resemble each other.
'Child' can mean a human being at most one day old or mean
a human being at most two days old or mean a human being at most three
days old .... Thus the logic of vagueness is a logic for
equivocators. Lewis' idea is that ambiguous statements are true
when they come out true under all disambiguations. But logicians
normally require that a statement be disambiguated *before*
logic is applied. The mere fact that an ambiguous statement comes out
true under all its disambiguations does not show that the statement
itself is true. Sentences which are *actually* disambiguated
may have truth-values. But the best that can be said of those that
merely *could* be disambiguated is that they *would*
have had a truth-value had they been disambiguated (Tye 1989).



Supervaluationism will converge with classical logic only if each word
of the supervaluated sentence is uniformly interpreted. For instance,
'Either a carbon copy of Teddy Roosevelt's signature is an
autograph or it is not the case that a carbon copy of Teddy
Roosevelt's signature is an autograph' comes out true only
if 'autograph' is interpreted the same way in both
disjuncts. Vague sentences resist mixed interpretations. However,
mixed interpretations are permissible for ambiguous sentences. As
Lewis himself notes in a criticism of relevance logic, 'Scrooge
walked along the bank on his way to the bank' can receive a
mixed disambiguation. When exterminators offer 'non-toxic ant
poison', we charitably switch relativizations within the noun
phrase: the substance is safe for human beings but deadly for
ants.



Even if one agrees that supervaluationism converges with classical
logic about theoremhood, they clearly differ in other respects.
Supervaluationism requires rejection of inference rules such as
contraposition, conditional proof and *reductio ad absurdum*
(Williamson 1994, 151-152). In the eyes of the
supervaluationist, a demonstration that a statement is not true does
not guarantee that the statement is false.



The supervaluationist is also under pressure to reject semantic
principles which are intimately associated with the application of
logical laws. According to Alfred Tarski's Convention T, a
statement '\(S\)' is true if and only if \(S\). In other
words, truth is disquotational. Supervaluationists say that being
supertrue (being true under all precisifications) suffices for being
true. But given Convention T, supertruth would then be disquotational.
Since the supervaluationists accept the principle of excluded middle,
the combination of Convention T and 'P or ~P' being
supertrue would force them to say '\(P\)' is supertrue or
'\(\neg P\)' is supertrue (even if '\(P\)'
applies a predicate to a borderline case). This would imply that
either '\(P\)' is true or '\(\neg P\)' is true
(Williamson 1994, 162-163). And that would be a fatal loss of
truth-value gaps for supervaluationism.



There is a final concern about the "ontological honesty"
of the supervaluationist's existential quantifier. As part of
his solution to the sorites paradox, the supervaluationist asserts,
"There is a human being who, for some \(n\), was a child when
\(n\) days old but not when \(n + 1\) days old." For this
statement comes out true under all admissible precisifications of
'child'. However, when pressed the supervaluationist adds
a codicil: "Oh, of course I do not mean that there really is a
sharp threshold for childhood."



After the clarification, some wonder how supervaluationists differs
from drastic metaphysical skeptics. In his nihilist days, Peter Unger
(1979) admitted that it is useful to talk *as if* there are
children. But he insisted that strictly speaking, vague terms such as
'child' cannot apply to anything. Unger was free to use
supervaluationism as a theory to explain our ordinary discourse about
children. (Unger instead used other resources to explain how we
fruitfully apply empty predicates.) But once the dust had cleared and
the precise rubble came into focus, Unger had to conclude that there
are no children.



Officially, the supervaluationist rejects the induction step of the
sorites argument. Unofficially, he seems to instead reject the
*base* step of the sorites argument.



Supervaluationists encourage the view that all vagueness is a matter
of linguistic indecision: the reason why there are borderline cases is
that we have not bothered to make up our minds. The method of
supervaluation allows us to assign truth-values prior to any
decisions. Expressivists think this is a mistake akin to assigning
truth-values to normative claims (MacFarlane 2016). They model
vagueness as practical uncertainty as to whether to treat a borderline
\(F\) as an \(F\). The deliberator may accept the tautologies of
classical logic as constraints governing competing plans for drawing
lines. I can accept "Either Donald is bald or not" without
accepting either disjunct. An existentially quantified sentence can be
accepted even when no instance is. A shrug of the shoulders signals
readiness to go either way, not ignorance as to which possible world
one inhabits.



