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
Stable URL: http://www.jstor.org/stable/20115033 .
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KIT FINE
VAGUENESS, TRUTH AND LOGIC1
This paper began with the question 'What is the correct logic of vague
ness?' This led to the further question 'What are the correct truth-condi
tions for a vague language?', which led, in its turn, to a more general
consideration of meaning and existence. The first half of the paper con
tains the basic material. Section 1 expounds and criticizes one approach
to the problem of truth-conditions. It is based upon an extension of the
standard truth-tables and falls foul of something called penumbral con
nection. Section 2 introduces an alternative framework, within which
penumbral connection can be accommodated. The key idea is to consider
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
if it is true for all ways of making it completely precise. The second half of
the paper deals with consequences, complications and comparisons. Sec
tion 4 considers the consequences that the rival approaches have for logic.
The favoured account leads to a classical logic for vague sentences; and
objections to this unpopular position are met. Section 5 studies the phe
nomenon of higher order vagueness :first, in its bearing upon the truth
conditions for a language that contains a definitely-operator or a hierar
chy of truth-predicates; and second, in its relation to some puzzles con
cerning priority and eliminability.
Some of the topics tie in with technical material. I have tried to keep
this at a minimum. The reader must excuse me if the technical under
current produces an occasional 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
such, 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.
These contrasts can be made very clear with the help of some artificial
Synthese 30 (1975) 265-300. All Rights Reserved
Copyright ? 1975 byD. Reidel Publishing Company, Dordrecht-Holland
266 KIT FINE
examples. Suppose that the meaning of the natural number predicates,
nicel5 nice2, and nice3, is given by the following clauses:
(1) (a) n isnice! ifn> 15
(b) n is not nice! if n < 13
(2) (a) n isnice2 if and only ifn> 15
(b) n is nice2 if and only if n > 14
(3) n is nice3 if and only if n > 15
Clause (1) is reminiscent of Carnap's (1952) meaning postulates. Clauses
(2) (a)-(b) are not intended to be equivalent to a single contradictory
clause; somehow the separate clauses should be insulated from one an
other. Then nice! is vague, its meaning is under-determined; nice2 is
ambiguous, itsmeaning is over-determined; and nice3 is highly general
or un-specific. The sentence 'there are infinitely many nice3 twin primes'
possibly undecidable but certainly not vague or ambiguous.
Any type of expression that is capable of meaning is also capable of
being vague; names, name-operators, predicates, quantifiers, and even
sentence-operators. The clearest, perhaps paradigm, case is the vague
predicate. A further characterization of vagueness will not, I think, be
theory-free; for itwill rest upon an account of meaning. In particular, if
meaning can have an extensional and intensional sense, then so can vague
ness. Extensional vagueness is deficiency of extension, intensional vage
ness deficiency of intension. Moreover, if intension is the possibility of
extension, then intensional vagueness is the possibility of extensional
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
remains intensionally vague in a world of hairy or hairless men. The
distinction is roughly Waismann's (1945) vagueness/open-texture one,
but without the epistemological overtones.
Extensional vagueness is closely allied to the existence of truth-value
gaps. Any (extensionally) vague sentence is neither true nor false; for any
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
uniquely referring name b for which the identity-sentence a = b is neither
true nor false. Some have thought that a vague sentence is both true and
false and that a vague predicate is both true and false of some object.
VAGUENESS, TRUTH AND LOGIC 267
However, this is a part of the general confusion of under- and over
determinacy. A vague sentence can be made more precise; and this opera
tion should preserve truth-value. But a vague sentence can be made to be
either true or false, and therefore the original sentence can be neither.
This battle of gluts and gaps may be innocuous, purely verbal. For
truth on the gap view is simply truth-and-non-falsehood on the glut view
and, similarly, falsehood is simply falsehood-and-non-truth. However, it
is the gap-inducing notion that is important for philosophy. It is the one
that directly ties in with the usual notions of assertion, verification and
consequence. The glut-inducing notion has a split sense; for it allows
truth to rest upon either correspondence with fact or absence of meaning.
Despite the connection, extensional vagueness should not be defined
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
deficiency is that they can be closed by an appropriate linguistic decision,
viz. an extension, not change, in the meaning of the relevant expression.
1. The truth-value approach
It is this possibility of truth-value gaps that raises a problem for truth
conditions. For the classical conditions presuppose Bivalence, the prin
ciple that every sentence be either true or false, and so they are not directly
applicable to vague sentences. In this, as in other, cases of truth-value
gap, it is tempting to treat Neither-true-nor-false, or Indefinite, as a third
truth-value and to model truth-value assessment along the lines of the
classical truth-conditions.
The details of this and subsequent suggestions will first be geared to a
first-order language. Only later do we consider the complications that
arise from extending the language. Let us fix, then, upon an intuitively
understood, but possibly vague, first-order language L. There are three
sources of vagueness inL: the predicates, the names, and the quantifiers.
To simplify the exposition, we shall suppose that only predicates are vague.
Indeed, it could be argued that all vagueness is reducible to predicate
vagueness. For possibly one can replace, without any change in truth
value, each vague name by a corresponding vague predicate and each
quantifier over a vague domain by an appropriately relativised quantifier
over amore inclusive but precise domain. We shall also suppose, though
268 KIT FINE
only to avoid talking of satisfaction, that each object in the domain has
a name.
We now 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
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. The valuation is to be truth-functional in the
sense that the truth-value of each type of compound sentence be a uni
form function of the truth-values of its immediate sub-sentences.
The possible truth-conditions can be subject to two natural constraints.
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
specifications are classical.
The second constraint is that definite truth-values be stable for im
provements in specification. Say that one specification u extends another
tifu assigns to an atomic sentence any definite truth-value assigned by t.
Then the Stability Condition S states that if a sentence has a definite
truth-value under a specification t it enjoys the same definite truth-value
under any specification u that extends t ;definite truth-values are preserved
under extension.
The two constraints work together: definite truth-values for a partial
specification must be retained upon the classical evaluation of any of its
complete extensions. Indeed, if quantifiers are dropped, the two con
straints are equivalent to the classical necessary truth and falsehood con
ditions:
(i) t=-?->=!?
1-B-*?B
(ii) h?&C-?h?andhC
1B&C-* IB or 4C.
VAGUENESS, TRUTH AND LOGIC 269
Similarly for the other truth-functional connectives. (I use
'
1=A9 for 'A is
true', '=}A9 for 'A is false', and '-?' for informal material implication.)
However, the conditions still allow some latitude in the formulation of
truth-conditions. One can move in the direction of minimizing or of maxi
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
tion of Bivalence. To 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,
clause (i) uniquely determines the conditions for negation, for the weak
and strong senses are excluded, and the above alternatives are the only
ones for commutative conjunction.
Is any account along truth-value lines acceptable? Any account that
satisfies the conditions F and S would always appear to make correct
allocations of definite truth-value. However, even the maximizing policy
fails tomake 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
junction P & R is false since the predicates 'is pink' and 'is red' are con
traries. But on themaximizing account the conjunction P & R is indefinite
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 P & R is false.
Thus a conjunction with indefinite conjuncts is sometimes indefinite and
sometimes false and so '&' is not truth-functional with respect to the
three truth-values, True, False and Indefinite.
