**Lecture 4: Shallow and Deep Paradoxes**

Today, I want to show that there are some interesting connections
between hard (or `deep’) paradoxes and some simpler ones. There is something encouraging about this –
perhaps some of the harder paradoxes we shall deal with in this course can be
reduced to simpler, more manageable ones.
But be cautious! Between any two
things X and Y there will be at least *one* similarity. But this does not entail that, if we have
found a solution to some problem about X, we shall automatically have a
solution to a problem about Y. We first
need to verify that the similarity between X and Y is *very *close, and
that the problem about Y is *very *similar to the problem about X

Here is a shallow paradox:
Consider a man who goes into a bookstore and asks the saleswoman `Where is the
self-help section?’ The saleswoman
replies that it would defeat his purpose if she revealed the answer. This is funny rather than deep because the
man was clearly interested in books about do-it-yourself or about medical
self-diagnosis, and not about books on finding one’s way around a new
bookstore. It would hardly be fair to
call so silly a story a paradox. However,
with a little tweaking of the parameters, we can get a problem that begins to
look genuinely paradoxical. Consider,
for example, a woman going into a psychiatrist and, to the psychiatrist’s
traditional enquiry `How can I help?’, she replies `The only help you can give
me is not to give me any help’. Here, if
the psychiatrist complies with her wishes and gives her no help then, in so
doing, he is giving her help of just the sort she mentions. On the other hand if he gives her help then,
if she is right, that is no help at all.
It seems that the psychiatrist can take one of two courses of action –
either giving or not giving help – but either way, he seems to end up,
impossibly, both helping and not helping the woman. We can extricate ourselves from this
difficulty, however, by seeing the argument as a *reductio ad absurdum*, a proof that the woman’s reply to the
psychiatrist was simply false. So, if
the puzzle is called a paradox, it too belongs at the shallow end. In each of the following examples, we move
from a shallow paradox to a deep one.
But always ask yourself the question: `Is there any important feature
present un the deep paradox that was not present in the shallow one? If your answer is `No’, then try solving the
simpler (`shallow’) paradox and see whether your solution carries over to the
deep one.

I have already
pointed out that there is a structural similarity between these two
paradoxes. A story is told about the *shrinking*. No barber could exist who both shaves and
does not shave himself and, only slightly less obviously, no barber can exist
who shaves all and only those villagers who do not shave themselves. For the bottom line will be that he shaves
himself if and only if he does not.
Thus, the answer here, as in the case of the psychiatrist’s patient we
considered earlier, is that the starting assumption is false. That assumption was that there *is* a
barber conforming to the stated description. By *reductio*,
there is no such barber. In other words,
we specify no barber by the condition `he who shaves all and only those who do
not shave themselves’. This is generally
accepted, hence it is now generally accepted that there is no deep paradox
about the eponymous Barber.

Now, compare the Barber with the Russell Paradox. They have, as we mentioned, a common stucture – they are of the same form. This can be seen by comparing the specification for the Russell Set with that of the Barber:

x is a member of R if and only if x is not a member of x

x is shaved by B if and only if x is not shaved by x

So just as we are happy to say
that there is no barber specified in this way, should we not be equally happy
to say that there is no Russell Set? The
answer is `No’, or, at least `Not immediately’ because we can see absolutely no
reason for denying that all the non-self-membered sets can be assembled into a
set. Sets, unlike barbers, are not
subject to the contingencies of physical existence. The point is clearly made by Sainsbury,
pp.108-9, and also by Alex Oliver: `the outstanding issue has not been
resolved, namely whether there is anything in our understanding of the concept *set
*which leads us to expect what contradiction shows us cannot happen. Russell’s paradox, for example, *was* a
paradox since it overturned the belief, essential to the naďve concept of *set*,
that every predicate (or concept) has an extension’ (`Hazy Totalities and
Indefinitely Extensible Concepts: An Exercise in the Interpretation of Dummett’s Philosophy of Mathematics’, *Grazer
Philosophische Studien* 55 (1998): 25-50, p.41).

*From Catch-22 to the Liar*

When we looked hard at the
Barber, we observed that what, at first sight, was a description of a barber,
turned out to be a contradiction which describes or specifies nothing. In Joseph Heller’s novel *Catch-22*, there is a clause that seems to specify the conditions
under which an airman can be excused combat duty. But there is a catch – it is a condition that
cannot be satisfied:

‘You mean there’s a catch?’

‘Sure there’s a catch,’ Doc Daneeka replied. ‘Catch-22. Anyone who wants to get out of combat duty isn’t really crazy.’

There was only one catch and that was Catch-22, which specified
that a concern for one’s own safety in the face of dangers that were real and
immediate was the process of a rational mind.
Orr was crazy and could be grounded.
All he had to do was ask; and as soon as he did, he would no longer be
crazy and would have to fly more missions. Orr would be crazy to fly more
missions and sane if he didn’t, but if he was sane he had to fly them. If he flew them he was crazy and didn’t have
to; but if he didn’t want to he was sane and had to. Yossarian was moved very deeply by the
absolute simplicity of this clause of Catch-22 and let out a respectful
whistle.

‘That’s some catch, that Catch-22,’ he observed.

‘It’s the best there is,’ Doc Daneeka agreed.

