Paradoxes (5): The Sorites Paradox
Paradoxes
Lecture Five
Vagueness: The Sorites Paradox
Rob Trueman
[email protected]
University of York
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
Vagueness: The Sorites Paradox
The Sorites Paradox
Two Forms of the Paradox
A Veridical Paradox?
Epistemicism
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
How Many Grains of Sand Make a Heap?
• 1 grain of sand does not make a heap
———————————————————-
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
How Many Grains of Sand Make a Heap?
• If 1 grain of sand does not make a heap, then 2 grains of sand
do not make a heap
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
How Many Grains of Sand Make a Heap?
• If 2 grains of sand do not make a heap, then 3 grains of sand
do not make a heap
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
How Many Grains of Sand Make a Heap?
999,999,997 steps later...
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
How Many Grains of Sand Make a Heap?
• If 999,999,999 grains of sand do not make a heap, then
1,000,000,000 grains of sand do not make a heap!
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
The Sorites Paradox
• This is known as the Sorites Paradox
– Soros is the ancient Greek for heap
• This is one Eubulides seven paradoxes
– The Liar
– The Masked Man
– The Electra
– The Overlooked Man
– The Horns
– The Heap (Sorites)
– The Bald Man (Phalakros)
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
The Bald Man Paradox
• A man with no hair is bald
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
The Bald Man Paradox
• If a man with no hair is bald, then a man with 1 hair is bald
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
The Bald Man Paradox
• If a man with 1 hair is bald, then a man with 2 hairs is bald
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
The Bald Man Paradox
• If a man with 2 hairs is bald, then a man with 3 hairs is bald
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
The Bald Man Paradox
9,997 steps later...
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
The Bald Man Paradox
• If a man with 9,999 hairs is bald, then a man with 10,000
hairs is bald!
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
The Same Paradox
• Clearly, the Heap Paradox is exactly the same as the Bald
Man Paradox
• As a result, philosophers now call both paradoxes, and all
paradoxes like them, Sorites Paradoxes
– Someone whose height is 0.5m is not tall; 1mm never makes
the difference between being tall and not being tall; so
someone whose height is 2m is not tall
– Something which weighs 1gm is not heavy; 0.5gm never makes
the difference between being heavy and not being heavy; so
something which weighs 10 tonnes is not heavy
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
A Paradox about Vagueness
• The Sorites Paradox is a paradox about vagueness
• Philosophers normally think of ‘bald’ as a vague predicate
• There are borderline cases of being bald
• These are cases where it isn’t clear whether someone counts
as bald or not
– These aren’t cases that come up because we can’t get a good
look at someone’s head
– We can know the exact number of hairs that they have, but
be unable to say whether or not they are bald
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
A Paradox about Vagueness
• The idea is that there is no sharp cut off between a number of
hairs that makes you bald, and a number that makes you
not-bald
– There are some numbers which make you clearly bald (0, 1,
2...)
– And some numbers that make you clearly not-bald (9998,
9999, 10000...)
– But there isn’t a sharp line between the two
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
A Paradox about Vagueness
• The Sorites Paradox exploit this lack of a sharp cut off
between being bald and being not-bald
• It is because there is no sharp cut off that every conditional of
this form looks true:
– If a man with n hairs is bald, then a man with n + 1 hairs is
bald too
• Since ‘bald’ is vague, there just can’t be a point at which a
bald man flips into being a not-bald man
Paradoxes (5): The Sorites Paradox
The Sorites Paradox
A Paradox about Vagueness
• So far, this is all seems like a lot of common sense
• But then the Sorites Paradox comes along and causes trouble
• The Sorites Paradox tries to convert our perfectly sensible
claim that there is no sharp cut off between being bald and
being not-bald into the silly claim that everyone is bald, no
matter how many hairs they have
• The Sorites Paradox thus poses a challenge to the very
concept of vague predicates
Paradoxes (5): The Sorites Paradox
Two Forms of the Paradox
Vagueness: The Sorites Paradox
The Sorites Paradox
Two Forms of the Paradox
A Veridical Paradox?
Epistemicism
Paradoxes (5): The Sorites Paradox
Two Forms of the Paradox
General Form A
• 0 grains of sand do not make a heap
• If n grains of sand do not make a heap, then n + 1 grains of
sand do not make a heap
• Therefore, for any n, n grains of sand do not make of heap
Paradoxes (5): The Sorites Paradox
Two Forms of the Paradox
General Form A
• A man with 0 hairs is bald
• If a man with n hairs is bald, then a man with n + 1 hairs is
bald
• Therefore, for any n, a man with n hairs is bald
Paradoxes (5): The Sorites Paradox
Two Forms of the Paradox
General Form A
• φ(0)
• ∀n(φ(n) → φ(n + 1))
(QP)
• ∴ ∀nφ(n)
– QP stands for Quantified Premise
• This form of argument will be familiar to those of you who
have studied much maths
• It is called mathematical induction
Paradoxes (5): The Sorites Paradox
Two Forms of the Paradox
Mathematical Induction
• Imagine you wanted to prove that all the numbers had some
property, φ
• You can’t just go through them all and prove that each has φ
– There are infinitely many numbers!