A similar argument also applies to the other logical connectives. For
example, the disjunction P v P is indefinite since it is equivalent to plain
P, which is indefinite; whereas the disjunction P v jR is true since the
predicates 'is pink' and 'is red' are complementary over the given colour
270 KIT FINE
range. Again, the conditional P id ? P is presumably not true, whereas
P 3 ? R is true. It ismore difficult to find examples for the quantifiers.
But for the universal quantifier, say, we may consider the sentence 'All
pretenders to the throne are the rightful monarch', where the domain of
quantification consists of several pretenders who are all borderline cases
of the predicate 'is a rightful monarch'. The whole sentence is false, yet
its immediate subsentences are indefinite.
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
quires a conjunction with indefinite conjuncts to be indefinite. Such ap
proaches are common and include those that are based upon degrees of
truth4 and those that satisfy a 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
with borderline cases or, more generally, among indefinite sentences.
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
examples require special predicates. Surely P& ? P is false even 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
arise, wholly or in part, from penumbral connection, truths on a penumbra
or penumbral truths. Then our argument is that no natural truth-value
approach respects penumbral truths. In particular, such an approach
cannot distinguish between 'red' and 'pink' as independent and as ex
clusive upon their common penumbra.
Placing the Indefinite on a par with the other truth-values is analogous
to basing modal logics on the three values Necessary, Impossible and
Contingent, or to basing deontic logic on the values Obligatory, For
bidden and Indifferent. For here, too, truth-functionality may be lost:
a conjunction of contingent sentences is sometimes contingent, some
times impossible; a conjunction 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
haps our understanding of sentential operators is, in some sense, finite.
VAGUENESS, TRUTH AND LOGIC 271
but this is not to say that it is based upon a finite substructure of truth
values.
2. AN ALTERNATIVE FRAMEWORK
How can we account for penumbral connection? Consider again the blob
that is on the border of pink and red and suppose that it is also a border
line case of the predicate 'small'. Why do we say that the conjunction
'The blob 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
a clear case of both the predicates 'pink' and 'red' but can be made a
clear case of both the predicates 'pink' and 'small'. In other words, the
difference in truth-value reflects a difference in how the predicates can
be made more precise.
Such a suggestion can be made precise within the following framework.
A (specification) space consists of a non-empty set of elements, the speci
fication-points, and a partial ordering ^ (also read :extends) on the set,
i.e. a reflexive, transitive and antisymmetric relation. A space is appro
priate if each point corresponds to a precisification, one point for each
precisification. We regard the ways of precisifying in a generous light and,
in particular, do not tie them to the expressions of any given language.
The nature of the correspondence is this: each point is assigned a speci
fication that is appropriate to the precisification to which it corresponds;
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
the partial ordering as the natural extension-relation on precisifications.
Then the suggestion is that truth-valuation be based, not upon the ap
propriate specification, but upon an appropriate specification space, i.e.
upon the specification-points that correspond to the different ways of
making the language more precise. The truth-valuation is to be uniform
in the sense that it only makes use of the specification-points at which the
given subsentences are true or false. There may, of course, be several
appropriate spaces, but then their differences should make no difference
to the truth-valuation.
This framework could be generalized in various ways. For example,
with each space could be associated a subset of points. The new space
272 KIT FINE
would be appropriate only if the subset determined a space that was
appropriate in the old sense. This would allow the truth-definition to call
upon specifications that did not correspond to precisifications. However,
such generalizations appear to have little intuitive foundation and will not
be considered further.
The account of appropriacy uses the intensional notion of precisifica
tion. A strictly extensional account could avoid this in various ways.
Perhaps the simplest is to identify the specification-points with the speci
fications themselves. Thus a specification space is, in effect, a collection
of specifications partially ordered by the natural extension-relation. A
space is appropriate if the specifications are the admissible ones. Un
officially, a specification is admissible if it is appropriate for some precisi
fication; officially, the notion of admissibility is primitive.
There are various conditions one can impose upon a specification space.
One is that it has a base-point, the appropriate specification-point. This
corresponds to the precisification of which all other precisifications are
extensions. Another is Completeability. It states that any point can be
extended to a complete point within the same space, i.e.
C.(Vi) (3w^ t) (u complete)
where a point is complete if its specification is complete.
There are also conditions one can impose upon the truth-definition. The
main ones are the appropriate modifications of the earlier fidelity and
stability conditions. Fidelity will state that the truth-values at a complete
point
are classical, i.e.
F.t)rA<r+ttA (classically) for t complete.
Stability will state that truth-values are preserved under extensions of
points within a given space, i.e.
S. t NA and t< u -> u f=A
tz\A and t< u -> u 4A.
As with the truth-value approach, there is the problem of how to tag
truth values to the different specifications. One can tend tominimize or to
maximize the amount of truth and falsehood tagging. Minimizing gives
nothing new. However, maximizing 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
VAGUENESS, TRUTH AND LOGIC 273
if and only if it is true at the appropriate specification-point, i.e. at all
complete and admissible specifications. Truth is super-truth, truth from
above5.
In contrast to the truth-value approach, there are now many interesting
intermediate truth-definitions. The most notable is the bastard intuition
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
quantification is constant, the clauses run like this :
I (i) t? - B <->(Vu^ t) (not-w?E)
(ii) t?B & C++t?B and t?C
(iii) t?Bv C++t?Bort?C
(iv) t?Bz*C*+<yw&t)(u?B-*u?C)
(v) t ? (3 x) B(x) <r+t? B(a) for some name a
(vi) t?(Vjc)B(x) <->(Vu> t) (3 v> u) (v ?B(d)) for each
name a.
There are two common factors in the rival approaches to truth-condi
tions. One is the insistence that the procedures for truth-valuation be
uniform. The other is the insistence that the appropriate form of stability
be satisfied.
These factors can be made explicit within an abstract theory of exten
sions. The standard Fregean theory has a principle of Functionality:
(1) The extension of a compound <?)(A?9..., Ak) is a
function/^
of the extensions of its parts 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 x'u...,xk extend extensions xu...,xk, respec
tively, then/^ applied to *i,...,;t? extends/^ applied to
#!,..., Xk.
This corresponds to the appropriate form of stability.
The most important common factor isMonotonicity. This was not
argued for in the previous section, but it can be given an intensional
foundation. First we must graft a theory of intensions onto the earlier
theory. We shall not be too concerned with the nature of intensions. A
274 KIT FINE
specific model can be obtained by indexing extensions with possible
worlds. (I presume that the specification-points remain constant from
world to world.) However, such amodel would suffer from familiar diffi
culties. For example, if the meaning of A is relevant to the meaning of
A v ?
A, then the vagueness of A should be relevant to the vagueness of
A v ? A. Thus A v ? A should be vague for vague A, though on the
super-truth account, say, it is equally and completely precise for all A.
The pure theory of intensions should contain the analogues of Func
tionality and Intensionality.
(3) The intension of a compound <?)(Ai9..., Ak) is a function F^
of the intensions of its parts Al9...9 Ak;
(4) If intensions X[,..., Xk extend intensions X[,..., Xk, respec
tively, then F^ applied to X[,..., Xk extends F$ applied to
Xl9..., Xk.
