Phil 2511: Paradoxes
Lecture 5: Sorites I: The
Problem and Some Standard Solutions
First – here’s the paradox of MINIAC that I should have included in the
notes to Lecture 4:
MINIAC
Take a
coin and designate one side `Yes' and the other `No'. Think of a question for which you would very
much like an answer, e.g. `Will I find the girl of my dreams?'. Toss the coin and note the answer it
gives. But how can we tell whether this
answer is true or false? Easy. Ask the
question `Will your present answer have the same truth value as your previous
answer?', flip the coin and note the response (`Yes'
or `No'). If the second response is
`Yes' then MINIAC's answer to the first question was
true; if the second response is `No' then the answer to the first question was
false, so now you know for sure whether you will find the girl of your
dreams. Proof: (I'll just do the proof
for the case where the second answer is `No.') Suppose the second answer is
`No'. This answer must be true or
false. If it's true then the
answer to the first question is false.
But, if the answer (`No') to the second question is false, then
the truth-value of the second question must be the same as that of the first,
so, again the answer to the first question is false. Therefore, if the second answer is `No' we
have proved that, whether or not that answer is true, the answer given by
MINIAC to the first question must be false. By similar reasoning we can prove that, if
MINIAC answers `Yes' to the second question its answer to the first question
must have been true. What's
paradoxical here, of course, is that one cannot guarantee to get correct
answers to momentous questions merely from two flips of a coin.
(Reference: T. Storer, `MINIAC: World's Smallest Electronic Brain', Analysis
22 (1961-2), pp.151-152.)
We now move
to a section of the course devoted to the Sorites
paradox. Last time, I suggested some
preliminary reading for this section of the course, and I hope that you have
now read at least some of it. If you
want to go deeper into the Sorites problem, I would
recommend a collection of readings: Rosanna Keefe and Peter Smith (eds), Vagueness: A Reader
(Cambridge MA, MIT Press, 1996), which includes a useful long introduction by
the editors, and also Rosanna Keefe, Theories of Vagueness (Cambridge
University Press, 2000). Her Chapter 1
covers more thoroughly than does this lecture questions concerning what
vagueness is and what types of solution to the Sorites
have been attempted. Timothy
Williamson’s Vagueness (London, Routledge,
1994) is very tough, but very rewarding.
1.
The
form of the Sorites paradox may be expressed as
follows: X1 is F; If Xi
is F then Xi+1 is also F, hence Xn
is F, where `F' is a vague predicate such that, for any pair of adjacent Xs, if
one member is F then so is the other.
This form of argument is paradoxical because we can choose premises that
are evidently true, the reasoning consists only of a simple mathematical
induction or, alternatively, of multiple applications of modus ponens, yet, for suitable n, the conclusion (e.g. that
0 is a large number, or that one grain of sand is a heap) is false. There are many instances of the Sorites. See, for
example, the tadpole/frog case devised by Cargile
(Keefe and Smith, p.89). Vague
predicates are tolerant. As Sainsbury puts it in his text (p.28): `Sorites reasoning depends on the supposition that vague
expressions are “tolerant”: small changes don’t affect the applicability of the
word. If someone is tall, so is a person
a millimeter shorter; if a collection is a heap, so is one otherwise similar
but with just one grain less’.
2.
Suppose
that I am leafing, right to left, through a thick book containing 2001 of the
flimsiest sheets. Near the beginning of
the book I should correctly judge that the right hand section (RH) is
fatter. And turning one flimsy page will
not alter that judgment. Now, in this
case, the principle `If Fatter(RH, LH) then
Fatter(RH-1 page, LH+1 page) fails when LH = 1000. In order to get a Sorites-type
paradox, we need to substitute `feels fatter' for `is fatter' in the above
principle. Such paradoxes depend
crucially on the employment of what have come to be called `observational'
terms - those the application of which does not wholly
rest on testable matters of fact. Thus,
while we know from chemistry that a water molecule minus one atom is not water,
chemistry cannot tell us whether Jones minus one atom is Jones, or is a person.
Neither the name `Jones' nor the sortal
`person' is a term of chemistry. The
predicate `is a person' is applied on the basis of discretionary judgments -
and the making of such a judgment may be difficult in the case of a Jones who
borders on the state of a vegetable.
3.