• Instead, you use mathematical induction:
• φ(0)
• ∀n(φ(n) → φ(n + 1))
• ∴ ∀nφ(n)
Paradoxes (5): The Sorites Paradox
Two Forms of the Paradox
Mathematical Overkill
• But we do not really need anything as powerful as
mathematical induction to set up a Sorites Paradox
• We can formulate them in a way that only makes use of
modus ponens
– If A then B; A; ∴ B
Paradoxes (5): The Sorites Paradox
Two Forms of the Paradox
General Form B
• 0 grains of sand do not make a heap
• If 0 grains of sand do not make a heap, then 1 grain of sand
does not make a heap
• If 1 grain of sand does not make a heap, then 2 grains of sand
do not make a heap
• If 2 grains of sand do not make a heap, then three grains of
sand do not make a heap
• ...
• If 999,999,999 grains of sand do not make a heap, then
1,000,000,000 grains of sand do not make a heap
• Therefore 1,000,000,000 grains of sand do not make a heap
Paradoxes (5): The Sorites Paradox
Two Forms of the Paradox
General Form B
• A man with 0 hairs is bald
• If a man with 0 hairs is bald, then a man with 1 hair is bald
• If a man with 1 hair is bald, then a man with 2 hairs is bald
• ...
• If a man with 9,999 hairs is bald, then a man with 10,000
hairs is bald
• Therefore a man with 10,000 hairs is bald
Paradoxes (5): The Sorites Paradox
Two Forms of the Paradox
General Form B
• φ(0)
• φ(0) → φ(1)
• φ(1) → φ(2)
• ...
• φ(m − 1) → φ(m)
• ∴ φ(m)
• The idea here is that m is a sufficiently large number that it is
obviously false to say φ(m)
– So if φ means bald, then m could be 10,000
Paradoxes (5): The Sorites Paradox
Two Forms of the Paradox
Another Advantage of B over A
• Presenting the Sorites Paradoxes in Form B is better than
presenting them in Form A partly because we only use modus
ponens in B
• But it is also better because each of the conditional premises
is very hard to deny
• On one end of the scale, the following obviously looks true:
– If 1 grain of sand does not make a heap, then 2 grains of sand
do not make a heap
• And on the other end of the scale, this looks true
– If 999,999,999 grains of sand do not make a heap, then
1,000,000,000 grains of sand do not make a heap
• Of course, 999,999,999 grains of sand do make a heap, and so
do 1,000,000,000
• But if 999,999,999 grains do not make a heap, then neither
do 1,000,000,000
Paradoxes (5): The Sorites Paradox
Two Forms of the Paradox
Denying a Conditional Premise
• In classical logic, denying A → B is the same as asserting
A ∧ ¬B
• So to block Form B of the Sorites Paradox by denying one of
the premises, we would have to assert an instance of:
– φ(n) ∧ ¬φ(n + 1)
– 12 grains of sand do not make a heap, but 13 grains do
– A man with 26 hairs is bald, but a man with 27 is not
• But on the face of things, these assertions like these look
absurd!
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
Vagueness: The Sorites Paradox
The Sorites Paradox
Two Forms of the Paradox
A Veridical Paradox?
Epistemicism
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
Quinean Classifications
A paradox is an apparently unacceptable conclusion
derived by apparently acceptable reasoning from
apparently acceptable premises. (Sainsbury’s definition
from Paradoxes, p. 1)
• Premise-flawed
– One of the premises turns out to be false
• Fallacious (“Falsidical”)
– The reasoning turns out to be faulty
• Veridical
– The conclusion turns out to be true
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
Could the Paradox be Veridical?
• Could we just say that the paradoxes are veridical?
• This definitely sounds strange!
– There are no heaps
• But this is Peter Unger’s (1979) view
• More generally, Unger denies that any of the “ordinary
objects” that we are used to thinking and talking about really
exist
• Ordinary objects are made up of sub-atomic particles, and so
we could run a Sorites-style argument on any ordinary object:
– 2 sub-atomic particles cannot be put together to make a table
– If n sub-atomic particles cannot be put together to make a
table, then neither can n + 1 sub-atomic particles
– So there are no tables
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
Vague Concepts are Faulty Concepts
• This might sound absolutely bizarre, but it is a lot less strange
if we think of it as the idea that vague concepts are inherently
flawed
• The reason that there are no heaps is because heap is a vague
concept, and vague concepts cannot be used to describe the
world
• An objection to Unger’s view:
– The conclusion of the Bald Man Paradox is that everyone is
bald!
