Vagueness, Truth and Logic
Author(s): Kit Fine
Source: Synthese, Vol. 30, No. 3/4, On the Logic Semantics of Vagueness (Apr. - May, 1975),
pp. 265-300
Published by: Springer
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KIT FINE
VAGUENESS,
This
paper
ness?' This
TRUTH
AND LOGIC1
'What is the correct logic of vague
began with the question
'What are the correct truth-condi
led to the further question
tions for a vague language?', which led, in its turn, to a more general
and existence. The first half of the paper con
of meaning
consideration
tains the basic material.
Section
1 expounds
one approach
an extension of the
and criticizes
to the problem of truth-conditions.
It is based upon
con
standard truth-tables and falls foul of something called penumbral
nection.
Section 2 introduces an alternative
framework, within which
The key idea is to consider
penumbral connection can be accommodated.
not only the truth-values that sentences actually receive but also the truth
values that they might receive under different ways of making them more
precise. Section 3 describes and defends the favoured account within this
framework. Very roughly, it says that a vague sentence is true if and only
it completely precise. The second half of
if it is true for all ways of making
the paper deals with consequences,
and comparisons.
Sec
complications
tion 4 considers the consequences
that the rival approaches have for logic.
The favoured account leads to a classical logic for vague sentences; and
to this unpopular position are met. Section 5 studies the phe
of higher order vagueness :first, in its bearing upon the truth
or a hierar
for a language that contains a definitely-operator
conditions
objections
nomenon
and second, in its relation to some puzzles con
chy of truth-predicates;
cerning priority and eliminability.
I have tried to keep
Some of the topics tie in with technical material.
this at a minimum.
The reader must excuse me if the technical under
an occasional
produces
unintelligible
ripple upon the surface.
Let us say, in a preliminary way, what vagueness
is. I take it to be a
semantic notion. Very roughly, vagueness
is deficiency of meaning. As
current
it is to be distinguished
from generality, undecidability,
and am
biguity. These latter are, if you like, lack of content, possible knowledge,
and univocal meaning,
respectively.
such,
These
contrasts
can be made
very clear with
the help of some artificial
Synthese 30 (1975) 265-300. All Rights Reserved
Copyright ? 1975 by D. Reidel Publishing Company, Dordrecht-Holland
266
KIT FINE
of the natural number
Suppose that the meaning
and
is
the
following clauses:
nicel5 nice2,
given by
nice3,
examples.
predicates,
(1) (a) n is nice! if n > 15
n is not nice!
(b)
if n <
13
(2) (a) n is nice2 if and only if n > 15
n is nice2
n is nice3
(b)
(3)
if and only
if and only
if n >
14
if n >
15
(1) is reminiscent of Carnap's
(1952) meaning postulates. Clauses
to a single contradictory
(2) (a)-(b) are not intended to be equivalent
clause; somehow the separate clauses should be insulated from one an
Clause
is under-determined;
nice! is vague, its meaning
nice2 is
its
is
and
is
over-determined;
meaning
ambiguous,
highly general
nice3
or un-specific. The sentence 'there are infinitely many nice3 twin primes'
but certainly not vague or ambiguous.
possibly undecidable
other.
Then
that is capable of meaning
is also capable of
type of expression
and even
being vague; names, name-operators,
predicates,
quantifiers,
case
The
is
the
clearest, perhaps paradigm,
vague
sentence-operators.
Any
predicate.
A
further
of vagueness will
characterization
an account of meaning.
not, I think, be
In particular,
if
and intensional sense, then so can vague
is deficiency of extension,
intensional vage
theory-free; for it will rest upon
meaning can have an extensional
ness. Extensional
vagueness
ness deficiency of intension. Moreover,
if intension is the possibility
of
is the possibility
then intensional
of extensional
vagueness
extension,
vagueness. Turn to the clear case of the predicate. A predicate F is ex
tensionally vague if it has borderline cases, intensionally vague if it could
have borderline cases. Thus 'bald' is extensionally
vague, I presume, and
a
of
in
world
remains intensionally
vague
hairy or hairless men. The
is roughly Waismann's
(1945)
overtones.
the epistemological
distinction
but without
vagueness/open-texture
one,
is closely allied to the existence of truth-value
vague sentence is neither true nor false; for any
gaps. Any (extensionally)
vague predicate F, there is a uniquely referring name a for which the sen
tence Fa is neither true nor false : and for any vague name a there is a
a = b is neither
uniquely referring name b for which the identity-sentence
true nor false. Some have thought that a vague sentence is both true and
Extensional
false and
vagueness
that a vague predicate
is both
true and false of some object.
VAGUENESS,
AND
TRUTH
267
LOGIC
of under- and over
this is a part of the general confusion
can
more
sentence
A
be
made
vague
precise; and this opera
determinacy.
tion should preserve truth-value. But a vague sentence can be made to be
However,
true or false, and therefore the original sentence can be neither.
This battle of gluts and gaps may be innocuous, purely verbal. For
either
on the glut view
truth on the gap view is simply truth-and-non-falsehood
it
falsehood
is
falsehood-and-non-truth.
However,
and, similarly,
simply
It is the one
notion that is important for philosophy.
is the gap-inducing
ties in with
that directly
the usual notions
and
of assertion, verification
has a split sense; for it allows
The glut-inducing
notion
consequence.
with fact or absence of meaning.
truth to rest upon either correspondence
extensional
should not be defined
the
vagueness
connection,
Despite
in terms of truth-value gaps. This is because gaps can have other sources,
such as failure of reference or presupposition. What distinguishes
gaps of
is that they can be closed by an appropriate
linguistic decision,
deficiency
of the relevant expression.
viz. an extension, not change, in the meaning
1. The
approach
truth-value
of truth-value gaps
It is this possibility
For the classical conditions
that raises a problem
for truth
the prin
presuppose Bivalence,
or
so
are
sentence
true
not
and
either
that
be
every
false,
they
directly
ciple
to vague sentences. In this, as in other, cases of truth-value
applicable
or Indefinite, as a third
gap, it is tempting to treat Neither-true-nor-false,
conditions.
truth-value
classical
and to model
truth-value
assessment
along
the lines of the
truth-conditions.
of this and subsequent suggestions will first be geared to a
that
first-order
language. Only later do we consider the complications
arise from extending
the language. Let us fix, then, upon an intuitively
but possibly vague, first-order
understood,
language L. There are three
The details
sources of vagueness
in L: the predicates,
the names, and the quantifiers.
To simplify the exposition, we shall suppose that only predicates are vague.
is reducible to predicate
Indeed, it could be argued that all vagueness
any change in truth
vagueness. For possibly one can replace, without
a
name
each
value,
vague
vague predicate and each
by
corresponding
an
over
a
relativised quantifier
vague domain by
quantifier
appropriately
over a more inclusive but precise domain. We shall also suppose, though
268
only
KIT
to avoid
talking of satisfaction,
FINE
that each object
in the domain
has
a name.
let a partial specification be an assignment of a truth-value True (T), False (F) or Indefinite (/) - to the atomic sentences of L; and
we call a specification appropriate if the assignment
is in accordance with
the intuitively understood meanings
of the predicates. Thus an appro
We
now
priate specification would assign True to 'Yul Brynner is bald', False to
'Mick Jagger is bald' and Indefinite to 'Herbert is bald', should Herbert
be a borderline case of a bald man. Then the present suggestion
is that
the truth-value of each sentence in L be evaluated on the basis of the
appropriate
specification.
sense that the truth-value
form function
The
valuation
is to be truth-functional
in the
of each type of compound
sentence be a uni
of the truth-values of its immediate sub-sentences.
can be subject to two natural constraints.
The possible truth-conditions
The first is that the conditions be faithful to the classical truth-conditions
whenever
these are applicable. Call a specification
complete if it assigns
only the definite truth-values, True and False. Then the Fidelity Condi
tion F states that a sentence is true (or false) for a complete specification
if and only if it is classically
true (or false); evaluations
over complete
are classical.
specifications
The second constraint
is that definite
truth-values be stable for im
in specification.
provements
Say that one specification u extends another
tifu assigns to an atomic sentence any definite truth-value assigned by t.
the Stability Condition
S states that if a sentence has a definite
under a specification
t it enjoys the same definite truth-value
u
under any specification
that extends t ;definite truth-values are preserved
Then
truth-value
under
The
extension.
two constraints
must
specification
extensions.
complete
straints are equivalent
ditions:
(i)
work
be retained
together: definite truth-values
upon the classical evaluation
are dropped,
the two con
Indeed, if quantifiers
to the classical necessary truth and falsehood con
t=-?->=!?
1-B-*?B
(ii)
h?&C-?h?andhC
1B&C-*
for a partial
of any of its
IB or 4C.
VAGUENESS,
TRUTH
AND
269
LOGIC
'
1=
A9 for 'A is
connectives.
(I use
Similarly for the other truth-functional
'-?'
'A
and
for
informal
material
A9
for
is
false',
true', '=}
implication.)
still allow some latitude in the formulation of
the conditions
However,
or of maxi
One can move in the direction of minimizing
truth-conditions.
mizing the degree to which sentences receive definite truth-values under
a given specification2. At the one extreme, the indefinite truth-value domi
nates: any sentence with an indefinite subsentence
is also indefinite. Sen
tences are only definite under a classical guarantee. At the other extreme,
the indefinite
truth-value
dithers:
a sentence
is definite
if its truth-value
is
unchanged for any way of making definite its immediate indefinite subsen
tences 3. In effect, the arrows in the clauses (i) and (ii) above are reversed
so that the only divergence from the classical conditions
lies in the rejec
illustrate, a conjunction with indefinite and false
conjuncts is indefinite on the first account, but false on the second. There
are intermediate possibilities,
but they are not very interesting. Indeed,
tion of Bivalence.