The Sorites is extremely important, not because it is an
intriguing puzzle, but because it brings into question
some of our most fundamental views about language. As Crispin Wright puts it (1987, p.231 =
Keefe and Smith, p.208) `the most fundamental task in the philosophy of
language is the achievement, in the most general terms, of an understanding of
the nature of language mastery. It is
here, if I am right, that the paradox has most to teach us'. Likewise, a paper of my own is called `The Sorites as a Lesson in Semantics’, Mind 97 (1988),
pp.447-455.
4.
Obviously,
most `solutions' to the paradox consist of rejecting either a premise or a
principle of inference used in the argument.
Some proponents of the second course say that mathematical induction is
invalid. This does not seem a promising
solution because (i) nothing independent of the Sorites would tempt us to think that mathematical induction
is invalid; (ii) the Sorites argument need not be
viewed as a mathematical induction but as a chain of modus ponens steps.
5.
Few
people would say that modus ponens is invalid, but some people have argued that a chain of applications of modus ponens
is invalid. This suggestion is as
hopeless as it looks -- modus ponens doesn't get worn out with repeated use. Several attempted types of solution are
explained and criticized by Roy Sorensen, `Vagueness, Measurement and
Blurriness', Synthese
75 (1988), pp.45-82, as well as in the general readings I have suggested.
6.
Sainsbury
spends most of his time considering two types of approach. I have found that students often
misunderstand this approach so I am distributing the 5 pages of his book where
he explains it. The `supervaluational'
account begins with the thought that there are penumbral objects that are not
comfortably called either V or not-V.
For example, there are small mounds to which we wouldn't confidently
assign or deny the title `heap'. But we
could invent a new sharper predicate `heap*' such that there is a sharp dividing line between those
objects that are heap*s and those that are not (see Sainsbury's
diagram, p.35). Call `heap*'
a sharpening of `heap'. There are many such sharpenings,
depending on where we divide the penumbral region. Stipulate that, for some object X, `X is a
heap' is true if and only if the sentence is true for all sharpenings of `heap',
and false if and only if it is false for all sharpenings.
It follows immediately that for an object within the penumbral region for the
application of a predicate F, it is, on the supervaluationist
scheme neither true that that object is F, not false that that object is F. In other words, the statement `That object is
F’ is neither true nor false. In
classical semantics, every statement is either true or false, so if we accept supervaluationism, we must abandon classical
semantics. Also, on the supervaluational account, the disjunctive statement `The
book is green or it is not green’, said of a book borderline between green and
blue’ receives the evaluation `true’ on the supervaluational
account, even though neither disjunct is true. Thus, the classical semantics for `or’ is
also rejected. We can show that a
statement like `b is F or it is not the case that b is F’ comes out true
on all precisifications. So the Law of Excluded Middle, and, in fact,
other laws of classical logic are preserved by supervaluationism
– but note the reservations expressed by Keefe and Smith (pp.29-32) regarding
the claim that supervaluationism retains classical
logic in toto.
7.
Now
take the conditional premise of the Sorites: If Xi
is F then Xi+1 is F. On the supervaluational account, this is false, for there will be some sharpening F* of F for
which, given an Xi within the penumbral
region, the antecedent is true and the consequent false. Sainsbury's main criticism of this approach
is that it has as a consequence that there is
some number n such that an n-grained collection is a heap, while an n-1 grained
collection is not; and this simply defies our intuitions which give rise to the
paradox in the first place. For further
criticisms of supervaluationism, see Keefe and Smith,
pp.32 –35, and note that `D’ is short for `definitely true’, and is defined by
Kit Fine is his seminal paper reprinted in Keefe and Smith, pp.119-150 (but do
not attempt this paper unless you are very confident of your ability in
logic). Note also that Keefe defends a
version of supervaluationism in Chaps. 7 and 8 of her excellent book.
8.
Before
we meet this Thursday, try to fully understand the supervaluational
theory. A quite
different solution – the `degrees of truth’ theory is contained in the next
section (2.6) of Sainsbury and, if you have time, read that. In brief, the `degrees of truth' solution is
as follows: where a and ß are adjacent items and a is more V than ß, then the degree of truth of `a is V' is higher than that of `ß is V', and the degree of truth
of the conditional `If a is V, then ß is V' is close to, but less
than 1. Since a Sorites
argument is a series of modus ponens, the cumulative
effect is non-validity. For a
sophisticated defence of a version of the `degrees’
theory, see Dorothy Edgington in Keefe and Smith, pp.294-316.