The expressivist is poised to explain how supervaluationism developed
into the most respected theory of vagueness. Frege portrayed vagueness
as negligent under-definition: One haphazardly introduces a necessary
condition here, a sufficient condition there, but fails to supply a
condition that is both necessary and sufficient. Supervaluationists
counter that indecision can be both intentional and functional.
Instead of committing ourselves prematurely, we fill in meanings as we
go along in light of new information and interests.



Psychologists may deny that we could really commit ourselves to the
complete precisifications envisaged by the supervaluationist (which
encompass an entire language). Many of these comprehensive
alternatives are too complex for memory. For instance, any
precisification admitting a long band of random verdicts will be
algorithmically complex and so not compressible in a rule. Realistic
alternatives are only modestly more precise than ordinary usage.



Nevertheless, the supervaluationist's conjecture about gradual
precisification is popular for the highly stipulative enterprise of
promulgating and enforcing laws (Endicott 2000). Judges frequently
seem to exercise and control discretion by means of vague language.
Uncertainties about the scope of discretion may arise from
higher-order vagueness (Schauer 2016).



Discretion through gap-filling pleases those who regard adjudication
as a creative process. It alarms those who think we should be judged
by laws rather than men. The doctrine of discretion through
indeterminacy has also been questioned on the grounds that the source
of the discretion is the generality of the legal terms rather than
their vagueness (Poscher 2012).



By David Lanius's (2019) reckoning, the only function for
vagueness in law is strategic. Drunk driving is discouraged by the
vagueness of 'drunk' in the way it is discouraged by
random testing of motorists. Scattershot enforcement becomes a welcome
feature of the legal system rather than a bug in the legal
programming. Motorists stay sober to avoid participation in a
punishment lottery.



Judicial variance is just what one expects if the judges are making a
forced choice on borderline cases. Judges cannot confess ignorance and
so are compelled to assert beyond their evidence. Little wonder that
what comes out of their mouth is affected by irrelevancies such as
what went in their mouths for lunch. This susceptibility to bias by
irrelevant factors (weather, order effects, anchoring) could be
eliminated by methodical use of a lottery. The lottery could be
weighted to the fact that borderline crimes vary in how close they
come to being clear crimes.



Hrahn Asgeirsson (2020, chapter 3) admits that the descriptive
question 'Is this drunk driving?' cannot be more reliably
answered by a judge than anyone else when it is a borderline case. But
he thinks the normative question, 'Should this be counted as
drunk driving?' could be better answered by those with superior
knowledge of legislative intent.


## 6. Subvaluationism



Whereas the supervaluationist analyzes borderline cases in terms of
truth-value gaps the dialetheist analyzes them in terms of truth-value
gluts. A glut is a proposition that is both true and false. The rule
for assigning gluts is the mirror image of the rule for assigning
gaps: A statement is true exactly if it comes out true on at least one
precisification. The statement is false just if it comes out false on
at least one precisification. So if the statement comes out true under
one precisification and false under another precisification, the
statement is both true and false.



To avoid triviality, the dialetheist must adopt a paraconsistent logic
that stops two contradictory statements from jointly implying
everything. The resulting "subvaluationism" is a dual of
supervaluationism.



The spiritual father of subvaluationism is Georg Hegel. For Hegel, the
basic kind of vagueness is conflict vagueness. The shopper in between
the wings of a revolving doorway is both inside the building and
outside the building. Degree vagueness is just a special case of the
conflict inherent in becoming. Any process requires a gradual
manifestation of a contradiction inherent in the original state. At
some stage, a metamorphosizing caterpillar is not a butterfly (by
virtue of what it was) and a butterfly (by virtue of what it will be).
Hegelians believed this dialectical conception of vagueness solved the
sorites and demonstrated the inadequacy of classical logic (Weber
2010). The Russian Marxist Georgi Plekhanov (1908 [1937]) proposed a
logic of contradiction to succeed classical logic (Hyde 2008,
93-5). One of his students, Henry Mehlberg (1958) went on to
substitute gaps for gluts. The first version of supervaluationism is
thus a synthesis, reconciling the thesis of classical logic with the
anti-thesis posed by the logic of contradiction.