The combined theory should link intensions to extensions. Each intension
X determined an extension x; and each extension is so determined. This
makes for two bridge principles:
(5) The intension F#(Xl9...9Xk) determines the extension
J4,(xu'"i xk)l
and
(6) X extends Y if and only if x extends y.
(5) states that a calculated intension determines the correspondingly cal
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
simplicity, take the case of k = 1.Now suppose x extends y. Then X ex
tends Y by (6);F^X) extendsF^Y) by (4); the extension determinedby
F^X) extends the extension determined by F^Y), by (6) again; and so
f^x) extendsf+(y) by (5).
The main assumptions behind this argument are (4) and (6). (4), with
(3), is tantamount to the assumption that an expression is made more
precise through making its simple terms more precise. This assumption
is correct for the language L. For the logical constants are transparent,
VAGUENESS, TRUTH AND LOGIC 275
as itwere, to vagueness; any precisification of a constituent shines through
into the compound. Indeed, the converse of the assumption also holds;
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
ness can be blamed onto constituents as opposed to constructions.
The second assumption says, roughly, that extension does not decrease
with an increase in intension. In particular, a sentence does not become
indefinite upon being made more precise. This is, perhaps, partly defini
tional of 'making more precise'. For what distinguishes this operation
from a mere change inmeaning is that it preserves truth-value. To pre
cisify is to rule out the possibility of certain truth-value gaps. In any case,
itwould be odd if definite truth-value could disappear upon precisifica
tion. Truth could 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
positive concordance of meaning and fact.
What is the rationale for these two assumptions? It lies, I think, in our
desire for an enduring use of language. Under the pressure of their own
use, the meanings of terms will need to change. The terms, in their old
sense, will not be adequate to express the new truths, pose the next ques
tions, make the right distinctions. Now clearly it is convenient that the
changes in meaning be conservative, that the true records before the
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.
These two assumptions tie in well with a dynamic conception of lan
guage. For language need not retain its identity upon arbitrary changes
in meaning; or rather, any such identity is a matter of degree 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 itwere, in the permanence of recorded
truth.
If language is like a tree, then penumbral connection is the seed from
which the tree grows. For it provides an initial repository of truths that
276 KIT FINE
are to be retained throughout all growth. Some of the connections are
internal. They concern the different borderline cases of a given predicate :
ifHerbert is to be bald, then so is the man with fewer hairs on his head.
But many other of the connections are external. They concern the common
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, with the expansion of each part pulling
the other parts into shape.
The two approaches to truth-conditions agree on requirements, but
differ on how 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
approach, the extension of a sentence is a truth-value True,
False or
Indefinite. Each truth-value extends itself; True and False extend Indef
inite; and that is all. The extensions and extending-relation for other
parts of speech can then be determined in a natural manner. It is then
easy to verify that the different accounts on the truth-value approach will
satisfy the principles and that Monotonicity is equivalent to the appro
priate form of stability.
On the specification space approach, there is a slight difficulty in inter
pretation. For the extension of B will extend that of A ifB corresponds
to A at a later stage of precisification. There are various ways of securing
this. One is to regard each expression as an ordered pair (A, t)9where A
is an ordinary expression and t is a specification-point that indicates the
stage of precisification. Another is to imagine that, in precisifying, an
expression is not endowed with a new sense but is succeeded by an ex
pression with that sense. Thus the language expands in an orderly manner
throughout the specification space, t can then be identified from the ex
pression as the first point at which it is introduced. Let us opt for the first
solution. Then the extension of a sentence (A91) is an ordered pair (U, V),
where u =
{U:t < u} and V= {ve U:v?A}. One extension (?/', P)extends
another (U, V) if U' ? U and V is V n U'. It is again then easy to verify
that the principles are satisfied and that Monotonicity is the appropriate
form of stability.
Note that on this approach, the extension is no longer a non-linguistic
VAGUENESS, TRUTH AND LOGIC 277
entity. For to each point is assigned a specification, which is language
based. Moreover, if there are external connections, this linguistic depen
dency cannot be avoided. For example, itwill 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.
On the super-truth account, the definitions can be simplified. The ex
tension of a sentence (A91) will be an ordered pair (U, V), where U =
=
{w.t^u and u is complete} and V= {veU:v?A}. The relation of ex
tension has then a similar definition. The partial specification-points can
be recovered from the complete ones so long as two further conditions
are satisfied. The first is that two points are identical if the complete
specifications assigned to their successors are the same. The second is
that for any non-empty set of complete points there is a point extended by
exactly those points in the set. For then each partial point can be identi
fied with a non-empty set of admissible specifications. The first condition
is harmless and the second can be justified. For the only constraint on
admission into the appropriate space is that a point can verify all the
original penumbral truths. But if they are true at all of the complete
points, they are true at any point extended by a certain subset of the
complete points.
The interpretation for the second approach has an important distin
guishing feature. On both approaches, the extension-relation is used to
formulate a constraint on extensions. But only on the second approach
does this relation enter into the extensions themselves. Thus how an ex
tension can be extended is already part of the extension. The extension of
an expression at a given point uniquely determines its extension at a
subsequent point.
The intensional analogue of this is that how an expression can be made
more precise is already part of itsmeaning. Let the actual meaning of a
simple predicate, say, be what helps determine its instances and counter
instances. Let itspotential meaning consist of the possibilities for making
itmore precise. Then the point is that the meaning of an expression is a
product of both its actual and potential meaning. In understanding a
language one has thereby understood how it can be made more precise;
one has understood, in terms of the earlier dynamic model, the possi
bilities for its growth.
278 KIT FINE
This difference in extension (or intension) implies a corresponding dif
ference in the notion of making more precise. On the first approach, to
extend is to resolve new cases or tomake new connections. For to exclude
subsequent specifications is also to extend. For example, suppose that
there are no penumbral connections between 'red' and 'scarlet'. Then to
require that all scarlet objects be red is to extend on the second ap
proach, but not on the first.
3. The super-truth theory
In this section we shall argue for the super-truth theory, that a vague
sentence is true if it is true for all admissible and complete specifications.
An intensional version of the theory is that a sentence is true if it is true
for all ways of making it completely precise (or, more generally, that an
expression has a given Fregean reference if it has that reference for all
ways of making it completely precise). As such, it is a sort of principle
of non-pedantry :truth is secured if it does not turn upon what one means.
Absence of meaning makes for absence of truth-value only if presence of
meaning could make for diversity of truth-value.
The theory is a partial vindication of the classical position. For the
truth-conditions are, if not classical, then classical at a remove. There is
but one rule linking truth to classical truth, viz. that truth is truth in each
of a set of interpretations. This rule is of general application and not
dependent upon the nature of language or interpretation. The actual
work is done by the clauses for truth in a single interpretation, and these
are classical.
The 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
false in any complete and admissible specification. For the bastard intu
itionistic account, on the other hand, P & R is false but Pv Ris indefinite.