To
clause
the conditions
for negation,
(i) uniquely determines
senses
are
and
the
and strong
above alternatives
excluded,
ones for commutative
conjunction.
for the weak
are the only
lines acceptable? Any account that
Is any account along truth-value
satisfies the conditions F and S would always appear to make correct
even the maximizing
However,
policy
fails to make many correct allocations of definite truth-value. For suppose
that a certain blob is on the border of pink and red and let P be the sen
tence 'the blob is pink' and R the sentence 'the blob is red'. Then the con
allocations
of definite
truth-value.
'is pink' and 'is red' are con
junction P & R is false since the predicates
account the conjunction P & R is indefinite
traries. But on the maximizing
since both of the conjuncts P and R are indefinite.
A more general argument applies to any three-valued approach, regard
less of whether
it satisfies the conditions F or S. For P & P is indefinite
since it is equivalent to plain P, which is indefinite, whereas
a conjunction with indefinite conjuncts is sometimes
Thus
sometimes
false and
so '&' is not
truth-functional
with
P & R is false.
indefinite
respect
and
to the
three truth-values, True, False and Indefinite.
A similar argument also applies to the other
example,
P, which
predicates
logical connectives. For
since it is equivalent to plain
the disjunction P v jR is true since the
P v P is indefinite
the disjunction
is indefinite; whereas
'is pink' and
'is red' are complementary
over the given colour
270
KIT
FINE
the conditional P id ? P is presumably
not true, whereas
range. Again,
?
P 3
R is true. It is more difficult to find examples for the quantifiers.
But for the universal quantifier,
say, we may consider the sentence 'All
to the throne are the rightful monarch',
where the domain of
are
consists
several
of
who
all borderline cases
pretenders
quantification
pretenders
'is a rightful monarch'.
of the predicate
The whole
are indefinite.
its immediate subsentences
sentence
is false, yet
Nor is there any safety in numbers. The argument can be extended to
cover any finite-valued
approach or any multi-valued
approach that re
a
to
indefinite
with
be
indefinite.
Such ap
conjunction
quires
conjuncts
are
common
are
and
include
those
that
based
upon degrees of
proaches
a
truth4 and those that satisfy
fidelity and stability condition with respect
to a trichotomy of True, False, and Indefinite truth-values.
The specific examples chosen should not blind us to the general point
that they illustrate. It is that logical relations may hold among predicates
cases or, more generally,
with borderline
indefinite sentences.
among
Given the predicate 'is red', one can understand
the predicate
'is non-red'
to be its contradictory:
the boundary of the one shifts, as it were, with
the boundary of the other. Indeed, it is not even clear that convincing
? P is false even
examples require special predicates.
Surely P&
though
P
is indefinite.
Let us refer to the possibility
that logical relations hold among indef
inite sentences as penumbral connection; and let us call the truths that
truths on a penumbra
arise, wholly or in part, from penumbral connection,
or penumbral
our argument
is that no natural truth-value
truths.
In
such an approach
penumbral
particular,
between
'red' and 'pink' as independent
and as ex
truths. Then
approach
respects
cannot distinguish
clusive upon their common penumbra.
is analogous
Placing the Indefinite on a par with the other truth-values
on
to basing modal
the
three values Necessary,
and
logics
Impossible
or to basing deontic
For
Contingent,
logic on the values Obligatory,
bidden and Indifferent. For here, too, truth-functionality
may be lost:
a conjunction
some
of contingent
sentences
is sometimes
contingent,
a conjunction
times impossible;
of indifferent
sentences
is sometimes
indifferent, sometimes forbidden. In all of these cases there appears to be
a dogmatic adherence to a framework of finitely many truth-values. Per
of sentential operators
is, in some sense, finite.
haps our understanding
VAGUENESS,
TRUTH
AND
but this is not to say that it is based upon
271
LOGIC
a finite
substructure
of truth
values.
2. AN
ALTERNATIVE
FRAMEWORK
can we account
for penumbral connection? Consider again the blob
of pink and red and suppose that it is also a border
'small'. Why do we say that the conjunction
line case of the predicate
How
that is on the border
is pink and red' is false but that the conjunction
'The blob is
pink and small' is indefinite? Surely the answer must rest on the fact that
inmaking the respective predicates more precise the blob cannot be made
'The blob
a clear case of both
the predicates
'pink' and 'red' but can be made a
the predicates
'pink' and 'small'. In other words, the
can
in truth-value reflects a difference
in how the predicates
difference
be made more precise.
clear case of both
Such a suggestion
can be made
precise within the following framework.
a non-empty
set of elements, the speci
consists
of
space
(specification)
a
and
partial ordering ^ (also read : extends) on the set,
fication-points,
i.e. a reflexive, transitive and antisymmetric
relation. A space is appro
a
one point for each
to
if
each
priate
precisification,
point corresponds
A
in a generous light and,
We regard the ways of precisifying
do not tie them to the expressions of any given language.
The nature of the correspondence
is this: each point is assigned a speci
to the precisification
fication that is appropriate
to which it corresponds;
precisification.
in particular,
points extend one another just in case they correspond to precisifications
that extend one another in the natural sense. Thus, at the very simplest,
the specifications could be regarded as the precisifications
themselves and
as
on
the natural extension-relation
the partial ordering
precisifications.
is that truth-valuation
be based, not upon the ap
an
but
upon
space, i.e.
propriate
specification,
appropriate
specification
to
the
that
the
different
upon
ways of
specification-points
correspond
Then
the suggestion
the language more precise. The truth-valuation
is to be uniform
making
at which the
in the sense that it only makes use of the specification-points
are
or
true
false.
subsentences
There
be several
of
course,
may,
given
spaces, but then their differences
appropriate
to the truth-valuation.
This
with
should make
no difference
framework
could be generalized
in various ways. For example,
a subset of points. The new space
each space could be associated
272
KIT FINE
a space that was
be appropriate
only if the subset determined
sense.
to call
in
the
old
This
would
the
allow
truth-definition
appropriate
would
to precisifications.
that did not correspond
However,
upon specifications
such generalizations
appear to have little intuitive foundation and will not
be considered further.
of appropriacy uses the intensional notion of precisifica
account could avoid this in various ways.
strictly extensional
the simplest is to identify the specification-points
with the speci
The account
tion. A
Perhaps
fications
themselves. Thus a specification
space is, in effect, a collection
ordered
the
natural extension-relation.
A
of specifications
by
partially
are
ones.
if
the
Un
is
the
admissible
space
specifications
appropriate
if it is appropriate for some precisi
is admissible
officially, a specification
is primitive.
fication; officially, the notion of admissibility
conditions one can impose upon a specification space.
This
is that it has a base-point,
the appropriate
specification-point.
are
to
all
the
of
which
other
corresponds
precisification
precisifications
It states that any point can be
extensions. Another
is Completeability.
extended to a complete point within the same space, i.e.
There are various
One
C.(Vi) (3w^ t) (u complete)
a point is complete if its specification is complete.
There are also conditions one can impose upon the truth-definition.
of the earlier fidelity
main ones are the appropriate modifications
where
stability
point
conditions.
are
classical,
will
Fidelity
and
at a complete
i.e.
F.t)rA<r+ttA
Stability will
points within
state that the truth-values
The
(classically)
state that truth-values
a given
space,
for t complete.
are preserved
under
extensions
of
i.e.
A
S. t NA and t < u -> u f=
->
u
u
t
A.
and
<
4
tz\A
As with
truth values
the truth-value
to the different
there is the problem of how
specifications. One can tend to minimize
approach,
to tag
or to
the amount of truth and falsehood
tagging. Minimizing
gives
new.
However, maximizing
nothing
gives something altogether different:
a sentence is true (or false) at a partial specification point if and only if
it is true (or false) at all complete extensions. A sentence is true simpliciter
maximize
AND
TRUTH
VAGUENESS,
273
LOGIC
if and only if it is true at the appropriate
specification-point,
complete and admissible
specifications. Truth is super-truth,
i.e. at all
truth from
above5.
approach, there are now many interesting
The most notable is the bastard intuition
intermediate truth-definitions.
istic account, which follows the intuitionistic conditions for ?, &, v, =>
?
and 3, and the classical definition of - 3
for V 6.Given that the domain of
In contrast
to the truth-value
run like this :
the clauses
is constant,
quantification
t? - B <->(Vu ^ t) (not-w?E)
t?B & C++t?B and t?C
I (i)
(ii)
(iii)
t?Bv
C++t?Bort?C
(iv)
(v)
t?Bz*C*+<yw&t)(u?B-*u?C)
t ? (3 x) B(x) <r+t? B(a) for some name a
(vi)
t ? (V jc)B(x) <->(Vu > t) (3 v > u) (v ?B(d)) for each
name
There
a.
are two common
tions. One
is the insistence
uniform. The other
to truth-condi
in the rival approaches
that the procedures
for truth-valuation
be
factors
is the insistence
that the appropriate
form of stability
be satisfied.
These
factors
sions. The
(1)
can be made
standard Fregean
The
of
extension
the
extensions
explicit within an abstract theory of exten
theory has a principle of Functionality:
of a compound
of
its parts
<?)(A?9..., Ak) is a function/^
Ax,...,Ak.
This corresponds to the appropriate form of Uniformity. We now suppose
that the extensions are partially ordered by a relation of extending and
add a principle of Monotonicity:
(2)
If
extensions
tively,
#!,...,
then/^
x'u...,xk
applied
extend
extensions
to *i,...,;t?
xu...,xk,
respec
applied
extends/^
to
Xk.
to the appropriate
form of stability.
corresponds
common
The most
factor
is Monotonicity.
important
in
for
the
section, but it can be given
argued
previous
This
foundation.
theory. We
graft a theory of intensions
shall not be too concerned with the nature
First we must
This was
an
onto
not
intensional
the earlier
of intensions.
