# [FOM] primer on vagueness

Stewart Shapiro shapiro.4 at osu.edu
Fri May 20 00:21:56 EDT 2005

```Harvey asked me to post a primer on the philosophical and logical issues
concerning vagueness.  The past few decades have seen an explosion of work
in this area, and I cannot do it all justice.  But I’ll try to go over the
main issues and battle lines, focusing on what I think will be of interest

There probably is no way to say what it is for an expression to be vague
without begging the question against some substantial view.  Intuitively, a
predicate is vague if it has, or can have borderline cases, instances in
which it is not determinate that the predicate applies, nor is it
determinate that it fails to apply.  A borderline case of “red” would be an
object whose color lies about halfway between red and pink.

Typical vague predicates give rise to the ancient *sorites paradox*,
sometimes called the *paradox of the heap*.  Here is an example:

Premise:  A man with no hair at all is bald.

Premise:  If a man has only n hairs on his head is bald, then so is a man
with n+1 hairs.

Conclusion:  A man with 50,000 hairs on his head is bald.

Clearly, the argument is valid (once supplemented with the obvious premises
about the natural numbers).  Its first premise is true and its conclusion
is false.  But its second premise *seems* to be true.  How can a single
hair make the difference between someone who is bald and someone is not?

Another version of the sorites uses tons of premises, instead of the
induction-like premise that invokes natural numbers.  If a man with 0 hairs
is bald, then so is a man with 1; if a man with 1 hair is bald, then so is
a man with 2; ...  This argument has 50,002 premises.

The traditional example uses the word “heap”, and turns on how many grains
make up a heap.

Intuitively, vague predicates are what Crispin Wright calls
“tolerant”.  Many of the predicates are applied on the basis of casual
observation, and small differences should not matter.

The problem is to give a semantics of natural-language vague predicates
which reveals what goes wrong (if something does) with sorites reasoning.

Just about every account agrees that something does go wrong with sorites
reasoning.  The only serious candidate is the inductive premise.  Most of
the different accounts have that premise coming out untrue (if not
false).  The burden that remains is to show why we find the inductive

Here are the main theses (or slogans) of some of the main theories of
vagueness:

I. Nihilism (Dummett)

Advocates of this view hold that the semantic rules for vague predicates of
natural languages are inconsistent.  The rules sanction the premises of the
sorites arguments, and yet they reject the conclusion.  Natural language is
incoherent.  Logic does not apply to it.

II. Epistemicism (Williamson, Sorenson)

This is the result of applying classical semantics to natural
languages.  Each predicate of natural language has a precise
extension.  There is a an exact nanosecond at which a person stops being a
child;  there is a real number n such that a person who is n feet tall is
short, but a person who is n+.00000001 feet tall is not short;  there is a
real number m such that m minutes after noon is noonish, but m+.00000001
minutes is not.

The view is called “epistemicism” since the sharp borderlines of vague
predicates cannot be known.  That is, it is impossible to know that values
of n and m, nor how many hairs make for non-baldness, how many grains of
sand make for a heap, etc.

Most theorists find this view fantastic.  How can natural language
predicates get extensions that are that precise?  The main advantage of the
view is that it maintains classical model-theoretic semantics, and
classical, two-valued logic.  In sorites arguments, the inductive premise
is simply false.

III. Supervaluationism (Fine, Keefe)

A given vague predicates does not have a single extension.  Instead, there
are a number of different ways of making it precise, which are consistent
with its meaning.  A sentence P that contains vague predicates is
super-true if it comes out true under all acceptable ways of sharpening the
predicate.  The sentence P comes out false if it comes out false under all
acceptable ways of sharpening the predicate.  P is indeterminate otherwise.

One advantage of supervaluationism is that it maintains much of classical
logic.  All tautologies are super-true.  But the semantics has some strange
properties.  Let t be a rose whose color is midway between red and
pink.  Then the sentence “t is red or t is pink” is super-true.  But is not
super-true that t is red, nor is it super-true that t is pink.

Something similar happens with the inductive premise of a sorites
argument.  Recall the instance above, formulated more carefully:

For every number x, if a man has only x hairs on his head is bald, then so
is a man with x+1 hairs.

This is super-false, since it comes out false on every acceptable way to
sharpen the predicate “bald”.  But there is no number n such that it is
super-false that “if a man has only n hairs on his head is bald, then so is
a man with n+1 hairs”.

It is generally agreed that this theory pushes the problem up to the
meta-language.  What makes a sharpening acceptable is itself vague.

