Introduction

Zeno the Greek lived in Elea (a town in what is now southern Italy) in the fifth century BC. The paradox for which he is best known today concerns the great warrior Achilles and a previously unknown tortoise. For some reason now lost in the folds of time, a race was arranged between them. Since Achilles could run much faster than the tortoise, the tortoise was given a head start. Zeno's astonishing contribution is a “proof” that Achilles could never catch up with the tortoise no matter how fast he ran and no matter how long the race went on.

The supposed proof goes like this. The first thing Achilles has to do is to get to the place from which the tortoise started. The tortoise, although slow, is unflagging: while Achilles is occupied in making up his handicap, the tortoise advances a little bit further. So the next thing Achilles has to do is to get to the new place the tortoise occupies. While he is doing this, the tortoise will have gone on a little bit further still. However small the gap that remains, it will take Achilles some time to cross it, and in that time the tortoise will have created another gap. So however fast Achilles runs, all the tortoise need do in order not to be beaten is keep going – to make some progress in the time it takes Achilles to close the previous gap between them.

In previous chapters, I have at many points argued that if something leads to a contradiction, then either it, or the relevant reasoning, must be rejected. The assumption that one must always reject contradictions has come under attack at various times in the history of philosophy. Recently, the attack has taken a subtle form and has harnessed impressive technical resources.

I shall discuss the following views:

1. (1) Some contradictions are true.

2. (2) For some contradictions, it is rational to believe that they are true.

The only version of (1) that I shall consider also holds that every contradiction is false; this is “dialetheism.” However, the view that some contradictions are both true and false does not add up to the view that some contradictions are acceptable, for one might go on to insist that anything perceived to be false should be rejected. If so, the assumption of the previous chapters of this book would remain unchallenged: we should reject anything which leads to a contradiction. So in this chapter I will discuss the conjunction of views (1) and (2). I shall call this combination “rational dialetheism.”

The rational dialetheist's claim that some contradictions are true is, under natural assumptions, equivalent to the claim that some sentences are both true and false. Any such sentence may be called a dialetheia.

Sorites paradoxes: preliminaries

Suppose two people differ in height by one-tenth of an inch (0.1″). We are inclined to believe that either both or neither are tall. If one is 6′ 6″ and the other is 0.1″ shorter than this, then both are tall. If one is 4′ 6″ and the other is 0.1″ taller, then neither is tall. This apparently obvious and uncontroversial supposition appears to lead to the paradoxical conclusion that everyone is tall. Consider a series of heights starting with 6′ 6″ and descending by steps of 0.1″. A person of 6′ 6″ is tall. By our supposition, so must be a person of 6′ 5.9″. However, if a person of this height is tall, so must a person one-tenth of an inch smaller; and so on, without limit, until we find ourselves forced to say, absurdly, that a 4′ 6″ person is tall (see question 3.1), indeed, that everyone is tall (see question 3.2).

In ancient times, a similar paradox was told in terms of a heap, and a Greek word for “heap” – soros – has given rise to the use of the word “sorites” for all paradoxes of this kind. Suppose you have a heap of sand.

Remark numbers refer to question numbers in the given chapter.

ZENO'S PARADOXES: SPACE, TIME, AND MOTION

1. 1.2 One possible argument is this. If there were as many as two things, say α and β, then we could consider the whole W formed by these two things. Then W has α and β as its parts. So if nothing has parts, there are not as many as two things, i.e. there is at most one thing.

2. 1.6 Yes, it does mean that the button will travel faster than the speed of light. Whether or not this is a logical possibility could be disputed, but it is fairly uncontroversial that it is not impossible apriori; that is, reasoning alone, unaided by experiment, cannot establish that nothing can travel faster than light.

3. 1.8 “Going out of existence at Z★” might mean “Z★ was the last point occupied” or, alternatively, “Z★ was the first point not occupied.” The latter serves Benacerraf's cause against the objection.

VAGUENESS: THE PARADOX OF THE HEAP

1. 3.11 He should deny the first premise. Since he believes that there are no heaps, he will think that it is false that a 10,000-grained collection is a heap. This shows that one should treat with care the taxonomy of responses described in the text. Unger's position, though in a sense unified, must be regarded as accepting the conclusion of some soritical arguments and denying the premises of others.