A
274
KIT
FINE
can be obtained
by indexing
specific model
that
the
worlds.
(I presume
specification-points
such a model would
world to world.) However,
extensions
remain
with possible
constant from
suffer from familiar diffi
culties. For example,
if the meaning
of A is relevant to the meaning
A v ? A, then the vagueness of A should be relevant to the vagueness
A v ? A. Thus A v ? A should be vague for vague A, though on
say, it is equally and completely precise for all
super-truth account,
The pure theory of intensions
and Intensionality.
should
contain
the analogues
of
of
the
A.
of Func
tionality
of a compound
<?)(Ai9..., Ak) is a function F^
of the intensions of its parts Al9...9 Ak;
If intensions X[,...,
Xk extend intensions X[,..., Xk, respec
to X[,...,
then
tively,
Xk extends F$ applied to
F^ applied
The
(3)
(4)
intension
Xl9...,
The combined
X determined
makes
Xk.
theory should link intensions to extensions. Each intension
an extension x; and each extension
is so determined. This
for two bridge principles:
The
(5)
intension
J4,(xu'"i
F#(Xl9...9Xk)
determines
the extension
xk)l
and
X extends
(6)
Y if and only
if x extends y.
the correspondingly
cal
intension determines
(5) states that a calculated
culated extension; and (6) states that one intension extends another if and
only if the extension determined by the one extends the extension deter
mined by the other.7
We can now derive (2), i.e. Monotonicity,
from (4), (5) and (6). For
= 1.Now
case
take
the
of
k
suppose x extends y. Then X ex
simplicity,
tends Y by (6);F^X) extendsF^Y) by (4); the extension determined by
F^X)
f^x)
extends
the extension
determined
by F^Y),
by (6) again;
and so
extendsf+(y) by (5).
behind this argument are (4) and (6). (4), with
assumptions
to the assumption
that an expression
is made more
(3), is tantamount
its simple terms more precise. This assumption
precise through making
The main
is correct
for the language L. For
the logical
constants
are transparent,
TRUTH
VAGUENESS,
AND
275
LOGIC
as itwere, to vagueness; any precisification
of a constituent shines through
converse
the
of the assumption
into the compound.
also holds;
Indeed,
an expression ismade more precise only through making
its simple terms
more precise. For the logical constants
are already perfectly precise.
Since the logical constants are also the grammatical particles, all vague
as opposed to constructions.
ness can be blamed onto constituents
The
with
second assumption
says, roughly, that extension does not decrease
an increase in intension. In particular, a sentence does not become
upon being made more
'making more precise'.
indefinite
tional
of
from a mere
precise. This is, perhaps, partly defini
For what distinguishes
this operation
is that it preserves truth-value. To pre
of certain truth-value gaps. In any case,
change in meaning
to
is
out the possibility
rule
cisify
it would be odd if definite truth-value
could disappear upon precisifica
then hold by default, in virtue of a lack of meaning.
It
could be a product of linguistic laziness and not be consequent upon a
tion. Truth
could
of meaning
and fact.
two
is the rationale for these
It lies, I think, in our
assumptions?
desire for an enduring use of language. Under
the pressure of their own
of terms will need to change. The terms, in their old
use, the meanings
positive
What
concordance
sense, will not be adequate to express the new truths, pose the next ques
that the
tions, make the right distinctions. Now clearly it is convenient
in meaning
be conservative,
that the true records before the
changes
change remain true after the change. We may wish, for example, to settle
a new case within a classificatory
scheme without upsetting the principles
of classification. But it is the two assumptions which guarantee that truth
value be preserved upon precisification
of terms, that allow for the stability
of recorded
truth within the required instability of meaning.
two assumptions
tie in well with a dynamic conception
guage. For language need not retain its identity upon arbitrary
or rather, any such identity is a matter of degree
in meaning;
These
of lan
changes
and de
pendent upon how much change there is. On the other hand, language
can retain its identity upon precisification
or conservative meaning change ;
for the two assumptions
result in a natural constraint upon change. The
identity of language
is visible,
as it were,
in the permanence
of recorded
truth.
If language is like a tree, then penumbral
connection
is the seed from
the tree grows. For it provides an initial repository of truths that
which
276
KIT
FINE
are to be retained
are
all growth. Some of the connections
throughout
internal. They concern the different borderline cases of a given predicate :
if Herbert
is to be bald, then so is the man with fewer hairs on his head.
are external. They concern the common
But many other of the connections
borderline
cases of different
predicates : if the blob is to be red, it is not
to be pink; if ceremonies are to be games, then so are rituals; if sociology
is to be a science, then so is psychology.
Thus penumbral
connection
results in a web that stretches across the whole of language. The language
itself must
grow
like a balloon,
into shape.
with
the expansion
of each part pulling
the other parts
The two approaches
to truth-conditions
but
agree on requirements,
the requirements are to be met. The agreement consists in
their satisfying the principles of an abstract theory of extension;
the dis
agreement consists in how they satisfy these principles. On the truth-value
the extension of a sentence is a truth-value - True, False or
approach,
differ on how
Indefinite.
Each
truth-value
extends
that
is all. The
extensions
and
itself; True and False extend Indef
and extending-relation
for other
inite;
parts of speech can then be determined
easy to verify that the different accounts
in a natural manner.
on the truth-value
It is then
approach will
to the appro
the principles and that Monotonicity
is equivalent
form
of
stability.
priate
On the specification
there is a slight difficulty in inter
space approach,
of
B
For
the
extension
that of A if B corresponds
will
extend
pretation.
a
are
to A at
There
various ways of securing
later stage of precisification.
satisfy
is to regard each expression as an ordered pair (A, t)9 where
is an ordinary expression and t is a specification-point
that indicates
to
of
Another
is
in
stage
imagine that,
precisification.
precisifying,
is not endowed with a new sense but is succeeded by an
expression
A
this. One
the
an
ex
that sense. Thus the language expands in an orderly manner
the
space, t can then be identified from the ex
throughout
specification
as
at
the
first
which
it is introduced. Let us opt for the first
pression
point
a
sentence (A91) is an ordered pair (U, V),
solution. Then the extension of
=
where u
One extension (?/', P)extends
{U:t < u} and V= {ve U:v?A}.
another (U, V) if U' ? U and V is V n U'. It is again then easy to verify
is the appropriate
that the principles are satisfied and that Monotonicity
pression with
form of stability.
Note
that on this approach,
the extension
is no longer a non-linguistic
VAGUENESS,
TRUTH
AND
277
LOGIC
entity. For to each point is assigned a specification, which is language
if there are external connections,
based. Moreover,
this linguistic depen
dency cannot be avoided. For example, it will be part of the extension of
'red' that its completion never overlaps with the completion of the exten
sion of 'pink'. Only
for the language
as a whole
will
extension
be non
linguistic.
can be simplified. The ex
On the super-truth account, the definitions
a
sentence (A91) will be an ordered pair (U, V), where U =
tension of
=
The relation of ex
and u is complete} and V= {veU:v?A}.
{w.t^u
can
tension has then a similar definition. The partial specification-points
be recovered from the complete ones so long as two further conditions
are satisfied.
The
first
is that two points
to their successors
are identical
if the complete
are the same. The second is
specifications
assigned
set of complete points there is a point extended by
that for any non-empty
set. For then each partial point can be identi
the
those
in
exactly
points
set of admissible specifications. The first condition
fied with a non-empty
can be justified. For the only constraint on
space is that a point can verify all the
if they are true at all of the complete
original
are
true
at
any point extended by a certain subset of the
points, they
is harmless
admission
and the second
into the appropriate
truths. But
penumbral
complete points.
The interpretation
for the second approach has an important distin
is used to
the extension-relation
guishing feature. On both approaches,
formulate a constraint on extensions. But only on the second approach
does this relation enter into the extensions
themselves. Thus how an ex
can be extended is already part of the extension. The extension of
an expression
at a
at a given point uniquely determines
its extension
tension
subsequent point.
The intensional
analogue of this is that how an expression can be made
of a
is
precise
already part of its meaning. Let the actual meaning
simple predicate, say, be what helps determine its instances and counter
more
instances. Let
its potential meaning
consist of the possibilities
for making
an
is a
the
is
that
the
of
Then
precise.
meaning
expression
point
a
of
its
actual
both
In
and
product
understanding
potential meaning.
more
one
has thereby understood
how it can be made
language
precise;
it more
one has understood,
for its growth.
bilities
in terms of the earlier dynamic
model,
the possi
278
KIT FINE
dif
(or intension) implies a corresponding
to
of making more precise. On the first approach,
is to resolve new cases or to make new connections.
For to exclude
This difference
ference
extend
in extension
in the notion
is also to extend. For example,
suppose that
subsequent
specifications
between
'red' and 'scarlet'. Then to
connections
there are no penumbral
require that all scarlet objects be red is to extend on the second ap
proach, but not on the first.
3. The
theory
super-truth
shall argue for the super-truth theory, that a vague
is true if it is true for all admissible and complete specifications.
intensional version of the theory is that a sentence is true if it is true
In this section we
sentence
An
of making
it completely precise (or, more generally,
that an
a
if
it
has
has
that
reference
reference
for all
given Fregean
expression
a
sort
As
it
is
of
it
of
ways
such,
making
completely precise).
principle
for all ways
: truth is secured if it does not turn upon what one means.
of non-pedantry
Absence
of meaning makes for absence of truth-value only if presence of
could make for diversity of truth-value.
meaning
of the classical position. For the
The theory is a partial vindication
not
if
then
classical at a remove. There is
truth-conditions
are,
classical,
one
to
classical
truth
rule
but
truth, viz. that truth is truth in each
linking
This rule is of general application
and not
of a set of interpretations.