– Far from being too faulty to apply to anything, the concept
bald seems to apply to everyone!
• Unger’s answer:
– Whereas the Heap Paradox is veridical, the Bald Man is
premise flawed
– No one is bald, not even someone with 0 hairs
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
Pairs of Paradox
• This might sound a bit arbitrary
– One Sorites paradox is veridical, the other premise-flawed
• But actually, there is nothing arbitrary here
• Sorites paradoxes come in pairs, with each member of the pair
being the “reverse” of the other
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
A Sorites Paradox in Reverse
• 1,000,000,000 grains of sand make a heap
—————————————————
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
A Sorites Paradox in Reverse
• If 1,000,000,000 grains of sand make a heap, then
999,999,999 grains of sand make a heap too
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
A Sorites Paradox in Reverse
999,999,996 steps later...
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
A Sorites Paradox in Reverse
• If 4 grains of sand make a heap, then 3 grains of sand make a
heap too
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
A Sorites Paradox in Reverse
• If 3 grains of sand make a heap, then 2 grains of sand make a
heap too
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
A Sorites Paradox in Reverse
• If 2 grains of sand make a heap, then 1 grain of sand makes a
heap too!
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
Veridical or Premise-Flawed?
• A pair of Sorites Paradoxes:
– 0 grains of sand do not make a heap; if n grains of sand do not
make a heap then n + 1 grains do not either; so there are no
heaps
– 1,000,000,000 grains of sand make a heap; if n grains of sand
make a heap, then so do n − 1 grains; so 0 grains of sand
make a heap
• Another pair:
– A man with no hair is bald; if a man with n hairs is bald, then
so is a man with n + 1 hairs; so a man with 10,000 hairs is bald
– A man with 10,000 hairs is not bald; if a man with n hairs is
not bald, then neither is a man with n − 1 hairs; so no one is
bald
• Unger’s view is that in every pair of Sorites paradoxes, one is
veridical and the other is premise-flawed
• The veridical one is the one that concludes that a given vague
concept (heap or bald) does not apply to anything
Paradoxes (5): The Sorites Paradox
A Veridical Paradox?
Vague Concepts are Faulty Concepts
• It seems to me that Unger’s view is coherent
• An even more extreme view was held by Frege, the inventor of
modern logic
– Frege (e.g. 1903, §56) insisted that no sentence of the form ‘a
is bald’ is true or false
– Vague concepts (and the predicates which express them) are
so defective that they cannot ever be truly or falsely applied to
anything
• But views like Frege’s and Unger’s are pretty radical
• We would need to find a whole new way of describing the
world without using vague concepts
• Could we speak a language without vagueness?
Paradoxes (5): The Sorites Paradox
Epistemicism
Vagueness: The Sorites Paradox
The Sorites Paradox
Two Forms of the Paradox
A Veridical Paradox?
Epistemicism
Paradoxes (5): The Sorites Paradox
Epistemicism
Sharp Cut-Offs
• The Sorites Paradoxes get going because vague concepts do
not seem to come with sharp cut-offs
– If n grains of sand do not make a heap, then neither do n + 1
– If a man with n hairs is bald, then so is a man with n + 1 hairs
• One very simple way of responding to these paradoxes do
have sharp cut-offs after all
• We could then deny QP from Form A of the paradoxes:
– ¬∀n(φ(n) → φ(n + 1))
– ∃n(φ(n) ∧ ¬φ(n + 1))
• And we could deny one of the conditional premises from Form
B:
– ¬(φ(n) → φ(n + 1))
– φ(n) ∧ ¬φ(n + 1)
Paradoxes (5): The Sorites Paradox
Epistemicism
Introducing Epistemicism
• This view might sound even weirder than Unger’s
– In fact, Unger argues for his position in part by pointing out
how strange it would be to believe in sharp cut-offs!
• Exactly how many hairs do you need to not be bald? 22? 23?
30? How could we ever answer this question?