III.  Fuzzy logic.  (Machina, Edgington)

This one involves truth values intermediate between truth and
falsehood.  The most common proposal is to use real numbers between 0 and
1.  That is, 0 is complete falsehood, 1 is complete truth.  .4 would be a
partial truth.  If s is borderline bald, leaning toward baldness, then the
statement “s is bald” might get truth value .623.

Another, less developed, proposal, is to use the elements of a Boolean
algebra as truth values.  This gives the theorist a shot a preserving
classical logic.

Below, I’ll insert the handout for some lectures I gave on
supervaluationist semantics and fuzzy logic.

IV.  Contextualism.  (Raffman, Shapiro, Graff)

The extensions of vague predicates shift with the context of
utterance.  Everyone agrees that the extensions of vague predicates vary
with factors like the comparison class and paradigm cases.   A family can
be wealthy with respect to professors in mathematics departments, but not
wealthy with respect to corporate executives.  A person can be short with
respect to NBA players but not short with respect to philosophers.  The
contextualist argues that extensions also vary in the course of a given
conversation (or psychological state), even if the comparison class is held
fixed.

There are a few proposals in this direction.  Some preserve classical logic
and some don’t.

If you want to read a bit more, without leaving your computer, here are a
couple of links to the Stanford Internet Encyclopedia of Philosophy.  The
third is to a closely related matter.

http://plato.stanford.edu/entries/vagueness/

http://plato.stanford.edu/entries/problem-of-many/

I’d be glad to supply more specific references, if anyone wants them.  Just
send me a (private) email.  I just finished a book that develops and
defends a contextualist account.  The model theory is much like that of
Kripke structures for intuitionistic languages.

Here are a couple of handouts, one on fuzzy logic and one on
supervaluationist model theory.

A crash course in many-valued model theory

If Harry is borderline bald, then perhaps it is natural to say that "Harry
is bald" is not fully true, but not quite false either.  Perhaps the
sentence has some intermediate truth value.

The idea behind the many-valued or degree-theoretic approach to vagueness
is that sentences can take any of a number of truth values.  The most
common collection of truth-values is the set of real numbers from 0 to 1
(inclusive).  The truth value 1 indicates complete truth and the truth
value 0 indicates complete and utter falsity.  So if "Harry is bald" is
true to degree .87, then he is almost bald.  If it is true to degree .5
then he is exactly halfway between determinately bald and determinately
non-bald, etc.  Other sets of truth values may be considered.

A fuzzy interpretation M of a formal language is a pair <d,I> where d is a
non-empty set, the domain of discourse (as usual), and I assigns
denotations and extensions to the non-logical terminology.  Let us assume
that we have no function letters in the language.  If a is a constant, then
Ia is a member of d (as usual).  If R is an n-place relation symbol and m1
. . . mn are members of the domain d, then IRm1 . . . mn is a real number
between 0 and 1.

A fuzzy interpretation is classical if the only truth-values it assigns are
0 and 1.

Let M = <d,I> be a fuzzy interpretation.  The semantics assigns a "truth
value" (i.e., a real number) to every sentence of the language.  If P is a
sentence, then let VP be its truth value in M.

For convenience, assume that the members of d have been added to the
language, as constants denoting themselves.  (So we can deal with sentences
without introducing variable assignments).

Only the atomic case and perhaps the negations are straightforward:

If R is an n-place relation letter and t1 . . . tn are terms, then VRt1 . .
. tn  = IRm1 . . . mn, which each mi is the denotation of ti.

V(not-P)  = 1-VP.

So if "Harry is bald" is half true, then so is "Harry is not bald".

One main issue concerns whether the other connectives should be
truth-functional, or what is sometimes called "degree-functional".  The
question is whether the truth value of a compound is determined completely
by the truth values of its parts.  I am convinced by Dorothy Edgington's
arguments that the semantics should not be truth-functional.  But I will
present the most common semantics, which is truth-functional:

V(P&Q)   is the minimum of VP    and VQ.

V(PvQ)   is the maximum of VP    and VQ.

if VP<=VQ, then V(if P then Q)        = 1,
if VP>VQ, then  V(if P then Q)        = 1-(VP-VQ).

The existential quantifier gives the least upper bound of its instances,
and the universal quantifier gives the greatest lower bound of its instances.

Notice that if M is classical (i.e., if the only truth values it assigns to
the atomic sentences are 0 and 1), then the semantics agrees with classical
semantics (exercise).

There are a number of different definitions of validity.  You can decide
which of them accords with your intuitions.