The actual
upon the nature of language or interpretation.
dependent
and these
work is done by the clauses for truth in a single interpretation,
are
classical.
super-truth view is better than the others for at least two reasons.
The first is that it covers all cases of penumbral connection. For example,
P & R is false and Pv Ris true since one of P and R is true and the other
The
in any complete and admissible
specification. For the bastard intu
on
P
the
& R is false but Pv Ris indefinite.
other hand,
itionistic account,
false
Indeed, one can argue that the super-truth view is the only one to ac
all penumbral
the following
clauses:
truths. For consider
commodate
A(i)
not-/l=^ -*(3u^t)(u4A)
=1
A -> (3 u ^
for A atomic
not-/
t) (u ? A)
,
vagueness,
truth
and
279
logic
t?-B++tiB
(ii)
tl-B*^t?B
(iii)
t?B & C*-*t?B and t?C
(iv)
t4B&C<-+
/1= (Vx) B(x)
or vjC)
(3 v^u)(vAB
(Vu^t)
+-> t ?
a
name
for
any
B(a)
H (Vx) B(x) <-?(Vu ^ t) (3 v^ u) (v =1
B(a) for some name a)
Clause (i) is a Resolution
Condition R for atomic sentences and states
that an indefinite atomic sentence can be resolved in either way upon im
in precision. The necessary
truth- and falsehood
conditions
provement
are
are to the effect that all truth-functional
to
be
For
redeemed.
pledges
example, clause (iii) for & requires that whenever
to point to a subsequent
possible
specification-point
or C is false.
All
of these clauses
sufficient falsehood
B 8c C
is false
at which
it is
either B
are reasonable
conditions
with the possible exception of the
for & and V. But these clauses are required
for such penumbral
falsehoods as 'The blob is pink and red'
or 'All pretenders
to the throne are the rightful monarch'.
Similar con
siderations apply to the other logical constants v, -* and 3. Now given
to account
F, S and C, the A clauses
super-truth account. Thus the claims of penumbral
to adopt our favoured view.
the ancillary
conditions
The second reason for preferring the super-truth
an optimizing
one's advantage
strategy: maximize
straints. The theory maximizes
the extent of truth
to the constraints F, S and C. The argument can be
are equivalent
to the
connection
force one
view
within
is that it follows
the given con
and falsehood
subject
put another way. The
Resolution
Condition R should hold for all sentences, so that any indef
inite sentence can be resolved in either one of two ways. The value of
of this bipolar resolution:
indefinite sentences lies in the possibility
they
are born, as it were, to be true or false. There is no point in withholding
truth from a sentence that can be made true by improving any improve
ment in precision. Now the super-truth account is the only one to satisfy
F, C, S and R. Thus placing the right value on indef
also forces one to adopt our favoured view.
These arguments are essentially claims of the following form: such and
such theory is the only one to satisfy the reasonable conditions X, Y and
Z. Such claims are of great importance, for they provide a point or ra
the four conditions
initeness
280
KIT FINE
if you want the conditions
for the theory in question:
then you
for different lan
accept the theory. All too often, truth-conditions
guages have been constructed with insufficient regard for rationale. Their
basis has often been a scanty set of intuitions. Thus a great advantage of
tionale
must
the present approach is its possession
One might object to the previous
which
presuppose
Completeability,
of a uniquely determining
rationale.
arguments on the grounds that they
is unreasonable.
there is
However,
a perfectly a priori argument for this condition.
Suppose that the 'limit'
of a chain of admissible
is
also
admissible.
This is a slight
specifications
on
for example,
connection:
it excludes
restriction
the re
penumbral
be finite (in an obvious sense) or that
that the specifications
quirement
they verify
decidable
theories.
Then
by Zorn's Lemma, any admissible
to a maximally
admissible
specification.
can
that each atomic sentence
always be settled in at least
can be extended
specification
Now
suppose
one of two ways,
i.e. that no atomic is ever always indefinite. This is a
Then it follows that the maximally
ad
very weak form of Resolution.
is
missible
complete.
specification
our arguments will still go through. In
Even without Completeability,
use an anticipatory account that makes
we
place of the super-truth theory
a sentence A true if?A
is true on the (bastard) intuitionistic
account,
true. In effect, we mould
i.e. if A is always going to be intuitionistically
a sentence whose
to the Resolution
intuitionism
truth can
Condition:
one
is already true. This account
is the maximal
always be anticipated
to satisfy Stability and the necessary A-clauses. The latter consist of A(i),
Resolution
for atomic sentences, and the left-to-right parts of A(ii)-(iv),
of truth-functional
for countable
do
Redemption
pledges. Moreover,
truth turns out to be a form of super-truth.8
mains, anticipatory
a sequence of specification points is complete if
(a)
each member
(b)
any sentence
Then
Say that
of the sequence extends its predecessor,
and
some
is settled by
member of the sequence.
account
is true on the anticipatory
iff it is true in all
i.e. in all limits of complete sequences.
specifications,
a sentence
generic
can be elim
over generic (complete) specifications
Thus quantification
over partial specifications. The generic
inated in favour of quantification
models figure as ideal points; they do not 'exist', but truth-values can be
lends itself to a nominalistic
calculated as if they did. This reformulation
TRUTH
VAGUENESS,
AND
281
LOGIC
are identified with the corre
partial specifications
of predicates. One requires that any borderline case
our control in the sense that it can be settled by making
the
The
interpretation.
collections
sponding
be under
predicate more precise. But one does not require that any predicate can
be made perfectly precise.
to Completeability
The objection
may really be a question about our
sentence.
itmay be asked, do we grasp all
of
How,
vague
understanding
of those complete and admissible
specifications,
necessary to determine truth-value?
the existence
of which
is
The first is that we under
are, I think, three main possibilities.
stand each of the predicates that make the given predicate perfectly pre
cise. We then grasp the complete and admissible specifications
indirectly,
as those appropriate
to the perfectly precise predicates.
There
Thus
a vague
sentence,
say:
The blob
is red
is like the scheme:
The blob isR
that we are able to
precise predicates
to
account
in understanding
is
this
that
objection
a vague predicate we may not understand all or, indeed, any of the pred
icates that make it perfectly precise.
is that we directly grasp all of the admissible and
The second possibility
where
'R9 stands
enumerate.
complete
in for perfectly
The main
Thus
specifications.
The blob
is like the open
the vague
sentence:
is red
sentence
:
The blob belongs
to R,
where R is a variable
that ranges over complete and admissible extensions
case
there will be restrictions on
'red'.
In
of
of penumbral
connection,
several variables ;and in case 'admissible' is vague, itwill give way to a third
order variable, and so on. But in any case the principle is the same: one
as being sets of a certain sort. The trouble with
grasps the specifications
this account is that 'admissible' contains a hidden quantifier over non
is one that is appropriate
extensional entities. An admissible specification
282
KIT FINE
for some precisification.
For example, an admissible and complete exten
sion for 'red' is one that is determined by a suitable pair of sharp boundary
shades; and a shade is, or corresponds
to, a property as opposed to a set.
Thus the third possibility
is that we grasp all of the perfect precisifi
cations. The sentence :
The blob
is now
is red
like the open-sentence
The blob has R,
where R is a variable
that ranges over all of the properties
that perfectly
'red'. The perfect properties are grasped, not individually, but
precisify
as a whole - in one go. There are, perhaps, two main ways in which this
can be done. First, they may be understood
from below, as the limits of
imperfect properties ;examples are provided by 'chair' and 'game'.
from above, in terms of some more
Second,
they may be understood
an
direct condition;
example is the sliding scale for 'red'.
relevant
Perhaps the main objection to this account is that grasping all properties
of a certain kind requires that one be able, in principle, to find a predicate
for one such property. But I do not see why any but a constructivist
accept this. One can quantify over a domain without being able
to specify an object from it. Surely one can understand what a precise
shade is without being able to specify one?
should
and contrast with am
bring out well the connection
and
are
sentences
biguity. Vague
ambiguous
subject to similar truth
conditions ;a vague sentence is true if true for all complete precisifications
;
an ambiguous
sentence is true if true for all disambiguations.
Indeed,
These
accounts
the only formal difference is that the precisifications may be infinite, even
connection.
is
indefinite, and may be subject to penumbral
Vagueness
on
a
and
ambiguity
grand
systematic scale.
how we grasp the precisifications
re
and disambiguations,
is
different.
is
understood
in
accordance
with
very
spectively,
Ambiguity
are distinguished;
the first account: disambiguations
to assert an ambig
uous sentence is to assert, severally9, each of its disambiguations.
Vague
ness is understood
in accordance with the third account: precisifications
However,
are extended
from a common
to assert a vague
sentence
basis and according
is to assert, generally,
to common
constraints
its precisifications.
Am
:
TRUTH
VAGUENESS,
AND
283
LOGIC
of several pictures, vagueness
like an
biguity is like the super-imposition
unfinished picture, with marginal notes for completion. One can say that
a super-imposed
are;
picture is realistic if each of its disentanglements
and one can say that an unfinished picture is realistic if each of its com
and completions
pletions are. But even if disentanglements
for one, how we see the pictures will be quite different.
4. The
logic
of
match
one
vagueness
This completes our discussion of the truth-conditions
for the language L.
now turn to logic and consider how the preceding analyses affect the
notions of validity and consequence.