From a formal point of view, there seems no more reason to prefer one
departure from classical logic rather than the other (Cobreros and
Tranchini 2019). Since Western philosophers abominate contradiction,
parity with dialetheism would diminish the great popularity of
supervaluationism.



A Machiavellian epistemicist will welcome this battle between the gaps
and the gluts. He roots for the weaker side. Although he does not want
the subvaluationist to win, the Machiavellian epistemicist does want
the subvaluationist to achieve mutual annihilation with his
supervaluationist doppelganger. His political calculation is:
Gaps + Gluts = Bivalence.



Pablo Cobreros (2011) has argued that subvaluationism provides a
better treatment of higher-order vagueness than supervaluationism. But
for the most part, the subvaluationists (and their frenemies) have
merely claimed subvaluationism to be at least as attractive as
supervaluationism (Hyde and Colyvan 2008). This modest ambition seems
prudent. After all, truth-value gaps have far more independent support
from the history of philosophy (at least Western philosophy). Prior to
the explosive growth of vagueness research after 1975, ordinary
language philosophers amassed a panoramic battery of analyses
suggesting that gaps are involved in presupposition, reference
failure, fiction, future contingent propositions, performatives, and
so on and so on. Supervaluationism rigorously consolidated these
appeals to ordinary language.



Dialetheists characterize intolerance for contradiction as a shallow
phenomenon, restricted to a twentieth-century Western academic milieu
(maybe even now being eclipsed by the rise of China). Experimental
philosophers have challenged the old appeals to ordinary language with
empirical results suggesting that glutty talk is as readily stimulated
by borderline cases as gappy talk (Alxatib and Pelletier 2011, Ripley
2011).


## 7. Contextualism



Just as contextualism in epistemology runs orthogonal to the familiar
divisions amongst epistemologists (foundationalism, reliabilism,
coherentism, etc.), there are contextualists of every persuasion
amongst vagueness theorists. They develop an analogy between the
sorites paradox and indexical sophistries such as:


* Base step: The horizon is more than 1 meter away.
* Induction step: If the horizon is more than \(n\) meters away,
then it is more than \(n + 1\) meters away.
* Conclusion: The horizon is more than a billion meters away.



The horizon is where the earth meets the sky and is certainly less
than a billion meters away. (The circumference of the earth is only
forty million meters.) Yet when you travel toward the horizon to
specify the \(n\) at which the induction step fails, your trip is as
futile as the pursuit of the rainbow. You cannot reach the horizon
because it shifts with your location.



All contextualists accuse the sorites monger of equivocating. In one
sense, the meaning of 'child' is uniform; the
context-invariant rule for using the term (its
"character") is constant. However, the set of things to
which the term applies (its "content") shifts with the
context. In this respect, vague words resemble indexical terms such
as: 'I', 'you', 'here',
'now', 'today', 'tomorrow'. When a
debtor tells his creditor on Monday 'I will pay you back
tomorrow' and then repeats the sentence on Tuesday, there is a
sense in which he has said the same thing (the character is the same)
and a sense in which he has said something different (the content has
shifted because 'tomorrow' now picks out Wednesday).



According to the contextualists, the rules governing the shifts
prohibit us from interpreting any instance of the induction step as
having a true antecedent and a false consequent. The very process of
trying to refute the induction step changes the context so that the
instance will not come out false. Indeed, contextualists typically
emphasize that each instance is true. Assent is mandatory.
Consequently, direct attacks on the induction step must fail. One is
put in mind of Seneca's admonition to his student Nero:
"However many you put to death, you will never kill your
successor."



Hans Kamp (1981), the founder of contextualism, maintained that the
extension of vague words orbits the speaker's store of
conversational commitments. He notes that the commitment pressures the
speaker to adjust past judgments to current judgments. This backward
smoothing undermines the search for a sharp transition. The diagnostic
potential of this smoothing is first recognized by Diana Raffman
(1994). Switching categories (e.g., from blue to green) between two
neighboring, highly similar items in a sorites series, without
installing a boundary, is enabled by a gestalt switch that occurs
between the two judgments. Switching categories comes at a cost,
however; so once the classifier has switched to a new category, she
will stay in the new category as long as she can. In particular, if
she now reverses direction and goes back down the series, she will
continue to classify as green some items she previously classified as
blue. This pattern of judgments, hysteresis, has the effect of
smoothing out the category shift so that the seeming continuity of the
series is preserved. Instead of merely inviting psychologists to
verify her predictions about hysteresis effects in sorites series,
Raffman (2014, 146-156) collaborates with two psychologists to
run an experiment that confirms them.