• This is where epistemicism comes in
– This is the view of Williamson (1994)
• According to epistemicism, there is a sharp cut-off between
being bald and not being bald, but we cannot know where it
is
Paradoxes (5): The Sorites Paradox
Epistemicism
Introducing Epistemicism
• It might be that you are bald if you have 2 hairs, but not if
you have 3
Paradoxes (5): The Sorites Paradox
Epistemicism
Introducing Epistemicism
• Or it might be that you are bald if you have 1 hair, but not if
you have 2
Paradoxes (5): The Sorites Paradox
Epistemicism
Introducing Epistemicism
• Although there is a sharp cut-off between being bald and not,
we can never know where it is
Paradoxes (5): The Sorites Paradox
Epistemicism
Vagueness as Epistemological
• More generally, vagueness is an epistemological phenomenon
• To say that a concept (or predicate) is vague is not to say
that it has no sharp cut-off between applying and not applying
• It is just to say that we do not, and cannot, know where the
cut-off is
• A “borderline” case for a vague concept (e.g. a man with 12
hairs) is just a case where we do not, and cannot, know
whether the concept applies
• However, there is still a fact of the matter whether the
concept applies
Paradoxes (5): The Sorites Paradox
Epistemicism
The Benefits of Epistemicism
• Most people respond to epistemicism with an incredulous stare
• Even with all the caveats about us not knowing where the line
is, it seems really odd to say that there is a sharp line between
being bald and not
• However, epistemicism provides a very simple response to the
Sorites Paradoxes
• Epistemicists just dismiss the paradoxes as premise-flawed
– Unlike Unger, the epistemicist does not need to say that there
is anything wrong with vague concepts, like heap
– And unlike the views we will discuss next week, we do not need
to introduce any clever logical or semantic machinery
• However, epistemicism does face some serious problems
Paradoxes (5): The Sorites Paradox
Epistemicism
Necessary Ignorance
• I mentioned earlier that according to epistemicism, we are not
merely ignorant of where the cut-off for a vague concept is
• We are necessarily ignorant
• We cannot know where the cut-off is
Paradoxes (5): The Sorites Paradox
Epistemicism
Necessary Ignorance
• Suppose that when we use the word ‘bald’, we are expressing
the concept bald1 , according to which a man is bald if he has
22 hairs, but not if he has 23
• Still, we could have been working with another concept, bald2 ,
according to which a man is bald if he has 23 hairs, not bald
if he has 24
• Moreover, we could not tell if we were using bald1 or bald2
• Either way, we would go on using the word ‘bald’ in exactly
the same way
Paradoxes (5): The Sorites Paradox
Epistemicism
Necessary Ignorance
• Now imagine that you happen to believe that a man is bald if
he has 22 hairs, and not if he has 23
• Given the supposition that we are using bald1 , your belief is
true
• But it still cannot be knowledge
• You could not tell if you were using bald1 or bald2
• Either way, you would come to believe that a man is not bald
if he has 23 hairs
• But this belief would have been false if we were using bald2
• So your belief can only be a lucky guess
Paradoxes (5): The Sorites Paradox
Epistemicism
Necessary Ignorance
• The idea that there are truths which we cannot know is
philosophically contentious
• According to verificationism, we can know every truth
• Admittedly, verificationism is not very popular at the moment
• But it is one thing to say that there are some, special truths
we cannot know
– e.g. the total state of the Universe
• It is another to say that we are necessarily ignorant of the
cut-off between being bald and not
Paradoxes (5): The Sorites Paradox
Epistemicism
Meaning is Use
• Why is it so weird to think that we are necessarily ignorant
about the cut-off between being bald and not?
• Well, words get their meanings from how we use them
• If we cannot know where the cut-offs are, then how do these
cut-offs come to be at all?
• According to the epistemicist, we could not tell whether we
were using concept bald1 or bald2
– Either way, everyone would continue to use the word ‘bald’ in
exactly the same way
• But in that case, how is it that our word ‘bald’ gets associated
with one of these concepts rather than the other?
Paradoxes (5): The Sorites Paradox
Epistemicism
Next Week
• This is definitely a big problem for epistemicism
• Maybe we can find a way of living with it if we don’t have a
choice
• But it is worth looking at some alternatives!
• Next week we will look at two more responses to the Sorites
Paradoxes:
– Supervaluationism
– Many truth-values
• Required reading:
– Paradoxes 3.5–7
Paradoxes (5): The Sorites Paradox
Epistemicism
For the Seminar
• Required reading:
– Williamson, T (1992) ‘Vagueness and Ignorance’, Aristotelian
Society Supplementary Volume 66: 145–77
• You can find this paper via the VLE
Paradoxes (5): The Sorites Paradox
Epistemicism
References
• Frege, G (1903) Grundgesetze der Arithmetik vol. II, partially
reprinted in Beaney (1997) A Frege Reader
• Sainsbury, R and Williamson, T (1997) ‘Sorites’ in Hale and
Wright (eds) A Companion to the Philosophy of Language
(Oxford: Blackwell)
• Unger, P (1979) ‘There are no ordinary things’, Synthese
41: 117–54
• Williamson, T (1994) Vagueness (London: Routledge)