We
On the truth-value approach, a formula is valid if it takes a designated
value for every specification.
If True is the sole designated value, then no
formulas are valid on any account
to the stability and
that conforms
fidelity conditions. For they require that any sentence is indefinite if all
are. If, somewhat unaccountably,
of its atomic subsentences
True and
Indefinite are the designated values, then validity is classical on any ac
count
that conforms
to the conditions.
For
if a sentence
is false for a
it is false for any of its complete specifications and so is not
specification,
Thus the truth-value approach
leads either to classical
valid.
classically
or
are
to
no
the
trivial
in
there
valid formulas at all.
which
logic
logic,
Formula B is a consequence of formula A if, for any specification, B
takes a designated value whenever A does. If True is the sole designated
of A on the minimal
account iff B is a
value, then B is a consequence
classical consequence
of A and any predicate (or sentence) letter in B is
account leads to a different consequence-relation
also in A. The maximal
with Ato B v ~B being the characteristic non-consequence.
If True and
Indefinite
account
are the designated values then B is a consequence of A on either
?
A is a consequence
of ? B with True as sole designated
iff
value.
the specification space approach, A is valid if it is true in all spec
ification spaces and B is a consequence
of A if, for any specification
space, B is true whenever A is. This approach gives rise to numerous
On
truth-conditions
lead to a
logics. For example, the bastard intuitionistic
of
extension
On
the
intuitionistic
other hand, the super-truth
slight
logic.
and anticipatory
accounts lead to classical logic. For if a formula is clas
284
KIT
FINE
it is true for all specification
sically valid, i.e. true in all classical models,
spaces, since it is true for each complete
specification within the space;
if a formula is true for all specification
and conversely,
spaces, it is classi
a
case
each
classical
is
since
model
of a specification
degenerate
cally valid,
that the consequence-relation
is
space. A similar argument establishes
classical for the language at hand. Thus the supertruth theory makes a
difference to truth, but not to logic.
that there is no special logic of vagueness?
Can we maintain
Let us
consider two objections against this, one against classical validity and the
other against classical consequence.
The first objection is that the Law of the Excluded Middle
sentence.
vague
man but
For
that Herbert
is a borderline
may fall for
case of a bald
suppose
'Herbert is bald v ?(Herbert
is bald)'
the disjunction
one of the disjuncts is true. But if the second disjunct is true
that
is true. Then
'Herbert is bald' is either true or false,
to
that
is a borderline case of a bald man.
the
Herbert
contrary
supposition
The first is that the clas
The argument here rests on two assumptions.
are correct. From this
for
'or'
truth-conditions
and
'not'
sical necessary
the first is false. So the sentence
it follows
Bivalence.
that the Law of the Excluded
The
Middle
implies the Principle of
cases give rise to
is that borderline
assumption
of Bivalence.
So from
i.e. to breakdowns
truth-values,
second
sentences without
it follows that LEM fails for such sentences.
both assumptions
It would be perverse to deny the force of this argument; both of its
are very reasonable. However,
I think that one can make
assumptions
out that the argument is a fallacy of equivocation.
If truth is super-truth,
to a space, then the necessary
for 'or' and
truth-conditions
'not' fail, though truth-value gaps can exist. If on the other hand, truth
is relative to a complete specification
hold but
then the truth-conditions
gaps cannot exist.
i.e. relative
An analogy with ambiguity may make the equivocation more palatable.
sentence is true if each of its disambiguations
is true. Now
An ambiguous
sentence 'John went to the bank'; let Jx and J2 be
let / be the ambiguous
its disambiguations,
viz. 'John went to the money bank' and 'John went
to the river bank'
; and suppose that John is after fish rather than money.
the disjunction / v ? / is true, for its disambiguations,
Jxv ?Jx
neither disjunct is true, for each dis
and Jx v ? J2 are true. However,
Thus a truth-value gap exists for assert
junct has a false disambiguation.
Then
VAGUENESS,
285
TRUTH AND LOGIC
the classical truth-conditions
hold
ible or unequivocable
truth, whereas
for truth as relative to a given disambiguation.
Mere ambiguity does not impugn LEM. So why should vagueness?
There is, however, a good ontological reason for disputing LEM. Suppose
I press my hand against my eyes and 'see stars'. Then LEM should hold
for the sentence S = 'I see many stars', if it is taken as a vague descrip
LEM should fail for S if it is
tion of a precise experience. However,
a
an
as
taken
intrinsically vague experience. Again
precise description of
set V is taken to be vague, then the sentence 'VeVv
if the universal
? Ve V9
I
is, imagine, not true. More generally, a set is vague if it is not
the case of every object that it either belongs or does not belong to the
set. One cannot but agree with Frege (1952, p. 159) that "the law of the
is really just another form of the requirement
that the
concept should have a sharp boundary".10
The second objection against the classical solution is that it gives rise
to the sorites-type of paradox. Consider
the following
instance, which is
:
said to go back to Eubulides
excluded middle
A man with no hairs on his head
If a man with
is bald
n hairs on his head
is bald
(n + 1) hairs on his head is bald.
.".A man with a million hairs on his head
The
conclusion
from
follows
of modus
the premisses
and universal
then a man
is bald.
the help
instantiation.
with
with
of a million
ponens
runs like this. The first premiss is true. The second
premiss is true: for if not, it is false; but then there is an n such than a
man with n hairs on his head is bald and a man with (n + 1) hairs on his
head is not bald; and so the predicate
'bald' is precise after all. The con
applications
The objection
now
clusion
is false. Therefore
This
contains
argument
truth of the second
vague
sentences.
The
the reasoning, which
is classical,
is at fault.
two non-sequiturs.
The first is that the non
premiss
implies its falsity; Bivalence may fail for
n
is that the existence of the hair-splitting
'bald' is precise. One need no more accept this
is bald or not bald implies that Herbert
is a
second
implies that the predicate
than accept that Herbert
clear case of a bald man.
In fact, on the super-truth view, the second premiss is false. This is
because a hair splitting n exists for any complete and admissable
specif
286
ication
KIT FINE
of
'is bald'.
is true may
I suspect that the temptation
to say that the second
have two causes. The first is that the value of a falsi
premiss
fying n appears to be arbitrary. This arbitrariness has nothing to do with
as such. A similar case, but not involving vagueness,
is : if n
vagueness
a
straws do not break
camel's back, then nor do (n + 1) straws. The second
cause is what one might call truth-value shift. This also lies behind LEM.
Thus A v ? A holds in virtue of a truth that shifts from disjunct to dis
junct for different complete specifications,
just as the sentence 'for some
a
n
a
man
man
hairs is bald but
with
with (n + 1) hairs is not' is true
n,
for an n that shifts for different complete specifications.
It is, perhaps, worth pointing out that no special paradoxes of vague
ness can arise on the super-truth view, at least for a classical language.
For suppose that intuitively false B is a classical consequence of intuitively
true A. Then for some complete and admissible
specification, A is true
and B is false, and this is a classical paradox within a second-order
lan
guage. This paradox can be brought
to correspond
if there are predicates
Thus the two objections
against
cannot
be sustained.
to the level of the original language
to the complete specification.
classical
logic for vague sentences
to deny that LEM is counter-intuitive.
considerations mitigate against it. In particular, an
I do not wish
It is just that external
adequate account of penumbral
logic be classical.
connection
appears
to require
that the
could, of course, still attempt to construct a logic that was more
faithful to unreformed
intuition. However,
such an attempt would soon
run into internal difficulties. One is that our unreformed
intuitions on
One
validity do not enable us to decide between the various ways of avoiding
if LEM goes, then so does A r>A or the standard
LEM. For example,
=>
in terms of v and ?. But which? Again,
if LEM goes,
definition of
?
?
or
of
de
then one of
the
Laws,
(A &
substitutability
Morgan's
A),
must go. Or again, if modus ponens holds but the logic is
A for-A
then either the (-?, v) or (->, ?) fragment is non-classical.
a departure from classi
Another
difficulty is that it is hard to motivate
cal logic. Perhaps the best that can be done is this. One interprets 'A or
B9 as 'clearly A v clearly B9, 'if A then B9 as 'clearly A zdB9, 'A and B9
not classical
as 'A & B99 and 'not A9 as 'clearly ?A9. The standard natural deduction
and conjunction will then hold. For
rules for disjunction,
implication
one
has
ifB is a consequence
still
the
Deduction
Theorem:
of A
example,
TRUTH
VAGUENESS,
AND
LOGIC
287
then 'if A then B9 is valid. Only negation bears the burden of non-classi
in a fairly plausible way between
cality. Also, this account discriminates
'P and not P9 and 'P and
The
and
conjunctions
disjunction.
conjunction
'P or not P9 and 'P or R9 are not true.
R9 are false, while the disjunctions
Shifts on conjuncts are allowed, shifts on disjuncts are not.
such an alternative does not, in any way, create a challenge
However,
have merely been re-interpreted
for classical logic. For the connectives
an extension
logic remains clas
logic. The underlying
reasons
at
for
least three
sical. There are, then,
adopting a classical solu
for which
of a truth-definition
tion. The first is that it is a consequence
within
of classical
there is strong independent evidence. The second is that it can account
for wayward
intuitions in an illuminating manner. And the last is that it
is simple and non-arbitrary.
vagueness
5. Higher-order
is penumbral
connection. Another
distinctive
feature of vagueness
is the possibility of higher-order vagueness. The vague may itself be vague,
or vaguely vague, and so on. For suppose that James has a few fewer
One
hairs on his head then his friend Herbert.
Then he may well be a border
case of a borderline case of
line case of a borderline case or a borderline
a borderline case of a bald man.