It looks as if an airman can get out of flying dangerous missions on condition that he is insane, for we have

1. Anyone can avoid flying missions if and only if he is insane

All you need
do is to establish your insanity. Now,
it defines you as being insane if you *don’t*
ask to be spared flying missions:

2. Anyone is insane if and only if he does not request to be taken off missions.

But you cannot be spared flying missions unless you request it:

3. Anyone who does not request it cannot avoid flying missions

Now, 1.,2. and 3. jointly entail

4. Anyone can avoid flying missions if and only if he cannot avoid flying missions

In symbols:

1* (x)(Ax ↔ Ix)

2* (x)(Ix ↔ ~Rx)

3* (x)(~Rx ↔ ~Ax)

entail

4* (x)(Ax ↔ ~Ax)

So we end up not with the condition one has to meet in order to avoid flying missions, but merely with a contradiction which specifies no condition at all. Notice that this is not the same as a condition that cannot be satisfied, such as `You can avoid flying missions if and only if you can trisect an arbitrary angle using only straightedge and compass’; it just does not amount to the expression of any condition at all.

*Protagoras and Euathlus*

The ancient paradox of Protagoras
and Euathlus turns out, perhaps surprisingly, to be related to Catch-22. The
situation here is that Protagoras, the father of Sophistry, puts his pupil
Euathlus through a training in law, and agrees not to be paid any fee for the
instruction until Euathlus wins his first case. Euathlus, completes the course
of instruction, but then, indolently, takes no cases. Eventually Protagoras
gets frustrated at not being paid, and sues him. So Euathlus’s first case is
this one — defending himself against Protagoras’ suit. If Euathlus loses the
case then, by the agreement he made with Protagoras, he does not have to pay
him (for he has to pay only after his first *win*). However, if Euathlus
wins, that means that Protagoras loses his suit to be paid; in other words,
Euathlus does not have to pay him. It seems that Protagoras cannot recover his
fee. On the other hand, it seems that Protagoras must recover his fee for, if
he wins the suit, the court will order in his favour, but if he loses — i.e.,
if Euathlus wins — then, by the terms of their agreement, he gets paid. This
paradox is somewhat simpler than Catch-22. For here there is a tension between
just two conditions — the one generously agreed to by Protagoras, that he gets
paid if and only if Euathlus wins:

2. ~P ↔ ~W

and the penalty code of the court which, in this particular case, enjoins

3. P ↔ ~W

(where `W’ stands for `Euathlus wins’ and `P’ for `Protagoras gets paid’). These two conditions entail

4. P ↔ ~P

The solution is, I think, that we
cannot infer that Protagoras can or that he cannot recover his fee. The case
could be decided either by the court’s rule or by Protagoras’ rule. But, since
these rules are in conflict, it cannot be decided by both together. In the same
way, a football match could not get started were it bound by both rules `The
side winning the toss kicks off’ and `The side that loses the toss kicks off’.
Note again our departure from classical principles, for, in classical logic,
from `p ↔ ~p’, *everything*
can be inferred.

*The Liar*

In order to
make the transition to the Liar Paradox, consider a statement `S is not true’,
where `S’ is the name of that very statement.
So here we have a statement that says of itself that it is not true. What would things have to be like for S to be
true? Well, consider that question
raised about a non-problematic statement like `On

A is true if and only if on

I have just given what are called the *truth-conditions*
for statement A. But now, if we employ
the same technique for giving the truth-conditions for S we get:

S is true if and only if S is not true.

Sound
familiar? It’s like Catch-22 all over
again. And just as, in that case, no
condition was specified for avoiding flying missions, so here no *statement* is specified – there just is
no statement S which could be both true and not true. We can prove this in a slightly more
convoluted way: Could `S’ be the name of
the statement `S is not true’? If we
assume that `S’ names a *true *statement,
then it obviously cannot be the name of the statement `S is not true’, for the
latter would (on the covering assumption) be *false*. On the other hand, if
we assume that `S’ names a *false*
statement then it obviously cannot be the name of the statement `S is not
true’, for the latter would (on the covering assumption) be *true.*
So `S’ cannot be the name of the statement that S is not true – in other
words, there can be no statement that says of itself that it is not true.

*From the Better Lover to MINIAC*

When two men are sharing one
woman, each man wants to know whether or not he is the better lover. He can ask, but there’s no guarantee that he
will get the right answer, since the woman may be a liar. Is there a single question that a man can ask
so that he can find out whether he is or is not the better lover, even though
the woman may be a habitual liar? The
answer is `Yes’ – he must ask the rather complicated question `When the other
guy asked you whether I was the better lover, was your answer to him
`Yes’?’ If the woman is a truth-teller
and answers `Yes’, this means that her answer to the other guy was `yes’, which
means that the questioner (call him Sam) is the better lover. Suppose, however that the woman is a habitual
liar. If she answers `Yes’, that means
that her answer to the other guy was `No’ and, since that too was a lie, the
truth is that Sam is the better lover.
So, whether the woman is a truth-teller or an habitual liar, her answer
of `Yes’ to Sam’s question establishes that Sam is the better lover. Likewise, if her answer to Sam’s question is
`No’, then, whether the woman is a truth-teller or an habitual liar, it follows
that Sam is not the better lover. This
answer is correct – there is no paradox.
Now, there *appears* to be a close connection between this non-paradox
and the paradox of 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.)

One difference, though, that
may be important, is that, in the *Better Lover* we make the assumption
that the woman *always* tells the truth or *always *lies. No such assumption is made in MINIAC.

**The next block of lectures will deal with the Sorites Paradox. In preparation for the lectures, please read
the section on the Paradox of the Heap in the course text, pp.69-76 and look at
Sainsbury, Chap. 2. Other useful
preliminary reading is:**

R.M. Sainsbury and Timothy Williamson, `Sorites’ in R. Hale and C.
Wright (eds), *A Companion to the Philosophy of Language *(Oxford,
Blackwell, 1997), pp.458-484.

**I shall also distribute a list of about 20 suggested essay topics from
which you choose one. But you are not
confined to that list – if there is some paradox that interests you in
particular, and you want to write on it,
then let me know and I will suggest to you some readings.**