Stewart Shapiro integrates Kamp's ideas with Friedrich
Waismann's concept of open texture. Shapiro thinks speakers have
discretion over borderline cases because they are judgment dependent.
They come out true in virtue of the speaker judging them to be true.
Given that the audience does not resist, borderline cases of
'child' can be correctly described as children. The
audience recognizes that other competent speakers could describe the
borderline case differently. As Waismann lyricizes, "Every
description stretches, as it were, into a horizon of open
possibilities: However far I go, I shall always carry this horizon
with me" (1945, 124).



American pragmaticism colors Delia Graff Fara's contextualism.
Consider dandelion farms. Why would someone grow weeds? The answer is
that 'weed' is relative to interests. Dandelions are
unwanted by lawn caretakers but are desired by farmers for food and
wine. Fara thinks this interest relativity extends to all vague words.
For instance, 'child' means a degree of immaturity that is
significant to the speaker. Since the interests of the speaker shift,
there is an opportunity for a shift in the extension of
'child'. Fara is reluctant to describe herself as a
contextualist because the context only has an indirect effect on the
extension via the changes it makes to the speaker's
interest.



How strictly are we to take the comparison between vague words and
indexical terms? Scott Soames (2002, 445) answers that all vague words
literally are indexical.



This straightforward response is open to the objection that the
sorites monger could stabilize reference. When the sorites monger
relativizes 'horizon' to the northeast corner of the
Empire State Building's observation deck, he seems to generate a
genuine sorites paradox that exploits the vagueness of
'horizon' (not its indexicality).



All natural languages have stabilizing pronouns, ellipsis, and other
anaphoric devices. For instance, in 'Jack is tired now and Jill
is too', the 'too' forces a uniform reading of
'tired'. Jason Stanley suggests that the sorites monger
employ the premise:



> 
> If that1 is a child then that2 is too, and if
> that2 is too, then that3 is too, and if
> that3 is too, then that4 is too, ... and
> then that\(\_i\) is too.
> 



Each 'that\(\_n\)' refers to the \(n\)th element in a
sequence of worsening examples of 'child'. The meaning of
'child' is not shifting because the first occurrence of
the term governs all the subsequent clauses (thanks to
'too'). If vague terms were literally indexical, the
sorites monger would have a strong reply. If vague terms only resemble
indexicals, then the contextualist needs to develop the analogy in a
way that circumvents Stanley's counsel to the sorites
monger.



The contextualist would also need to address a second technique for
stabilizing the context. R. M. Sainsbury (2013) advises the sorites
monger to present his premises in apparently random order. No pair of
successive cases raises an alarm that similar cases are being treated
differently. Unless the hearer has extraordinary memory, he will not
feel pressured to adjust the context.



The contextualist must find enough shiftiness to block every sorites
argument. Since vagueness seeps into every syntactic category, critics
complain that contextualism exceeds the level of ambiguity
countenanced by linguists and psycholinguists.



Another concern is that some sorites arguments involve predicates that
do not give us an opportunity to equivocate. Consider a sorites with a
base step that starts from a number too large for us to think about.
Or consider an inductive predicate that is too complex for us to
reason with. One example is obtained by iterating 'mother
of' a thousand times (Sorensen 2001, 33). This predicate could
be embedded in a mind-numbing sorites that would never generate
context shifts.



Other unthinkable sorites arguments use predicates that can only be
grasped by individuals in other possible worlds or by creatures with
different types of minds than ours. More fancifully, there could be a
vague predicate, such as Saul Kripke's "killer
yellow", that instantly kills anyone who wields it. The basic
problem is that contextualism is a psychologistic theory of the
sorites. Since arguments can exist without being propounded, they
float free of attempts to moor them to arguers.