Indeed, one can argue that the super-truth view is the only one to ac
commodate all penumbral truths. For consider the following clauses:
A(i) not-/l=^ -*(3u^t)(u4A)
not-/ =1A ->
(3 u ^ t) (u ? A) ,
for A atomic
vagueness, truth and logic 279
(ii) t?-B++tiB
tl-B*^t?B
(iii) t?B & C*-*t?B and t?C
t4B&C<-+ (Vu^t) (3 v^u)(vAB or vjC)
(iv) /1=(Vx) B(x)
+-> t ?B(a) for any name a
H (Vx) B(x) <-?(Vu^ t) (3 v^ u) (v =1B(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
provement in precision. The necessary truth- and falsehood conditions
are to the effect that all truth-functional pledges are to be redeemed. For
example, clause (iii) for & requires that whenever B 8c C is false it is
possible to point to a subsequent specification-point at which either B
or C is false.
All of these clauses are reasonable with the possible exception of the
sufficient falsehood conditions for & and V. But these clauses are required
to account 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
the ancillary conditions F, S and C, the A clauses are equivalent to the
super-truth account. Thus the claims of penumbral connection force one
to adopt our favoured view.
The second reason for preferring the super-truth view is that it follows
an optimizing strategy: maximize one's advantage within the given con
straints. The theory maximizes the extent of truth and falsehood subject
to the constraints F, S and C. The argument can be 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
indefinite sentences lies in the possibility of this bipolar resolution: they
are born, as itwere, to be true or false. There is no point inwithholding
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
the four conditions F, C, S and R. Thus placing the right value on indef
initeness 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
280 KIT FINE
tionale for the theory in question: if you want the conditions then you
must accept the theory. All too often, truth-conditions for different lan
guages have been constructed with insufficient regard for rationale. Their
basis has often been a scanty set of intuitions. Thus a great advantage of
the present approach is its possession of a uniquely determining rationale.
One might object to the previous arguments on the grounds that they
presuppose Completeability, which is unreasonable. However, there is
a perfectly a priori argument for this condition. Suppose that the 'limit'
of a chain of admissible specifications is also admissible. This is a slight
restriction on penumbral connection: for example, it excludes the re
quirement that the specifications be finite (in an obvious sense) or that
they verify decidable theories. Then by Zorn's Lemma, any admissible
specification can be extended to a maximally admissible specification.
Now suppose that each atomic sentence can always be settled in at least
one of two ways, i.e. that no atomic is ever always indefinite. This is a
very weak form of Resolution. Then it follows that the maximally ad
missible specification is complete.
Even without Completeability, our arguments will still go through. In
place of the super-truth theory we use an anticipatory account that makes
a sentence A true if?A is true on the (bastard) intuitionistic account,
i.e. if A is always going to be intuitionistically true. In effect, we mould
intuitionism to the Resolution Condition: a sentence whose truth can
always be anticipated is already true. This account is the maximal one
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),
Redemption of truth-functional pledges. Moreover, for countable do
mains, anticipatory truth turns out to be a form of super-truth.8 Say that
a sequence of specification points is complete if
(a) each member of the sequence extends its predecessor, and
(b) any sentence is settled by some member of the sequence.
Then a sentence is true on the anticipatory account iff it is true in all
generic specifications, i.e. in all limits of complete sequences.
Thus quantification over generic (complete) specifications can be elim
inated in favour of quantification over partial specifications. The generic
models figure as ideal points; they do not 'exist', but truth-values can be
calculated as if they did. This reformulation lends itself to a nominalistic
VAGUENESS, TRUTH AND LOGIC 281
interpretation. The partial specifications are identified with the corre
sponding collections of predicates. One requires that any borderline case
be under our control in the sense that it can be settled by making the
predicate more precise. But one does not require that any predicate can
be made perfectly precise.
The objection to Completeability may really be a question about our
understanding of vague sentence. How, itmay be asked, do we grasp all
of those complete and admissible specifications, the existence of which is
necessary to determine truth-value?
There are, I think, three main possibilities. The first is that we under
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.
Thus a vague sentence, say:
The blob is red
is like the scheme:
The blob isR
where 'R9 stands in for perfectly precise predicates that we are able to
enumerate. The main objection to this account is that in understanding
a vague predicate we may not understand all or, indeed, any of the pred
icates that make it perfectly precise.
The second possibility is that we directly grasp all of the admissible and
complete specifications. Thus the vague sentence:
The blob is red
is like the open sentence :
The blob belongs to R,
where R is a variable that ranges over complete and admissible extensions
of 'red'. In case of penumbral connection, there will be restrictions on
several variables ;and in case 'admissible' isvague, itwill give way to a third
order variable, and so on. But in any case the principle is the same: one
grasps the specifications as being sets of a certain sort. The trouble with
this account is that 'admissible' contains a hidden quantifier over non
extensional entities. An admissible specification is one that is appropriate
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 red
is now like the open-sentence
The blob has R,
where R is a variable that ranges over all of the properties that perfectly
precisify 'red'. The perfect properties are grasped, not individually, but
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
relevant imperfect properties ;examples are provided by 'chair' and 'game'.
Second, they may be understood from above, in terms of some more
direct condition; an example is the sliding scale for 'red'.
Perhaps themain 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
should 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 iswithout being able to specify one?
These accounts bring out well the connection and contrast with am
biguity. Vague and ambiguous sentences are 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,
the only formal difference is that the precisifications may be infinite, even
indefinite, and may be subject to penumbral connection. Vagueness is
ambiguity on a grand and systematic scale.
However, how we grasp the precisifications and disambiguations, re
spectively, is very different. Ambiguity is understood in accordance with
the first account: disambiguations are distinguished; 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
are extended from a common basis and according to common constraints :
to assert a vague sentence is to assert, generally, its precisifications. Am
VAGUENESS, TRUTH AND LOGIC 283
biguity is like the super-imposition of several pictures, vagueness like an
unfinished picture, with marginal notes for completion. One can say that
a super-imposed picture is realistic if each of its disentanglements are;
and one can say that an unfinished picture is realistic if each of its com
pletions are. But even if disentanglements and completions match one
for one, how we see the pictures will be quite different.
4. The logic of vagueness
This completes our discussion of the truth-conditions for the language L.
We now turn to logic and consider how the preceding analyses affect the
notions of validity and consequence.
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 that conforms to the stability and
fidelity conditions. For they require that any sentence is indefinite if all
of its atomic subsentences are. If, somewhat unaccountably, 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
specification, it is false for any of its complete specifications and so is not
classically valid. Thus the truth-value approach leads either to classical
logic or to the trivial logic, in which there are no valid formulas at all.
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
value, then B is a consequence of A on the minimal account iff B is a
classical consequence of A and any predicate (or sentence) letter in B is
also in A. The maximal account leads to a different consequence-relation
with Ato B v ~B being the characteristic non-consequence. If True and
Indefinite are the designated values then B is a consequence of A on either
account iff
?
A is a consequence of ? B with True as sole designated
value.
On 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
logics. For example, the bastard intuitionistic truth-conditions lead to a
slight extension of intuitionistic logic. On the other hand, the super-truth
and anticipatory accounts lead to classical logic. For if a formula is clas
284 KIT FINE
sically valid, i.e. true in all classical models, it is true for all specification
spaces, since it is true for each complete specification within the space;
and conversely, if a formula is true for all specification spaces, it is classi
cally valid, since each classical model is a degenerate case of a specification
space. A similar argument establishes that the consequence-relation is
classical for the language at hand. Thus the supertruth theory makes a
difference to truth, but not to logic.
