Phil 2511: Paradoxes

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.

From The Barber to Russell

I have already pointed out that there is a structural similarity between these two paradoxes. A story is told about the village of Alcala and the barber in that village who shaves all and only those who do not shave themselves. Does this barber shave himself? My mental image of Alcala is of a sleepy settlement in which at least twenty adult males dwell, most too listless to shave themselves. Now imagine that bubonic plague takes the lives of all the male inhabitants, sparing only the barber. Well, he shaves exactly the persons who don’t shave themselves, and he himself is the only person surviving who needs a shave. So he shaves himself if and only if he doesn’t shave himself ……clearly nobody could perform such a feat. This illustrates a manoeuvre which is a very useful one for dealing with paradoxes: reduce to the simplest case, ensuring that nothing vital is lost in the reduction. We cut out the `noise’ – in this case the other male villagers. We can imagine their number shrinking to two, one or zero and the shape of the problem remains the same. I’ll call this technique 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 naïve 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 Thursday, February 5th, 2004, Laurence is lecturing in M167’. Call that statement `A’. The answer to the question of how things have to be like for A to be true is simple:

A is true if and only if on Thursday, February 5th, 2004, Laurence is lecturing in M167.

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.

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.

Laurence Goldstein, `How to Boil a Live Frog’, Analysis 60 (2000), pp.170—178.

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.