This
ator
feature of vagueness
6D9 for 'it is definitely
for 'it is indefinite
IA
=
can be expressed with the help of the oper
'/'
the case that'. Let us define the operator
that' by:
df-
DA&-
D
-A.
is in analogy to the definition of the contingency operator in modal
logic. But note that 'D', unlike the adjective 'definite' or the truth-value
n
This
=
'/', is biased towards the truth. ThenInFa
11...IFaexpresses
a denotes is an ?-th order borderline case of F. For example,
is bald).
for James is expressed by ://(James
the first of the two possibilities
designator
that what
same possibility
can be put in terms of the truth-predicate.
One
nor
true
:
true
sentence
neither
is
false'
is
bald
is
neither
'James
the
says
can
nor false. Thus higher-order
in
the
be
vagueness
material
expressed
The
mode, with the help of the definitely-operator,
the help of the truth-predicate.
or in the formal mode,
with
288
KIT FINE
would appear to belie undue scepticism over the
of higher-order vagueness. For if IFa can be true, then so surely
can IIFa, IIIFa, and so on. Or again, if a can denote a borderline case of
the predicate F, then surely the sentence Fa can be a borderline case of the
The
above notations
existence
'is neither true nor false'. In both instances higher-order
vague
predicate
ness is a species of first-order vagueness : in the first instance, the higher
a statement of indefinite
order consists in the correct application
of/to
and
the
the
consists
in the truth-predicate
in
ness,
second,
higher-order
a
cases.
This makes
sudden discontinuity
borderline
in the
possessing
orders
unreasonable.
appear
can
In any case, artificial examples of higher-order
vague predicates
cases
are
to
which
borderline
be
One might
be constructed.
stipulate
not
not.
if
in
clear and which
natural
all, vague predicates
Indeed, most,
some, such as 'red', have a
vague. Though
language are higher-order
higher concentration
such as 'few', appear
order
of 'lateral' or first-order vagueness, whilst others,
or higher
to have a higher concentration
of'vertical'
vagueness.
can we
characterize higher-order
vagueness? We shall consider
two equivalent forms of this question. The first is :what are the truth
The second is:
conditions
for a language with the definitely-operator?
for a language with a hierarchy of truth
what are the truth-conditions
predicates? To answer the first question, we let L' be the result of en
How
riching the original language L with the operator D, and, to simplify the
answer, we first take care of the case of mere first-order vagueness. We
in turn; though, in view of
consider the truth-value and rival approaches
earlier criticisms, the former consideration
On the truth-value approach, D should
?DA<^?A
ADA <-?not
The
DA
is an act of generosity.
clauses:
satisfy the following
A
1=
for
language will no longer satisfy the stability condition,
is false for A indefinite, but true for A true. Indeed, all three-valued
extended
can be defined in terms of maximal &,?,/)
and a con
stant for the Indefinite, whilst all three-valued
functions
satisfying sta
? and constants
a
can
for the
in
terms
of
defined
maximal
be
,
bility
truth-functions
Indefinite
and the True.11
On the specification
space approach, D can receive the following
clause
:
TRUTH
VAGUENESS,
AND
LOGIC
289
<-
i.e. the admissible
specifi
ws?A9 where ws is the base-point,
In
S.
of
this
the
case, Stability will still
space
specification
cation-point,
the proper form for stability is:
hold for the enriched language. However,
w?DA
w ?A (for the space S) and w < v -
v ? A (for the space R)9
from S by taking v as the base point. Now B may be
at w but true at v. So A = DB will be false at w (in S) but true
where R is obtained
indefinite
at v (in R). Thus internal Stability holds, but external Stability does not.
is that the clause for D ignores any im
The reason for this divergence
in
that
may have taken place. If it were not ig
provement
specification
nored,
the clause would
be:
?->w ? A
<r*not ? w?A
w ?DA
w ADA
to that for the operator
is analogous
'Now' in tense
or
to
reference
clause
for
the
certain
logic (see Prior, 1968)
rigid designa
tors as inKaplan
(unpublished) or Kripke (1972). In all of these cases, the
reference (or truth-value) of an expression at an arbitrary point is given in
The original
clause
of a simpler expression at a privileged point, the
or the present time or the actual world. Ref
specification
terms of the reference
appropriate
erence is, as it were,
frozen at the privileged point.
is that the sentences of V are
aspect of the distinction
not necessarily made more precise through making
their predicates more
that
is
'bald'
first
order
vague. Then the sentence:
only
precise. Suppose
The
intensional
Definitely
Herbert
is bald
'bald' more precise. Indeed,
precise through making
is, in the relevant way, already perfectly precise. More gen
sentence will suffer from H-th order vagueness only
the compound
is not made
more
the sentence
erally,
if its constituent
sentence
The definitely-operator
example, the sentence:
Casanova
suffers from (n + l)-th order vagueness.
is not the only one to behave in this way. For
believes
that he has had many mistresses
may be a precise report of a vague belief or a vague report of a precise
belief. In the latter case, the sentence can be made more precise through
making
'many' more precise; but in the former case, it cannot.
290
KIT
FINE
A compound
sheds, as it were, the n-th order vagueness of its constit
uent and comes under the control of its (n + l)-th order vagueness. The
is formally
similar to that behind Frege's distinction
be
phenomenon
tween direct and indirect reference. Reference
may depend upon indirect
indirect reference upon indirectly indirect reference, and so on.
zero-order vagueness may depend upon first-order vagueness,
vagueness upon second-order vagueness, and so on. If indirect
reference,
Similarly,
first-order
to be sense, and indirect sense to be itself, then the
reference hierarchy has essentially only two terms. The vagueness hier
archy will, of course, have as many terms as there are orders of vagueness.
The logics for D on the different accounts are quite distinctive.
Indeed,
is taken
reference
one might say that the characteristic
is not a
logical feature of vagueness
a
non-classical
notion. For the truth-value
non-classical
ap
logic but
choice of
proach there are six logics in all, one for each independent
or minimal and of {T} or {T91} as the set of designated values.
for all choices, the unacceptable
shall not go into details. However,
v
is valid. On the super-truth view, the
formula D(BvC)^>
(DB
DC)
set of valid formulas is given by the modal
system S5. This is because a
maximal
We
sentence
is true at a complete
at the base
if and only if it is true
specification-point
which holds if and only if the sentence is
specification-point,
true at all of the complete points.
On all of the accounts,
the Deduction
Theorem
does not hold
for the
This again distinguishes
the presence of D from its
consequence-relation.
DA is not valid.
absence. In particular, DA is a consequence
of A but A :=>
For
A
the truth of A guarantees
the truth of DA9 but the indefiniteness of
=>
A
of
DA.
the
Thus in one sense A and DA are equiv
implies
falsity
alent, for to assert A is to assert DA ;while, in another sense, A and DA
are not equivalent, for to assert ? A is not to assert ? DA. With
the ex
with {T,I} as designated values, the
and
consequence
validity is given by: B is a con
relationship
=>
B is valid. In the presence of higher
sequence of A if and only of DA
of
takes the form: B is a consequence
order vagueness,
the relationship
A if and only if the set {? A, B9 DB, DDB,...}
is not satisfiable.
It is worth noting that the truth of DA =>A is not completely
straight
ception
of the truth-value
accounts
between
forward.
For
a sort of penumbral
on the super-truth view,
it involves
of vagueness. Thus
for the predicates
of A must
be a member
connection
between
orders
any complete
specification
of the first-order
space that
helps to determine
the truth-predicate.
now
This point is even clearer for
ismade a clear case of 'true', then
a clear case of P. There is a penum
If the sentence Fa
also be made
'true' and F.
between
bral connection
291
LOGIC
of DA.
the truth-value
of a must
the denotation
AND
TRUTH
VAGUENESS,
how higher-order
vagueness affects the truth
the truth-value approach, we can no longer be
satisfied with the trichotomy True, False and Indefinite. For example,
DA will be true if A is definitely true but indefinite if A is indefinitely true.
We must
consider
for D. On
conditions
Thus D will not be truth-functional
In order to determine
respect to the three truth-values.
of DA we need to know whether
with
the truth-value
A is definitely, indefinitely or definitely-not
So we
under the scope of a D-operator.
need
indefinitely
qualifications
apply definitely,
In general, a truth-value of order n ^ 0 is
degree, i.e. number of arguments, n. Thus
F and / - are the 0-order functions. That
or definitely-not,
and so on.
a 3-valued truth-function/of
the ordinary truth-values
T,
sentence A has 'truth-value' /
means
that
for
y =f(x?9...,
any
xn).
values
ordinary
the
is
0Xi
xl9...,
operator
true. But DA may itself come
to know whether
these
has
xn,
0XnOXn_i...OXxA
to
corresponding
value
xi9i=l,...,n.
Thus Or is D, O i is / and 0F is D ?.
A sentence can contain any finite number of nested D's. So we must
also define an infinite-order value. This may be regarded as an infinite
such that :
sequence f0/1/2...
(a)
P
(b)
/
is an i-th order value
(Xq9 xl9...9
t* Pfor
(b) is a compatibility
has
value
xi9
then/'
any
Xi-i,j
x09
xl9...9
(x09
condition:
must
not
iff1"1
say
that
..., Xi-i))
xl9
x?_l51
=
0,1,
says
^
2,....
that 0Xi_10Xi_2...0XQA
OXiOx._l...OX0A
has
value
F.
are more involved. Suppose second-order
func
truth-conditions
are
to
B
and
C
Then
what
second-order
g
tions/and
assigned
respectively.
ac
function h =/u
g should be assigned to B & C upon the maximal
=
=
T.
count? Let us illustrate the construction of A by putting x0
I and xx
The ordinary truth-value of DI(B & C) equals that for D [IB &(DC v
The
v IC) v IC &(DB v /C)], which equals that forDIB &(DDC v DIC) v
v DIC &(DDB v DIC). So that if/(/, T) = g(T,T) = T and/(P, T) =
=
g (I, T) = /, say, then h(I, T) = /.