Can we maintain that there is no special logic of vagueness? 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 may fall for
vague sentence. For suppose that Herbert is a borderline case of a bald
man but that the disjunction 'Herbert is bald v ?(Herbert is bald)'
is true. Then one of the disjuncts is true. But if the second disjunct is true
the first is false. So the sentence 'Herbert is bald' is either true or false,
contrary to the supposition that Herbert is a borderline case of a bald man.
The argument here rests on two assumptions. The first is that the clas
sical necessary truth-conditions for 'or' and 'not' are correct. From this
it follows that the Law of the Excluded Middle implies the Principle of
Bivalence. The second assumption is that borderline cases give rise to
sentences without truth-values, i.e. to breakdowns of Bivalence. So from
both assumptions it follows that LEM fails for such sentences.
It would be perverse to deny the force of this argument; both of its
assumptions are very reasonable. However, I think that one can make
out that the argument is a fallacy of equivocation. If truth is super-truth,
i.e. relative to a space, then the necessary truth-conditions for 'or' and
'not' fail, though truth-value gaps can exist. If on the other hand, truth
is relative to a complete specification then the truth-conditions hold but
gaps cannot exist.
An analogy with ambiguity may make the equivocation more palatable.
An ambiguous sentence is true if each of its disambiguations is true. Now
let / be the ambiguous sentence 'John went to the bank'; let Jx and J2 be
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.
Then the disjunction / v ? / is true, for its disambiguations, Jxv ?Jx
and Jx v ?
J2 are true. However, neither disjunct is true, for each dis
junct has a false disambiguation. Thus a truth-value gap exists for assert
VAGUENESS, TRUTH AND LOGIC 285
ible or unequivocable truth, whereas the classical truth-conditions hold
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
tion of a precise experience. However, LEM should fail for S if it is
taken as a precise description of an intrinsically vague experience. Again
if the universal set V is taken to be vague, then the sentence 'VeVv
? Ve V9 is, I 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
excluded middle 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 :
A man with no hairs on his head is bald
If a man with n hairs on his head is bald then a man with
(n+ 1) hairs on his head is bald.
.".A man with a million hairs on his head is bald.
The conclusion follows from the premisses with the help of a million
applications of modus ponens and universal instantiation.
The objection now 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
clusion is false. Therefore the reasoning, which is classical, is at fault.
This argument contains two non-sequiturs. The first is that the non
truth of the second premiss implies its falsity; Bivalence may fail for
vague sentences. The second is that the existence of the hair-splitting n
implies that the predicate 'bald' is precise. One need no more accept this
than accept that Herbert is bald or not bald implies that Herbert is a
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 KIT FINE
ication of 'is bald'. I suspect that the temptation to say that the second
premiss is true may have two causes. The first is that the value of a falsi
fying n appears to be arbitrary. This arbitrariness has nothing to do with
vagueness as such. A similar case, but not involving vagueness, is : if n
straws do not break a camel's back, then nor do (n+ 1) straws. The second
cause iswhat 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
n, a man with n hairs is bald but a man with (n+ 1) hairs is not' is true
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 the level of the original language
if there are predicates to correspond to the complete specification.
Thus the two objections against classical logic for vague sentences
cannot be sustained. I do not wish to deny that LEM is counter-intuitive.
It is just that external considerations mitigate against it. In particular, an
adequate account of penumbral connection appears to require that the
logic be classical.
One 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
validity do not enable us to decide between the various ways of avoiding
LEM. For example, if LEM goes, then so does A r>A or the standard
definition of => in terms of v and ?. But which? Again, if LEM goes,
then one of
?
(A&
?
A), de Morgan's Laws, or the substitutability of
A for-A must go. Or again, ifmodus ponens holds but the logic is
not classical then either the (-?, v) or (->, ?) fragment is non-classical.
Another difficulty is that it is hard to motivate a departure from classi
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
as 'A& B99 and 'not A9 as 'clearly ?A9. The standard natural deduction
rules for disjunction, implication and conjunction will then hold. For
example, one still has the Deduction Theorem: ifB is a consequence of A
VAGUENESS, TRUTH AND LOGIC 287
then 'if A then B9 is valid. Only negation bears the burden of non-classi
cality. Also, this account discriminates in a fairly plausible way between
conjunction and disjunction. The conjunctions 'P and not P9 and 'P and
R9 are false, while the disjunctions 'P or not P9 and 'P or R9 are not true.
Shifts on conjuncts are allowed, shifts on disjuncts are not.
However, such an alternative does not, in any way, create a challenge
for classical logic. For the connectives have merely been re-interpreted
within an extension of classical logic. The underlying logic remains clas
sical. There are, then, at least three reasons for adopting a classical solu
tion. The first is that it is a consequence of a truth-definition for which
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.
5. Higher-order vagueness
One distinctive feature of vagueness is penumbral connection. Another
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
hairs on his head then his friend Herbert. Then he may well be a border
line case of a borderline case or a borderline case of a borderline case of
a borderline case of a bald man.
This feature of vagueness can be expressed with the help of the oper
ator 6D9for 'it is definitely the case that'. Let us define the operator '/'
for 'it is indefinite that' by:
IA =
df- DA&- D -A.
This is in analogy to the definition of the contingency operator inmodal
logic. But note that 'D', unlike the adjective 'definite' or the truth-value
n
designator '/', is biased towards the truth. ThenInFa =
11...IFaexpresses
that what a denotes is an ?-th order borderline case of F. For example,
the first of the two possibilities for James is expressed by ://(James isbald).
The same possibility can be put in terms of the truth-predicate. One
says : the sentence 'James is bald is neither true nor false' is neither true
nor false. Thus higher-order vagueness can be expressed in the material
mode, with the help of the definitely-operator, or in the formal mode, with
the help of the truth-predicate.
288 KIT FINE
The above notations would appear to belie undue scepticism over the
existence 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
predicate 'is neither true nor false'. In both instances higher-order vague
ness is a species of first-order vagueness : in the first instance, the higher
order consists in the correct application of/to a statement of indefinite
ness, and in the second, the higher-order consists in the truth-predicate
possessing borderline cases. This makes a sudden discontinuity in the
orders appear unreasonable.
In any case, artificial examples of higher-order vague predicates can
be constructed. One might stipulate which borderline cases are to be
clear and which not. Indeed, most, if not all, vague predicates in natural
language are higher-order vague. Though some, such as 'red', have a
higher concentration of 'lateral' or first-order vagueness, whilst others,
such as 'few', appear to have a higher concentration of'vertical' or higher
order vagueness.
How can we characterize higher-order vagueness? We shall consider
two equivalent forms of this question. The first is :what are the truth
conditions for a language with the definitely-operator? The second is:
what are the truth-conditions for a language with a hierarchy of truth
predicates? To answer the first question, we let L' be the result of en
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
consider the truth-value and rival approaches in turn; though, in view of
earlier criticisms, the former consideration is an act of generosity.