One slogan is that validity is the necessary preservation of truth.  Since
(complete) truth is the truth value 1, then we might define strict validity
as the necessary preservation of truth value.  So an argument is strictly
valid iff there is no fuzzy interpretation that makes its premises 1
(completely true) and its conclusion less than 1.

Every strictly valid argument is classically valid, but the converse
fails.  There are no strictly logical truths.  However, classical
tautologies can never get a truth value less than .5.

Why should validity be limited to the preservation of complete truth?  We
often reason with sentences that (on this view) are less then completely
true, and need a notion of validity for those.  We can say that an argument
is valid if it preserves whatever truth value the premises may have:

Say that an argument is fuzzy-valid if there is no fuzzy interpretation M
such that the truth value of the conclusion is less than the greatest lower
bound of the truth values of the premises.

Edgington has an interesting definition of validity.  Define the unverity
of a sentence P in a fuzzy interpretation M to be 1-VP in  M.  Then an
argument is E-valid if the unverity of the conclusion is no smaller than
the sum of the unverities of the premises.  (But as noted above, she uses a
different, non-truth-functional semantics.)

Now, what of sorites?

Suppose we have a series of 1000 men lined up.  The first m1 has no hair
whatsoever, and so is clearly bald.  The last m1000 has a fine head of
hair.  Moreover, each man in the list has only slightly more hair than the
one before (and his hair is arranged in roughly the same way).  Let B be
the predicate for bald.  So a reasonable fuzzy interpretation would be one
in which, say,  Bm1  =   Bm2  = . . . =  Bm60  = 1, and  Bm61  = .999.  The
truth values fall slowly from there, until  Bm925 =0, and everything after
that is completely false.  Suppose that the difference in truth value
between adjacent formulas is never more than .002.

Recall the two versions of the sorites argument:

Bm1
for all i<1000(Bmi ->  Bmi+1)
therefore Bm1000

Bm1
Bm1 ->  Bm2
Bm2 ->  Bm3
. . .
Bm999 ->  Bm1000
therefore Bm1000

In both cases, the first premise has truth value 1, and the conclusion has
truth value 0.  The first few conditionals and the last few conditionals
each have truth value 1.  Given the assumption about the differences
between adjacent formulas, each of the conditionals has truth value of at
least .998, and some are not fully true.  So the second premise of the
first argument has truth value somewhere between .998 and .999.

On the first definition of validity (as the necessary preservation of
complete truth), both arguments are valid.  The only inferences used are
modus ponens and (for the first argument) universal elimination, and both
are valid on that conception of validity.  But neither argument is sound,
since it has premises that are less than fully true.

One puzzling feature is that we can have a valid argument whose premises
are almost completely true (at least .998), but which has a completely
false conclusion.

On the second definition of validity (as the preservation of minimal truth
value), neither argument is valid.  Modus ponens fails.

I think that the Edgington definition of validity is the most satisfying
here.  She shows that (on her semantics), all classically valid arguments
are valid in her sense.  In the second argument, the many small
"unverities" in the premises add up to 1.  But on her semantics, the
conditional in the first argument ends up completely false.

Here is the other handout:

A crash course in supervaluation model theory

The first modification to ordinary model theory introduced here is the
notion of a *partial interpretation*.  This is a pair <d,I>, where d is a
non-empty set (the domain of discourse) and I is a function.

Let R be an n-place relation letter.  In the partial interpretation
M=<d,I>, IR is a  pair <p,q> such that p and q are disjoint sets of
n-tuples of members of the domain.  The idea is that p is the extension of
R in the partial interpretation, the n-tuples of objects that the relation
(determinately) applies to; q is the anti-extension of R, the n-tuples of
objects that the relation (determinately) fails to apply to.  If IR=<m,n>,
then define IR+= m and IR-=n.  From now on, we write “interpretation” for
“partial interpretation”.

Any n-tuples from the domain of discourse d that are in neither IR+ nor IR-
are borderline cases of R in the partial interpretation M.  If there are no
such borderline cases, then we say that R is *sharp* in M.  A partial
interpretation M is completely sharp or classical if every relation in the
language is sharp in M.  A completely sharp interpretation corresponds to a
classical interpretation since, in that case, the anti-extension of each
predicate is just the complement of the extension.  So we sometimes call a
completely sharp partial interpretation “classical”.

We introduce a “three-valued” semantics on partial interpretations.  The
values are T (truth), F (falsehood), and I (indeterminate).  It does not
make much difference formally, but it is more convenient if we do not think
of I as a truth-value on a par with truth and falsehood.  Rather, we take
“I” to indicate the lack of a (standard) truth value in the given partial
interpretation.