## 8. Global Indeterminacy without Local Indeterminacy?



Contextualists base their holism about vagueness on top-down
psychology. Holism can also be logical. We encountered this in the
supervaluationist's rejection of truth-functionality. Instead of
calculating the truth-values of all compound statements bottom-up from
the truth-values of their component statements, Kit Fine (1975)
assigned truth-values top-down by counting a statement as true if it
comes out true under all admissible precisifications of the entire
language.



Fine's (2020) new holism parachutes in from R. M.
Sainsbury's (1996) characterization of vagueness in terms of
boundarylessness. A precise predicate such as
'thirty-something' organizes cases into pigeonholes.
Positive cases commence at thirty and cease at thirty nine. But
paradigms and foils of 'thirty-ish' operate as poles of a
magnet. Candidates for 'thirty-ish' resemble iron filings.
This force field prevents completion of a forced-march sorites
argument (in which we are marched from a definite instance of a
predicate to a definite non-instance). Judgment must be suspended
somewhere in the series. But it does not follow that there is any
specific case such that one must suspend judgment about it.



Borderline status has the mobility of a bubble in a level. The
precisificationist's mistake is to remove the bubble.
Peirce's mistake is to freeze the bubble into *intrinsic*
uncertainty. This is the dual of the error Peirce attributes to
Rene Descartes' method of universal doubt. Doubting
everything is impossible because doubt only gets leverage against a
fulcrum of presuppositions. (Ludwig Wittgenstein's analogy is
with the hinges of a door that hold fast in order that the door of
doubt can open and close.) These presuppositions are indubitable in
the sense that they cannot be challenged *within* the inquiry.
When the skeptic claims to doubt everything, he winds up doubting
nothing. The skeptic's sham doubts are belied by the absence of
any interruption of his habits. A genuine doubt triggers inquiry.
Inquiry requires presupposition. These background beliefs must be
taken for granted. But the necessary existence of certainty does not
entail any intrinsic certitudes. What must be presupposed for one
inquiry need not be presupposed for another. By Fine's
reckoning, Peirce should have issued a parallel rejection of intrinsic
uncertitudes. Within any inquiry, there must be some suspension of
judgment. But it does not follow that any proposition must resist all
inquiry.



Consider a literary historiographer who, weary of debate about the
identity of the first book, demonstrates that 'book' is
vague by assembling a slippery slope ranging from early non-books to
paradigm books. According to Peirce, the demonstration succeeds only
if the spectrum contains some intrinsically doubtful cases. They are
what makes 'book' vague. Fine's negative thesis is
that this reductionist explanation relies on impossible explanans. A
booklet can no more be an intrinsically borderline book than it can be
intrinsically misshelved. Fine's positive thesis is that
slippery slopes suffice to demonstrate the vagueness of
'book'. There are borderline books only in the way there
are heavy books. In contrast to 'mass',
'heavy' requires relativization to a gravitational field.
'Borderline' requires relativization to a range of
cases.



Many-valued logicians have long contended that local indeterminacy is
incompatible with the law of excluded middle. Fine's (1975)
supervaluationism was designed to reconcile the two. But now Fine
concedes the incompatibility. Unlike the many-valued logician, Fine
blames all the inconsistency on local indeterminacy. Indeed, Fine
continues to think the law of excluded middle is undeniable (which is
not to say it must be affirmed). What Fine does deny is
*conjunctive* excluded middle: (\(B \vee \neg B) \wedge (C \vee
\neg C) \wedge (D \vee \neg D) \wedge \ldots .\) For this principle
implies that there is a sharp cut-off in a forced-march sorites
paradox.



Since conjunctive excluded middle is a theorem of classical logic,
Fine proposes a compatibility semantics that bears a family
resemblance to intuitionism. Fine reconstructs a pedigree by modifying
Saul Kripke's semantics for intuitionistic logic. Timothy
Williamson (2020) objects that the resulting logic condemns even more
innocent inferences than intuitionism. Others defend local
indeterminacy by saying that there are paradigm cases of intrinsically
uncertain propositions such as those opening section one. Paradigms
such as the decapitation riddle concern qualitative vagueness rather
than the quantitative vagueness that generates a forced-march sorites
argument. Kit Fine is at work on a future book that may address the
worry that his account is incomplete or disjunctive.