On the truth-value approach, D should satisfy the following clauses:
?DA<^?A
ADA <-?not 1=A
The extended language will no longer satisfy the stability condition, for
DA is false for A indefinite, but true for A true. Indeed, all three-valued
truth-functions can be defined in terms of maximal &,?,/) and a con
stant for the Indefinite, whilst all three-valued functions satisfying sta
bility can be defined in terms of maximal a ,
? and constants for the
Indefinite and the True.11
On the specification space approach, D can receive the following clause :
VAGUENESS, TRUTH AND LOGIC 289
w?DA <ws?A9
where ws is the base-point, i.e. the admissible specifi
cation-point, of the specification space S. In this case, Stability will still
hold for the enriched language. However, the proper form for stability is:
w ?A (for the space S) and w < v - v ?A (for the space R)9
where R is obtained from S by taking v as the base point. Now B may be
indefinite at w but true at v. So A = DB will be false at w (in S) but true
at v (in R). Thus internal Stability holds, but external Stability does not.
The reason for this divergence is that the clause for D ignores any im
provement in specification that may have taken place. If it were not ig
nored, the clause would be:
w ?DA ?->w ?A
w ADA <r*not ? w?A
The original clause is analogous to that for the operator 'Now' in tense
logic (see Prior, 1968) or to the reference clause for certain 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
terms of the reference of a simpler expression at a privileged point, the
appropriate specification or the present time or the actual world. Ref
erence is, as itwere, frozen at the privileged point.
The intensional aspect of the distinction is that the sentences of V are
not necessarily made more precise through making their predicates more
precise. Suppose that 'bald' is only first order vague. Then the sentence:
Definitely Herbert is bald
is not made more precise through making 'bald' more precise. Indeed,
the sentence is, in the relevant way, already perfectly precise. More gen
erally, the compound sentence will suffer from H-th order vagueness only
if its constituent sentence suffers from (n+ l)-th order vagueness.
The definitely-operator is not the only one to behave in this way. For
example, the sentence:
Casanova 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 itwere, the n-th order vagueness of its constit
uent and comes under the control of its (n+ l)-th order vagueness. The
phenomenon is formally similar to that behind Frege's distinction be
tween direct and indirect reference. Reference may depend upon indirect
reference, indirect reference upon indirectly indirect reference, and so on.
Similarly, zero-order vagueness may depend upon first-order vagueness,
first-order vagueness upon second-order vagueness, and so on. If indirect
reference is taken 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,
one might say that the characteristic logical feature of vagueness is not a
non-classical logic but a non-classical notion. For the truth-value ap
proach there are six logics in all, one for each independent choice of
maximal or minimal and of {T}
or
{T91} as the set of designated values.
We shall not go into details. However, for all choices, the unacceptable
formula D(BvC)^> (DB v DC) is valid. On the super-truth view, the
set of valid formulas is given by the modal system S5. This is because a
sentence is true at a complete specification-point if and only if it is true
at the base specification-point, which holds if and only if the sentence is
true at all of the complete points.
On all of the accounts, the Deduction Theorem does not hold for the
consequence-relation. This again distinguishes the presence of D from its
absence. In particular, DA is a consequence of A but A :=>DA is not valid.
For the truth of A guarantees the truth of DA9 but the indefiniteness of
A implies the falsity of A =>DA. Thus in one sense A and DA are equiv
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
ception of the truth-value accounts with {T,I} as designated values, the
relationship between consequence and validity is given by: B is a con
sequence of A if and only of DA =>B is valid. In the presence of higher
order vagueness, the relationship takes the form: B is a consequence of
A if and only if the set {? A, B9 DB, DDB,...} is not satisfiable.
It isworth noting that the truth of DA =>A is not completely straight
forward. For it involves a sort of penumbral connection between orders
of vagueness. Thus on the super-truth view, any complete specification
for the predicates of A must be a member of the first-order space that
VAGUENESS, TRUTH AND LOGIC 291
helps to determine the truth-value of DA. This point is even clearer for
the truth-predicate. If the sentence Fa ismade a clear case of 'true', then
the denotation of a must also be made a clear case of P. There is a penum
bral connection between 'true' and F.
We must now consider how higher-order vagueness affects the truth
conditions for D. On the truth-value approach, we can no longer be
satisfied with the trichotomy True, False and Indefinite. For example,
DA will be true ifA is definitely true but indefinite ifA is indefinitely true.
Thus D will not be truth-functional with respect to the three truth-values.
In order to determine the truth-value of DA we need to know whether
A is definitely, indefinitely or definitely-not true. But DA may itself come
under the scope of a D-operator. So we need to know whether these
qualifications apply definitely, indefinitely or definitely-not, and so on.
In general, a truth-value of order n ^ 0 is a 3-valued truth-function/of
degree, i.e. number of arguments, n. Thus the ordinary truth-values -
T,
F and / - are the 0-order functions. That sentence A has 'truth-value' /
means that for any ordinary values xl9..., xn, 0XnOXn_i...OXxA
has value
y =f(x?9..., xn). 0Xi
is the operator corresponding to xi9i=l,...,n.
Thus Or isD, O i is / and 0F isD ?.
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
sequence f0/1/2... such that :
(a) P is an i-th order value
(b) / (Xq9 xl9...9 Xi-i,j (x09 xl9 ..., Xi-i)) ^
t* Pfor any x09 xl9...9 x?_l51
=
0,1, 2,....
(b) is a compatibility condition: iff1"1 says that 0Xi_10Xi_2...0XQA
has value xi9 then/' must not say that
OXiOx._l...OX0A
has value F.
The truth-conditions are more involved. Suppose second-order func
tions/and g are assigned to B and C respectively. Then what second-order
function h =/u g should be assigned to B & C upon the maximal ac
count? Let us illustrate the construction of Aby putting x0
= I and xx
= T.
The ordinary truth-value of DI(B & C) equals that for D [IB &(DC v
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 FINE
(n+ 1) arguments and a O-order truth-value z, we let/z be the function
defined by:
Jz\xi9'"9 xn) =/\z-> xl9...9 xn)
.
We now define operations f,fvg9fngby induction on the degree n of
/and g :
n = 0. T=F9F=T9I
= I
fvg
=
Tifforg
= T
=
Fiff=g
= F
fng
=
Tiff=g
= T
=
Fiffoxg
= F
(fv g)T=fTvgT
(fvg)F=fFngF
(/u 9)i
=
(fi n (gFu g?))u (#, n (/F u/7))
(/^0)r=/r^0r
(/^?,)f=/f^^f
(/n ?Oj=
(// ^(#r u #,)) u (#, n(/T u/,))
There are similar definitions for the minimal account.
The clauses should now go as follows. If infinite-order values/0/1...
and g?gi ...are assigned to B and C respectively, then (f? n g0) (f1 n
n^...is assigned to (B & C), f0/1 ...to -P, andf?ff ...to P>P.
It is reasonable to impose several further conditions upon what func
tions can be values. For example,
one can
require that/(;*;0,..., x?_ l9 z) #
# Pfor some value z, or that if/(x0, >xn-1 ?*n)
=:
rthen/(x0,..., xn_ 1?
z)
= P for z ?=xn. In case there ismerely vagueness to order k9 one should
require of the infinite-order values that f1(x0,xi9...9 x1^l9z)
= T for
z=fl~1(x09 xl9...9 *!_!) and =Potherwise, 1>&.