Assume that we have expanded the language to include the members of the
domain as new constants (so we do not have to futz with variable assignments).

If a and b are terms, then a=b is true in M if and only if the denotation
of a in M is identical to the denotation of b in M; a=b is false in M
otherwise.  Notice that a=b is never indeterminate.

Let R be an n-place predicate letter, and t1, ...tn terms.  For each i, let
mi be the denotation of ti under M.  Then Rt1...tn is true in M if
<m1...mn> is in IR+; Rt1...tn is false in M if <m1...mn> is in IR-; and
Rt1...tn is indeterminate in M otherwise.

We use the strong, internal negation.  Its truth table is the following:

P	not-P
T	F
F	T
I	I

For the other connectives, we employ the strong Kleene truth-tables:

P&Q	T	F	I		PvQ	T	F	I
T	T	F	I		T	T	T	T
F	F	F	F		F	T	F	I
I	I	F	I		I	T	I	I

P->Q	T	F	I
T	T	F	I
F	T	T	T
I	T	I	I

A universally quantified sentence is true if all of its instances are true,
it is false if it has a false instance, and it is indeterminate
otherwise.  Existentially quantified sentences are analogous.

So far, we do not have a plausible model for a semantics of
vagueness.  Suppose, for example, that we have a patch whose color is on
the borderline between red and orange.  To capture this, we would make
statement Rt that the patch is red indeterminate; and we would make the
statement Ot that the patch is orange indeterminate.  In such a partial
interpretation, both RsvOs and not-(Rs&Os) would both be
indeterminate.  But this seems implausible.  Even if it is indeterminate
that the patch is red and it is indeterminate that it is orange, it surely
true that it is either red or orange.  After all, what other color could it
have?  And it is surely true that it is not both red and orange--nothing
can be red (all over) and orange (all over) at the same time.  Moreover,
Rs->Rs is also indeterminate in the given interpretation.  Even if Rs can
go either way, it surely is determinately true that if s is red, then s is red.

For a second example, consider a pair of men, r, s.  Both are borderline
bald, but r has a bit more hair than s (arranged the same way).  A partial
interpretation that reflects this situation would have Br->Bs
indeterminate.  But surely, if we sharpen in order to make Br true (as we
can, presumably), then we would thereby made Bs true, since s has even less
hair than r.  So, intuitively, Br->Bs should come out determinately true.

Let M1=<d1,I1> and M2=<d2,I2> be partial interpretations.  Say that M2
sharpens M1 if:

(1) d1=d2,

(2) the interpretation functions I1 and I2 agree on each constant and
function letter.

(3) for each relation letter R, I1+Ris a subset ofI2+R and I1-Ris a subset
of I2-R.

The idea is that M2 sharpens M1 if and only if the two interpretations have
the same domain and agree on the constants and function letters, and M2
extends both the extension and anti-extension of each relation.  The two
interpretations agree on the clear or determinate cases of M1, but M2 may
decide some borderline cases of M1.

Kit Fine [1975] calls M2 a “precisification” of M1.  We have that each M1
sharpens itself (although that does sound awkward).

Theorem 1.  Suppose that M2 sharpens M1.  If a sentence P is true (resp.
false) in M1, then P is true (resp. false) in M2.

The proof of Theorem 1 is a straightforward induction on the complexity of P.

Recall our decision to not think of “indeterminate” as a truth value, but
rather as lack of a (standard) truth value.  In these terms, it follows
from this result that the semantics is monotonic.  As we sharpen, we do not
change the truth values of formulas that have truth values; we only give
truth values to formulas that previously lacked them.

Notice that since this nice result is an induction on the complexity of
formulas, it depends on the particular expressive resources in the object
language.

Recall that a partial interpretation is completely sharp if every relation
in the language is sharp in the interpretation.  A straightforward
induction shows that a completely sharp interpretation corresponds to a
classical interpretation.

So far, things are pretty rigorous.  Now we wax intuitive for a bit.  Not
every sharpening is legitimate.  Suppose we have a predicate for “rich” and
there are two members of the domain, Jon and Joe, who are borderline rich
(so that neither fall in the extension nor the anti-extension of
“rich”).  Suppose that Joe’s total worth is a bit greater than the total
worth of Jon.  A sharpening that declares Jon rich and fails to declare Joe
rich (or declares that he is not rich) would clearly be unacceptable—it is
incompatible with the meaning of “rich”.  Similarly, a sharpening that
declares a man bald and declares a man with less hair (arranged similarly)
to be not bald is likewise unacceptable.