## 9. Is All Vagueness Linguistic?



Supervaluationists emphasize the distinction between words and
objects. Objects themselves do not seem to be the sort of thing that
can be general, ambiguous, or vague (Eklund 2011). From this
perspective, Georg Hegel commits a category mistake when he
characterizes clouds as vague. Although we sometimes speak of clouds
being ambiguous or even being general to a region, this does not
entitle us to infer that there is metaphysical ambiguity or
metaphysical generality.



Supervaluationists are seconding a fallacy attribution dating back to
Bertrand Russell's seminal article "Vagueness"
(1923). This consensus was re-affirmed by Michael Dummett (1975) and
ritualistically re-avowed by subsequent commentators.



In 1978 Gareth Evans focused opposition to vague objects with a short
proof modeled after Saul Kripke's attack on contingent identity.
If there is a vague object, then some statement of the form '\(a
= b\)' must be vague (where each of the flanking singular terms
precisely designates that object). For the vagueness is allegedly due
to the object rather than its representation. But any statement of
form '\(a = a\)' is definitely true. Consequently, \(a\)
has the property of being definitely identical to \(a\). Since \(a =
b\), then \(b\) must also have the property of being definitely
identical to \(a\). Therefore '\(a = b\)' must be
definitely true!



Evans agrees that there are vague identity statements in which one of
the flanking terms is vague (just as Kripke agrees that there are
contingent identity statements when one of the flanking terms is a
flaccid designator). But then the vagueness is due to language, not
the world.



Despite Evans' impressive assault, there was a renewal of
interest in vague objects in the 1980s. As a precedent for this
revival, Peter van Inwagen (1990, 283) recalls that in the 1960s,
there was a consensus that all necessity is linguistic. Most
philosophers now take the possibility of essential properties
seriously.



Some of the reasons are technical. Problems with Kripke's
refutation of contingent identity have structural parallels that
affect Evans' proof. Evans also relies on inferences that
deviant logicians challenge (Parsons 2000).



In the absence of a decisive *reductio ad absurdum*, many
logicians feel their role to be the liberal one of articulating the
logical space for vague objects. There should be 'vague objects
for those who want them' (Cowles and White 1991). Logic should
be ontologically neutral.



Since epistemicists try to solve the sorites with little more than a
resolute application of classical logic, they are methodologically
committed to a partisan role for logic. Instead of looking for
loopholes, we should accept the consequence (Williamson 2015).



Some non-enemies of vague objects also have an ambition to consolidate
various species of indeterminacy (Barnes and Williams 2011). Talk of
indeterminacy is found in quantum mechanics, analyses of the open
future, fictional incompleteness, and the continuum hypothesis.
Perhaps vagueness is just one face of indeterminacy.



This panoramic vision contrasts with the continuing resolution of many
to tether vagueness to the sorites paradox (Eklund 2011). They fear
that the clarity achieved by semantic ascent will be muddied by
metaphysics.



But maybe the mud is already on the mountain top. Trenton Merricks
(2001) claims that standard characterizations of linguistic vagueness
rely on metaphysical vagueness. If 'Donald is bald' lacks
a truth-value because there is no fact to make the statement true,
then the shortage appears to be ontological.



The view that vagueness is always linguistic has been attacked from
other directions. Consider the vagueness of maps (Varzi 2001). The
vagueness is pictorial rather than discursive. So one cannot conclude
that vagueness is linguistic merely from the premise that vagueness is
representational.



Or consider vague instrumental music such as Claude Debussy's
"The Clouds". Music has syntax but too little semantics to
qualify as language. There is a little diffuse reference through
devices such as musical quotation, leitmotifs, and homages. These
referential devices are not precise. Therefore, some music is vague
(Sorensen 2010). The strength and significance of this argument
depends on the relationship between music and language. Under the
musilanguage hypothesis, language and music branched off from a common
"musilanguage" with language specializing in semantics and
music specializing in the expression of emotion. This scenario makes
it plausible that purely instrumental music could have remnants of
semantic meaning.



Mental imagery also seems vague. When rising suddenly after a
prolonged crouch, I 'see stars before my eyes'. I can tell
there are more than ten of these hallucinated lights but I cannot tell
how many. Is this indeterminacy in thought to be reduced to
indeterminacy in language? Why not vice versa? Language is an
outgrowth of human psychology. Thus it seems natural to view language
as merely an accessible intermediate bearer of vagueness.