The most natural choice for the designated value is the sequence d?d1...,
where dl is the i-th order value such that dl(x09 xl9...9 xi-1)
= T if
x0
=
jcj
= =
Xi_ ?
= Pand = Potherwise. However, I have not worked
out the logics that result from this or other choices.
Itwould be a bad mistake to fit the values into a discrete linear ordering.
For example, one might try to work with the truth-values P = true,
/* = indefinite to degree k9 k>09 and P=/? = false and declare that
VAGUENESS, TRUTH AND LOGIC 293
DB had value Ik ifB had value Ik+1 and value T(F) ifA had T(F). Such
an account would ignore important distinctions. For suppose that we
move our blob on the border of pink and red to the pink side of the
colour spectrum. Then the sentence P might be indefinitely true but def
initely not false, though the above ordering could express no such dis
tinction. It would be an even worse mistake to treat the values as a
continuous or densely ordered set, say the real closed interval between
0 and 1, as in Zadeh (1965). More distinctions would go. For example,
one could no longer express the fact that Herbert was a clear borderline
case of a bald man.
We must now consider how the rival approach fares for V under
conditions of higher-order vagueness. The general set-up is extremely
complicated, so let us consider the special case of the super-truth theory.
To simplify further, we identify specification-points with specifications.
Now suppose we pick upon an admissible complete specification for the
language L. If the language suffers from first-order vagueness, this specif
ication is not unique and we may pick upon an admissible set of com
plete specifications. If the language also suffers from second-order vague
ness, this set is not unique and we may pick upon an admissible set of
sets, and so on. After (n+ 1) such choices, we obtain what might be called
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
boundary is then a sequence s?s1 ...sn such that sl is an z-th order space,
i< n, and sJesJ
+
i,j <n; and a co-order boundary is an infinite sequence
505,1...such that each s?s1...si is an /-order boundary, i=1,2,.... A
boundary is admissible if each of its terms are and we suppose that the
members of an admissible (n+ l)-order space are also admissible.
We can now define the truth of L '-sentences relative to an co-order
boundary, or boundary for short. The clauses for the logical constants
are standard. The clause for '/>' is :
b?D(f)o(V boundaries c) (bRcoc?B),
where b =
b0b1 ...Re =
c0cl... if bieci+l9i
=
09 l,...The justification of
the clause is this: D<?> is true at b if (?>is true for all admissible ways of
drawing the boundaries; but the admissible zero-order boundaries are
the c0ebl9 the admissible first-order boundaries the c0cx such that c0ect
294 KIT FINE
and cieb2, and so on. Assertible or absolute truth is, in accordance with
the super-truth view, truth in all admissible boundaries.
The above clause has the form of the necessity clause in the standard
relational semantics for modal logic. However, the 'accessibility' relation
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
logic is the modal system T. Further restrictions on R could be obtained
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
bi+1
=
{bi} for i> n. This corresponds to there being at most n-th order
vagueness.
So much for the truth-conditions of L'. We must now consider the
truth-conditions for a language with a hierarchy of truth-predicates. We
let the meta-language M? of level 0 be the original language L9 the meta
language Mn+1 of level n + 1 be the result of adding the truth-predicate
forMn toMn (with appropriate means for referring to the sentences of
Mn)9 and the meta-language M of infinite level be the union, in an ob
vious sense, of the previous languages M?9 M1, M29....
In one way, it is simpler to provide truth-conditions forM& than for
V. For each of themeta-languages ismerely another first-order language.
So any account for the original language L should, when properly gen
eralized, lead to an account for each of the meta-languages.
However, the details for the general case are very complicated. For the
truth predicate for L will be defined in terms of the following predicates,
say: x is an admissible ?-specification; x extends y; the atomic L-sentence
A is true (false, indefinite) at x. So the truth-predicate for M1 will be
defined in terms of the corresponding primitives for the language M. But
then, in particular, the third primitive must tell us whether it is true, false
or indefinite at an Af-specification that an atomic L-sentence is true, false
or indefinite at an L-specification. The whole process must then be succes
sively repeated for the other meta-languages. Ifwe imagine that the truth
conditions for L are given in the form of a (labelled) tree, then those for
M1 are given by a tree whose nodes are trees that 'grow' throughout the
bigger tree, and those forM2 by a tree whose nodes are ordinary trees,
and so on.
However, for particular approaches the details may be much simpler.
On the truth-value approach, the truth-predicate for L is defined solely
VAGUENESS, TRUTH AND LOGIC 295
in terms of the primitives: the atomic sentence A is true (false, indefinite).
Since truth-value is determined relative to a unique appropriate specifi
cation, the admissible specifications drop out of view. The truth-predicate
forM
*
is then defined in terms of the primitive: the atomic M1 -sentence
A is true, false, or indefinite. But the atomic M-sentences will now include
the atomic L-sentences and the sentences of the form:
'A9 is true (false, indefinite),
where 6A9is an atomic L-sentence. Similarly for the other meta-languages.
On the super-truth view, the truth-predicate for L is defined in terms
of the primitive: 6x is a complete and admissible L-specification'. The
assignments of truth-values can be regarded as internal to the specifications
and so left out of view. The truth-predicate for M1 is then defined in
terms of the predicate: * is a complete and admissible M-specification.
But such a specification will consist of an L-specification and an assign
ment of an extension to the predicate 6x is a complete and admissible
L-specification'. Similarly for the other meta-languages.
Higher-order vagueness gives rise to two puzzles, to which it is difficult
to give convincing answers. The first arises from the systematic correla
tion between the sentences of L' and M .This is provided by the equiv
alence :
'A9 is true ?- It is definitely the case that A.
For a sentence of L' can be converted into one ofM upon successively
replacing innermost 'DA9 by "A* is truen', for n an appropriate level
indicator. Accordingly, there should also be a conversion of truth-condi
tions. Since we have already given independent truth-definitions for L'
and Mv>9 this conversion should provide a check on correctness. I cannot
give details, but let us observe that there will also be a conversion of
conditions. For example, the conditions, given for no vagueness of order
(n+ 1) in L' will correspond to the conditions which guarantee that the
truth-predicate forMn+X has no borderline cases.
The puzzle is: should we regard 'DA9 as merely elliptical for "A* is
true'? This would be to regard the definitely-operator as a device for
incorporating the meta-language into the object-language. The device
would, strictly speaking, be improper since it ignores use/mention and
type distinctions; but it would be harmless if no quantifiable variables
296 KIT FINE
occurred within the scope of '/)'. On the semantic side, it is a matter of
whether the extended spaces or truth-values have an independent status
or whether they are merely fanciful formulations of the ordinary spaces
or values, but for a richer language. An analogous question is whether
necessity is best regarded as an operator on or predicate of sentences.
The ellipsis view has the general advantage of replacing a non-exten
sional operator with an extensional predicate. It has the general disad
vantage of involving an incorrect reference to language. Suppose 'bald'
has first-order vagueness and the borderline cases are just those people
with 40 to 60 cranial hairs. Then 'It is indefinite that Herbert is bald' is
synonymous with 'Herbert has between 40 and 60 cranial hairs', but this
latter sentence is not synonymous with any claim about a sentence being
true. The indefiniteness of vague sentences is as much amatter of fact as
the truth or falsehood of precise ones.