Fine [1975] uses the term “penumbral connection” for analytic (or all but
analytic) truths for the terms in question.  An example of a penumbral
connection is that if someone is rich then (other things equal) anyone with
more money is also rich.  Another penumbral connection is that if someone
is not bald then someone with more hair (arranged the same way) is also not
bald.  Another is that nothing is completely red and completely orange.

Of course, formal languages do not have such analytic (or almost analytic)
truths.  The formulas are just sequences of characters, and have no meaning
beyond the stated truth conditions for the logical terminology and perhaps
any premises and axioms that are explicitly added.  So, as a first
approximation, we assume a collection of conditions on partial
interpretations.  Suppose, for example, that the object language has a
predicate B, to represent the English word “bald”, and a binary relation R,
such that Rab represents the statement that a is less hairy than b.  Then a
partial interpretation M is not acceptable if there are objects m,n in the
domain of discourse such that m is in the extension of B, M satisfies Rnm,
but n is not in the extension of B.  Similarly, M is not acceptable if m is
in the anti-extension of B, M satisfies Rmn and n is not in the
anti-extension of B.

Say that a partial interpretation M is acceptable if it satisfies all of
its penumbral connections, and we say that a partial interpretation M2 is
an acceptable sharpening of a partial interpretation M1 if they are both
acceptable on the same collection of penumbral connections and M1˜M2.

Let P be a sentence and let M be a partial interpretation.  Say that P is
super-true in M if (1) there is no acceptable sharpening M’ of M such that
P is false in M’ and (2) there is at least one acceptable sharpening M’ of
M such that P is true in M’.

While we are at it, define P to be super-false in M similarly.

The idea, of course, is that P is super-true if it comes out true under
every acceptable way of making the predicates sharp.  That is, if we
sharpen, P comes out true if it gets a truth value at all (and we make sure
that P does get a truth value in some acceptable sharpenings).

Let P be a sentence, and assume, for now, that each acceptable partial
interpretation has at least one acceptable, completely sharp
sharpening.  In light of monotonicity (Theorem 1), P is super-true if and
only if P is true in every acceptable, completely sharp sharpening of
M.  Thus, in a sense, the definition of super-truth (and super-falsity)
depends only on the semantics of completely sharp interpretations, and, in
effect, those are ordinary, classical interpretations.  So the notion of
super-truth does not invoke the three-valued semantics above.

Most defenders of the supervaluation approach insist that the primary (or
perhaps the only) notion of truth is super-truth. The mantra is that truth
is super-truth.  Accordingly, they define validity as necessary
preservation of super-truth:

An argument is externally valid if every partial interpretation that makes
its premises super-true also makes its conclusion super-true.

Call this external validity.  Under the prevailing assumption about
acceptable, completely sharp interpretations, this definition does not
depend on the three-valued semantics introduced above.  All of the relevant
action takes place in completely sharp interpretations.

Theorem 2.  External validity coincides with classical (two-valued)
consequence.

So the supervaluationist accepts classical logic for vague expressions.

What of sorites?  Suppose we have a series of 1000 men lined up.  The first
m1 has no hair whatsoever, and so is clearly bald.  The last m1000 has a
fine head of hair.  Moreover, each man in the list has only slightly more
hair than the one before (and his hair is arranged in roughly the same
way).  Let B be the predicate for bald.  So a reasonable partial
interpretation would be one in which, say, m1,..., m247 are all in the
extension of B, and m621, ..., m1000 are in the anti-extension.  The
penumbral connection would be that mi is in the extension of B and if j<i,
then mj is in the extension of B; and if mi is in the anti-extension of B
and if i<j, then mj is in the anti-extension.  The various acceptable
(complete) sharpenings would put the border somewhere between m247 and m621.

Recall the two versions of the sorites argument:

Bm1
for every i<1000(Bmi->Bmi+1)
Therefore, Bm1000

Bm1
Bm1->Bm2
Bm2->Bm3
...
Bm999->Bm1000
Therefore, Bm1000

Both of these are classically valid, and so they are valid on the
supervaluational approach.  But the conclusions are super-false (in the
envisioned partial interpretation), and the first premise is
super-true.  So at least one other premise fails to be super-true.

In the first argument, the only other premise is the inductive
one.  Actually, it is super-false.  In each complete sharpening of the
partial interpretation, there is a sharp boundary, an i such that Bmi but
not-Bmi+1.

In the second argument, many of the conditionals fail to be super-true
(although none are super-false).  If mi is in the borderline region (in the
original partial interpretation), then there is a sharpening in which
Bmi->Bmi+1 is false.

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