[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 
to logicians.  I’ll give some links for further reading.

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 
premise so appealing, when it leads to paradox.

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

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 

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.




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:

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

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 

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

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

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 

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) 

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:

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

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.

More information about the FOM mailing list