Also the ellipsis view has the particular disadvantage of making for a
sudden discontinuity between first- and second-order vagueness. First
order vagueness is a matter of ordinary predicates having borderline
cases, but second-order vagueness is amatter of the truth-predicate having
borderline cases. There is, of course, a correlation between the second
order vagueness of ordinary predicates and the first-order vagueness of
the truth-predicate. But we feel that the latter arises from the former,
and not vice versa. The truth-predicate is supervenient upon the object
language; there can be no independent grounds for its having 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 itwere, 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).
Thus 'truer' is primary; 'true' is secondary and to be defined with the
help of the definitely-operator.
VAGUENESS, TRUTH AND LOGIC 297
The second puzzle arises from the demand for a perfectly precise meta
language. So far, we have only demanded of our truth-conditions that
they provide correct allocations of truth. To respect the truth-value gap,
to account for penumbral connection, to yield the correct logic; these
are all special cases of this more general demand. However, one may also
require that the meta-language not be vague or, at least, not so vague in
its proper part as the object-language. Thus itwill not do to subject truth
to the standard equivalences:
'A9 is true if and only if A.
For then truth will be truthT; the truth-conditions will be classical; and
the vagueness of the truth-predicate will exactly match that of the object
language.
What we require is that the true/false/indefinite trichotomy be relatively
firm. Ideally, the truth of the disjunction 'A is true, false or indefinite'
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,
regardless of whether it is clear that a particular complete specification is
admissible. Rather it is that the infirmity raises another problem for
truth-conditions.
This raises the puzzle: is there a perfectly precise meta-language? Cer
tainly, each of the meta-languages Mn could be vague. One could take
the whole construction into the transfinite and have, for each ordinal a,
ameta-language Ma or strong definitely-operator D a.But the same prob
lem would arise anew. At no point does it seem natural to call a halt to
the increasing orders of vagueness.
However, if a language has a semantics in terms of higher-order bound
aries, then it also has a firm truth-predicate. For the boundaries will be
based upon a set of admissible specifications and we can let truth (or
falsehood) be truth (or falsehood) in all such specifications. Anything
that smacks of being a borderline case is treated as a clear borderline
case. The meta-languages become precise at some, but no pre-assigned,
ordinal level. The only alternative to this is that the set of admissible
specifications is itself intrinsically vague. There would then be a very
298 KIT FINE
intimate connection between vague language and reality: what language
meant would be an intrinsically vague fact.
If higher-order vagueness terminates at some stage a then vagueness
can, in a sense, be eliminated. For each sentence A can be replaced by a
perfectly precise sentence DaA that entails it. However, this method is
unsatisfactory in several ways. First, one may not be able to specify the a.
Second, even if one can, one may not be able to make much sense of D*.
Our intuitions 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
aries to boundaries. Finally, the method is too uniform to be dis
criminate. Penumbral connections may be lost: our blob, for example,
is not definitely red or definitely pink. Indeed, the question of making
predicates perfectly precise
12 is independent of whether higher-order
vagueness terminates. The predicate 'small', as applied to numbers, may
suffer from endless higher-order vagueness; yet it can still be made per
fectly precise.13
University of Edinburgh
NOTES
1 I should like to thank Gordon Baker for numerous stimulating conversations on the
topics of this paper. My ideas would not have taken their present form without his
help. I should also like to thank Michael Dummett for some valuable remarks in a
discussion of the paper.
2 Kleene's (1952, pp. 329 and 332) 'weak/strong' and 'regular' correspond to our
'minimal/maximal' and 'stable', though his motivation for introducing the terms is
different from ours.
3
Frege (1952, p. 63) and Hallden (1949) have adopted the minimal truth-value ap
proach, though Frege would not be happy in regarding Indefinite as a third truth-value.
K?rner (1960, p. 166) and ?qvist (1962) have espoused the maximal approach.
4
See the work of Zadeh (1965) and others. It is not clear that one can make much
sense of degrees of truth within a closed interval for 'multi-dimensional' vagueness, as
in 'chair' and 'game'. It is even less clear that any semantical sense can be given to the
notion. Possibly there is a confusion with the higher order vagueness of Section 5.
5 Van Fraassen (1968) has already made much of the super-truth notion, though with
different applications inmind. He has also drawn out the consequences for logic and
considered the possibility of minimizing and maximizing truth-value (the conservative/
radical distinction of van Fraassen (1969).)
6 The semantics for intuitionistic logic comes from Kripke (1965). The bastard account
can be found inFitting (1969).
VAGUENESS, TRUTH AND LOGIC 299
7 The Fregean theory and its extension have a nice algebraic formulation. The usual
theory states that there is a homomorphism from the word algebra into the algebra of
intensions, and from the algebra of intensions into the algebra of extensions, and hence
a homomorphism from the word algebra into the algebra of extensions. The extended
theory states that the extension and intension algebras both possess amonotonie partial
ordering, which is respected by the homomorphism. It follows that (1) and (2) are im
plied by (3)-(6).
8 The argument is Cohen's (1966).
9 To assert, severally, sentencesPi,...,Pjc is not to assert the conjunction or, for that
matter, the disjunction of the sentences. For the conjunctive assertion is false if one of
the sentences is false, whereas the multiple assertion is false only if each of the sen
tences is false; and the disjunctive assertion is true if one of the sentences is true, whereas
the multiple assertion is true only if each of the sentences is true. These distinctions
may have a useful application to the cluster theory of names. For suppose predicates
Fi,...,Fic underly the name a. Then the assertion of </>(a) can be regarded as the multi
ple, as opposed to the conjunctive or disjunctive, assertion of <?(the Fi-er),..., <?(the
Ffc-er). A truth-value gap results in case some of the predicates individuate and others
do not.
10
Philosophers have been unduly dismissive over intrinsically vague entities. This atti
tude may derive, in part, from the view that any piece of empirical reality is isomorphic
to a mathematical structure; since the structure is precise, so is the reality. Thus, the
blurred outline becomes isomorphic to a set of points in Euclidean space. However, I
am not even sure that all mathematical entities are precise. Perhaps one could develop
an intuitive theory of vague sets. Hopefully, itwould not even be interpretable within
standard set theory; so that the sceptic could not then treat vague sets on the onion
model, as a 'fa?on de parler'.
11
For references to other results on functional completeness, see Rescher (1969).
12 This question is usually settled upon covertly constructivist lines. Our powers of
perceptual discrimination are limited ; therefore we cannot settle whether an object has
such and such exact shade. But could not a non-constructivist take 'red', say, to mean
the colour of that, where 'that' refers to a perfectly uniform shade? The inability to
know would not affect the ability to mean.
13
(Added in the proofs). After writing the paper, I discovered that the super-truth
account of vague languages had also been espoused by the following authors :H. Kamp
in 'Scalar Adjectives', to appear in the Proceedings of the Cambridge Linguistics Confer
ence; D. Lewis in 'General Semantics', Synthese 22 (1970), 18-67; M. Przetecki in The
Logic of Empirical Theories, Routledge and Kegan Paul, London, 1969; and P. A.
Williams inChapter 1 of his doctoral thesis. The view seems to go back to H. Mehlberg's
The Reach of Science, Toronto University Press, Toronto, 1956.
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