This calculation can be made precise
as follows. Given
a function/of
292
KIT
(n + 1) arguments
defined by:
and a O-order
Jz\xi9'"9
xn)
FINE
xl9...9
=/\z->
z, we
truth-value
let/z
be the function
.
xn)
now define operations f,fvg9fngby
g :
/and
= I
n = 0. T=F9F=T9I
= T
=
fvg
Tifforg
=
= F
Fiff=g
=
= T
Tiff=g
fng
=
= F
Fiffoxg
We
induction
on the degree n of
g)T=fTvgT
(fv
(fvg)F=fFngF
(/u 9)i = (fi n (gFu g?)) u (#, n (/F u/7))
(/^0)r=/r^0r
(/^?,)f=/f^^f
?Oj= (// ^(#r u #,)) u (#, n(/T u/,))
(/n
are similar definitions for the minimal
account.
clauses should now go as follows. If infinite-order values/0/1...
then (f? n g0) (f1 n
and g?gi ...are assigned to B and C respectively,
There
The
n ^...is
...to -P,
...to P>P.
assigned to (B & C), f0/1
andf?ff
to impose several further conditions upon what func
It is reasonable
tions
#
can
be
values.
some
Pfor
value
For
one
example,
z, or
that
can
if/(x0,
require
>xn-1
?*n)
that/(;*;0,...,
=:
rthen/(x0,...,
x?_
l9 z) #
xn_
1?
= P for z ?=
xn. In case there ismerely vagueness to order k9 one should
= T for
values that f1(x0,xi9...9
x1^l9z)
require of the infinite-order
z)
xl9...9
z=fl~1(x09
The most
where
x0
=
natural
dl
is the
=
=
jcj
Xi_
*!_!)
and
1>&.
=Potherwise,
choice for the designated value is the sequence d?d1...,
i-th order value such that dl(x09 xl9...9 xi-1) = T if
=
?
Pand
= Potherwise.
However,
I have
out the logics that result from this or other choices.
It would be a bad mistake to fit the values into a discrete
For example,
/* = indefinite
one might
try to work with
to degree k9 k>09
and P=/?
not
worked
linear ordering.
P = true,
the truth-values
= false and declare that
VAGUENESS,
293
TRUTH AND LOGIC
DB had value Ik if B had value Ik+1 and value T(F) if A had T(F). Such
an account would
For suppose that we
ignore important distinctions.
move
our blob
on the border
of pink and red to the pink side of the
sentence
P might be indefinitely true but def
the
colour spectrum.
initely not false, though the above ordering could express no such dis
to treat the values as a
tinction. It would be an even worse mistake
Then
or densely ordered set, say the real closed interval between
as
in Zadeh (1965). More distinctions would go. For example,
1,
no
one could
longer express the fact that Herbert was a clear borderline
case of a bald man.
continuous
0 and
now consider how the rival approach
fares for V under
of higher-order
The
vagueness.
general set-up is extremely
so
us
case
let
consider the special
of the super-truth theory.
complicated,
To simplify further, we identify specification-points
with specifications.
Now
suppose we pick upon an admissible complete specification for the
We
must
conditions
language L. If the language suffers from first-order vagueness, this specif
set of com
ication is not unique and we may pick upon an admissible
If the language also suffers from second-order vague
plete specifications.
ness,
set of
this set is not unique and we may pick upon an admissible
we
such
what
be
obtain
called
+
choices,
(n
1)
might
sets, and so on. After
an
H-th-order
boundary.
Let us be more precise. A zero-th order space is a complete specification
and a (n + l)-th order space is a set of w-th order spaces. A w-th order
is then a sequence s?s1 ...sn such that sl is an z-th order space,
boundary
+
i< n, and sJesJ i,j <n; and a co-order boundary
is an infinite sequence
that
505,1...such
each
s?s1...si
is
an
/-order
boundary,
i =1,2,....
A
is admissible
if each of its terms are and we suppose that the
boundary
an
of
members
admissible
(n + l)-order space are also admissible.
We can now define the truth of L '-sentences relative to an co-order
or boundary
for short. The
boundary,
are standard. The clause for '/>' is :
boundaries
b?D(f)o(V
where
b =
the clause
drawing
the c0ebl9
b0b1
...Re
=
c0cl...
clauses
for the logical
constants
c) (bRcoc?B),
if bieci+l9i
=
09
l,...The
justification
of
is this: D<?> is true at b if (?>is true for all admissible ways of
are
the boundaries;
but the admissible
zero-order boundaries
the admissible
first-order
boundaries
the c0cx such that c0ect
294
KIT FINE
or absolute truth is, in accordance with
and cieb2, and so on. Assertible
the super-truth view, truth in all admissible boundaries.
The above clause has the form of the necessity clause in the standard
the 'accessibility' relation
relational semantics for modal logic. However,
R is not primitive but is determined
from the structure of the boundary
points. This structure is such that R is reflexive; and, in fact, the resulting
system T. Further restrictions on R could be obtained
logic is the modal
by restricting the possible boundary points. For example, given any n ^ 0,
one could require that each boundary b = b0b1 ...tapers after n9 i.e. that
=
> n. This corresponds
to there being at most n-th order
bi+1
{bi} for i
vagueness.
So much
of L'. We must now consider the
a hierarchy of truth-predicates. We
of level 0 be the original language L9 the meta
for the truth-conditions
for a language with
truth-conditions
let the meta-language
M?
of
level n +
language Mn+1
1 be the result of adding the truth-predicate
for Mn to Mn (with appropriate means for referring to the sentences of
M
of infinite level be the union, in an ob
Mn)9 and the meta-language
vious
sense,
of
the
previous
languages
M?9
M1,
M29....
In one way, it is simpler to provide truth-conditions
for M& than for
ismerely another first-order language.
V. For each of the meta-languages
So any account for the original language L should, when properly gen
eralized, lead to an account for each of the meta-languages.
the details for the general case are very complicated. For the
However,
truth predicate for L will be defined in terms of the following predicates,
x extends y; the atomic L-sentence
say: x is an admissible ?-specification;
for M1 will be
is true (false, indefinite) at x. So the truth-predicate
the
in terms of the corresponding
for
language M. But
primitives
us
must
whether
it is true, false
third
tell
in
the
then,
particular,
primitive
A
defined
or indefinite at an Af-specification
or indefinite at an L-specification.
that an atomic L-sentence
The whole
is true, false
process must then be succes
If we imagine that the truth
sively repeated for the other meta-languages.
conditions for L are given in the form of a (labelled) tree, then those for
the
M1 are given by a tree whose nodes are trees that 'grow' throughout
bigger tree, and those for M2 by a tree whose nodes are ordinary trees,
and
so
on.
the details may be much simpler.
for particular approaches
However,
for L is defined solely
the
On the truth-value approach,
truth-predicate
VAGUENESS,
TRUTH
AND
295
LOGIC
the atomic sentence A is true (false, indefinite).
in terms of the primitives:
relative to a unique appropriate
is determined
truth-value
specifi
the
admissible specifications drop out of view. The truth-predicate
cation,
*
is then defined in terms of the primitive: the atomic M1 -sentence
for M
Since
A is true, false, or indefinite. But the atomic M-sentences
the atomic L-sentences
and the sentences of the form:
'A9 is true (false,
will now include
indefinite),
6A9is an atomic L-sentence.
Similarly for the other meta-languages.
for L is defined in terms
the super-truth view, the truth-predicate
6x is a complete and admissible L-specification'.
The
of the primitive:
as
can
to
be
internal
the
of
truth-values
regarded
assignments
specifications
where
On
and
so left out of view. The
terms of the predicate:
But such a specification
is then defined in
for M1
truth-predicate
is a complete and admissible M-specification.
and an assign
will consist of an L-specification
*
to the predicate
6x is a complete and admissible
Similarly for the other meta-languages.
vagueness gives rise to two puzzles, to which it is difficult
of an extension
ment
L-specification'.
Higher-order
to give convincing answers. The first arises from the systematic correla
.This is provided by the equiv
tion between the sentences of L' and M
:
alence
'A9 is true ?- It is definitely
a sentence
the case that A.
into one of M
upon successively
n
an
"A*
is
'DA9
for
level
replacing
by
truen',
appropriate
there should also be a conversion of truth-condi
indicator. Accordingly,
for L'
tions. Since we have already given independent
truth-definitions
For
of L' can be converted
innermost
I cannot
and Mv>9 this conversion
should provide a check on correctness.
a
us
let
that
also
of
observe
there
will
be
conversion
but
details,
give
For example, the conditions,
given for no vagueness
to
L'
the
in
will
conditions which guarantee
(n + 1)
correspond
for Mn+X has no borderline cases.
truth-predicate
conditions.
of order
that the
The puzzle is: should we regard 'DA9 as merely elliptical for "A* is
as a device for
true'? This would be to regard the definitely-operator
into
The device
the
the
incorporating
meta-language
object-language.
would,
strictly
type distinctions;
and
speaking, be improper since it ignores use/mention
if no quantifiable
but it would be harmless
variables
296
KIT
FINE
the scope of '/)'. On the semantic side, it is a matter of
status
spaces or truth-values have an independent
or whether
are
of the ordinary spaces
they
merely fanciful formulations
or values, but for a richer language. An analogous
is whether
question
an
as
or
on
is
best
of
sentences.
operator
regarded
necessity
predicate
occurred
within
whether
the extended
the general advantage of replacing a non-exten
an
It has the general disad
sional operator
extensional
predicate.
'bald'
vantage of involving an incorrect reference to language. Suppose
cases are just those people
has first-order vagueness and the borderline
The
view has
ellipsis
with
with
40 to 60 cranial hairs. Then
with
synonymous
latter sentence
true. The
'It is indefinite
'Herbert has between
is not synonymous
of vague
indefiniteness
with
that Herbert
is bald'
is
40 and 60 cranial hairs', but this
any claim about a sentence being
sentences
is as much
a matter
of precise ones.
the ellipsis view has the particular disadvantage
between first- and second-order
discontinuity
of fact as
the truth or falsehood
Also
sudden
of making
for a
First
vagueness.
of ordinary predicates
is a matter
vagueness
having borderline
cases, but second-order vagueness is a matter of the truth-predicate having
cases. There is, of course, a correlation between the second
borderline
order
order vagueness of ordinary
But we
the truth-predicate.
predicates and the first-order vagueness of
feel that the latter arises from the former,
and not vice versa. The truth-predicate
language; there can be no independent
is supervenient upon
for its having
grounds
the object
borderline
cases.
Indeed, I think that 'D' is a prior notion to 'true' and not conversely.
For let 'truer' be that notion of truth that satisfies the Tarski-equivalence,
even for vague sentences :
'A9 is trueT if and only
if A.
The vagueness of 'truer' waxes and wanes, as it were, with the vagueness
of the given sentence; so that if a denotes a borderline case of F then Fa
is a borderline
case of 'truer\ Then
the ordinary
notion
of truth is given
by the definition:
x is true = dfDefinitely
(x is truer).
'true' is secondary
'truer' is primary;
of
the
definitely-operator.
help
Thus
and to be defined with
the
VAGUENESS,
297
TRUTH AND LOGIC
second puzzle arises from the demand for a perfectly precise meta
of our truth-conditions
that
language. So far, we have only demanded
correct
of
To
allocations
truth.
the
truth-value
gap,
respect
they provide
to yield the correct logic; these
to account for penumbral
connection,
The
one may also
are all special cases of this more general demand. However,
not be vague or, at least, not so vague in
require that the meta-language
Thus it will not do to subject truth
its proper part as the object-language.
to the standard
equivalences:
'A9 is true if and only
For
if A.
then truth will be truthT; the truth-conditions
will be classical; and
of the truth-predicate will exactly match that of the object
the vagueness
language.
What
firm.
we require is that the true/false/indefinite
trichotomy be relatively
'A is true, false or indefinite'
the truth of the disjunction
Ideally,
should imply the truth of one of its disjuncts. It is not that the infirmity
of this trichotomy
in any way impugns the correctness of the previous
accounts.
In particular, validity is still classical on the super-truth view;
for classically valid A is true in all complete and admissible specifications,
it is clear that a particular complete specification
is
regardless of whether
it is that the infirmity raises another problem
for
Rather
admissible.
truth-conditions.
This raises the puzzle: is there a perfectly precise meta-language?
Cer
Mn could be vague. One could take
tainly, each of the meta-languages
the whole construction
into the transfinite and have, for each ordinal a,
a meta-language
Ma or strong definitely-operator
D a. But the same prob
lem would
arise anew. At no point
orders of vagueness.
does
it seem natural
to call a halt
to
the increasing
if a language has a semantics in terms of higher-order bound
However,
For the boundaries will be
aries, then it also has a firm truth-predicate.
based upon a set of admissible
and we can let truth (or
specifications
be truth (or falsehood)
in all such specifications. Anything
falsehood)
case is treated as a clear borderline
that smacks of being a borderline
case. The meta-languages
become precise at some, but no pre-assigned,
to this is that the set of admissible
ordinal level. The only alternative
specifications
is itself
intrinsically
vague.
There would
then be a very
298
KIT FINE
between vague language and reality: what language
be an intrinsically vague fact.
If higher-order
terminates at some stage a then vagueness
vagueness
a
in
be
For each sentence A can be replaced by a
eliminated.
can,
sense,
intimate
meant
connection
would
this method
is
precise sentence DaA that entails it. However,
one
a.
not
to
in
several
be
able
the
may
ways.
First,
unsatisfactory
specify
Second, even if one can, one may not be able to make much sense of D*.
perfectly
seem to run out after the second or third orders of vague
ness. Perhaps this is because our understanding
of vague language is, to
a large extent, confused. One sees blurred boundaries,
not clear bound
to be dis
too
the
is
uniform
aries to boundaries.
method
Finally,
Our
intuitions
connections may be lost: our blob, for example,
or
red
definitely pink. Indeed, the question of making
12 is
of whether
higher-order
perfectly precise
independent
terminates. The predicate
'small', as applied to numbers, may
Penumbral
criminate.
is not definitely
predicates
vagueness
suffer from endless higher-order
vagueness;
yet
it can still be made
per
fectly precise.13
University
of Edinburgh
NOTES
1 I should
topics
on the
Baker
for numerous
conversations
like to thank Gordon
stimulating
not have
this paper. My
form without
his
ideas would
taken their present
in a
I should
for some valuable
also like to thank Michael
Dummett
remarks
of
help.
discussion
2 Kleene's
of
the paper.
to our
and
pp. 329 and 332)
'regular'
correspond
'weak/strong'
the terms
his motivation
for introducing
is
and
'stable',
though
'minimal/maximal'
different
from ours.
3
the minimal
truth-value
ap
(1949) have adopted
(1952, p. 63) and Hallden
Frege
as a third truth-value.
not be happy
in regarding
Indefinite
proach,
though Frege would
K?rner
the maximal
(1962) have espoused
approach.
(1960, p. 166) and ?qvist
4 See the work
It is not clear that one can make much
of Zadeh
(1965) and others.
(1952,
a closed
as
interval for 'multi-dimensional'
of truth within
of degrees
vagueness,
sense can be given
to the
It is even less clear that any semantical
and 'game'.
with
the higher
order vagueness
5.
there is a confusion
of Section
notion.
Possibly
5 Van Fraassen
of the super-truth
notion,
though with
(1968) has already made much
for logic and
different
in mind. He has also drawn out the consequences
applications
sense
in 'chair'
and maximizing
the possibility
of minimizing
radical distinction
of van Fraassen
(1969).)
6 The semantics
for intuitionistic
logic comes from Kripke
considered
can be found in Fitting (1969).
truth-value
(1965).
(the conservative/
The
bastard
account
VAGUENESS,
TRUTH AND LOGIC
299
7 The
have a nice algebraic
The usual
formulation.
theory and its extension
Fregean
states that there is a homomorphism
from the word algebra
into the algebra
of
into the algebra of extensions,
and from the algebra of intensions
and hence
intensions,
a homomorphism
from the word algebra
into the algebra
of extensions.
The extended
theory
states
theory
that the extension
which
ordering,
and
intension
algebras
the homomorphism.
is respected
by
is Cohen's
(1966).
a monotonie
both possess
partial
It follows
that (1) and (2) are im
plied by (3)-(6).
8 The
9 To
argument
assert,
severally,
the disjunction
matter,
the sentences
is false,
tences
is false;
sentencesPi,...,Pjc
of the sentences.
is not
For
the multiple
whereas
the disjunctive
assertion
if each
is true only
assertion
and
to assert
the conjunction
or, for that
the conjunctive
assertion
is false if one of
if each of the sen
assertion
is false only
is true if one of the sentences
is true, whereas
the multiple
have a useful
of the sentences
is true. These
distinctions
to the cluster
For suppose
application
theory of names.
predicates
as the multi
the assertion
the name a. Then
of </>(a) can be regarded
Fi,...,Fic
underly
or disjunctive,
to the conjunctive
assertion
of <?(the Fi-er),...,
ple, as opposed
<?(the
in case some of the predicates
individuate
and others
gap results
Ffc-er). A truth-value
do not.
10
over intrinsically
have been unduly
dismissive
vague entities. This atti
Philosophers
may
tude may derive,
to a mathematical
blurred
outline
am not
even
an
that any piece of empirical
is isomorphic
reality
so is the reality. Thus,
the structure
the
is precise,
to a set of points
in Euclidean
I
space. However,
are precise.
one could develop
entities
that all mathematical
Perhaps
in part,
from
the view
since
structure;
becomes
isomorphic
sure
sets. Hopefully,
of vague
it would
not even be interpretable
within
so that the sceptic could not then treat vague
set theory;
sets on the onion
as a 'fa?on de parler'.
model,
11 For
on functional
to other
see Rescher
references
results
completeness,
(1969).
12 This
is usually
settled upon
constructivist
lines. Our powers
of
covertly
question
are limited ; therefore we cannot
an object has
discrimination
settle whether
perceptual
intuitive
theory
standard
exact shade. But could not a non-constructivist
take 'red', say, to mean
'that' refers to a perfectly
to
uniform
shade? The
that, where
inability
not affect the ability
to mean.
know would
13
in the proofs).
After
the paper,
I discovered
that the super-truth
(Added
writing
:H. Kamp
account
of vague
had also been espoused
authors
languages
by the following
in 'Scalar Adjectives',
to appear
in the Proceedings
of the Cambridge
Linguistics
Confer
in 'General Semantics',
22 (1970),
in The
ence; D. Lewis
Synthese
18-67; M. Przetecki
such
and
the colour
such
of
and Kegan
Theories,
Logic
of Empirical
Routledge
Paul, London,
Williams
1 of his doctoral
in Chapter
thesis. The view seems to go back
The Reach of Science,
Toronto
1956.
Press, Toronto,
University
1969; and P. A.
to H. Mehlberg's
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