ACTING RATIONALLY

1. 4.3 If a person's utilities can be measured in cash value terms, then his or her utilities are “commensurable”: of any two possible circumstances, either one has more utility than the other, or else they are of equal utility. However, if we think of very disparate “utilities,” it may be not merely that we do not know how they compare, but that there is no such thing as how they compare. For an early defense of commensurability see Rashdall (1907, ch. 2). For a more recent discussion see Nussbaum (1986, esp. pp. 107ff.).

2. […]

The main change in this edition is the addition of a new chapter on moral paradoxes, which I was inspired to write by reading Smilansky's excellent book Ten Moral Paradoxes. To Saul Smilansky I owe thanks for encouragement and comments. The new chapter is numbered 2 and subsequent chapters are renumbered accordingly. I placed the new chapter early in the book in the belief that the discussion is more straightforward than just about any of the other chapters.

I have made some small changes elsewhere, notably to the chapter on vagueness (now chapter 3) and to the suggested reading. Since the second edition appeared in 1995, the internet has transformed many aspects of our life. There are now many websites which help people doing philosophy at every level. The Stanford Encyclopedia of Philosophy (plato.stanford.edu) should be the first place to turn if your curiosity has been aroused by this text. Someone with internet access can now do serious research in philosophy, even without the advantage of a university library.

My thanks to Daniel Hill for many useful suggestions for this edition.

Crime Reduction

Suppose that crimes of a certain category (e.g. car-jacking) are completely eliminated by prescribing an extraordinarily severe penalty (e.g. death). The penalty is so severe that it is 100 percent effective as a deterrent: car-jacking (or whatever crime we consider) never occurs, and so is never punished (so the prescribed severe penalties are never in fact imposed). It seems that we are forced to make conflicting judgments about this imaginary situation:

Good: A crime has been eliminated. There are no bad side-effects: no car-jackers are executed (which might indeed be unjust), for there are no car-jackers.

Bad: A crime has been associated with a punishment of unjust severity. This makes for an unjust society. Even if injustice is a means to a good end (crime reduction) it is still unjust, and should be condemned.

Both views are apparently reasonable; but as they conflict, it appears we cannot hold both.

It may be objected that the imaginary situation is unrealistic, and so need not be taken seriously: we cannot be expected to have consistent judgments about such situations. England tried using capital punishment for petty theft in the seventeenth century, and the deterrent effect was far from complete. Many crimes are committed on impulse, under the influence of alcohol or drugs, out of desperation, or in the false belief that they will go unpunished, and are thus isolated from the threat of penalties. We simply could not eliminate car-jacking by prescribing capital punishment for this crime.

The paradoxes to be discussed in this chapter are probably the hardest of all, but also the most fecund. Russell's paradox about classes, which he discovered in 1901, led to an enormous amount of work in the foundations of mathematics. Russell thought that this paradox was of a kind with the paradox of the Liar, which in its simplest form consists in the assertion “I am now (hereby!) lying.” The Liar paradox has been of the utmost importance in theories of truth. Everything to do with these paradoxes is highly controversial, including whether Russell was right in thinking that his paradox about classes and the Liar paradox spring from the same source (see section 6.9).

Russell's paradox

If Socrates is a man, then he is a member of the class of men. If he is a member of the class of men, then he is a man. Can classes be members of classes? The answer would seem to be Yes. The class of men has more than 100 members, so the class of men is a member of the class of classes with more than 100 members. By contrast, the class of the Muses does not belong to the class of classes having more than 100 members, for tradition has it that the class of the Muses has just nine members.

Newcomb's paradox

You are confronted with a choice. There are two boxes before you, A and B. You may either open both boxes, or else just open B. You may keep what is inside any box you open, but you may not keep what is inside any box you do not open. The background is this.

A very powerful being, who has been invariably accurate in his predictions about your behavior in the past, has already acted in the following way:

The paradox consists in the fact that there appears to be a decisive argument for the view that the most rational thing to do is to open both boxes; and also a decisive argument for the view that the most rational thing to do is to open just box B. The arguments commend incompatible courses of action: if you take both boxes, you cannot also take just box B. Putting the arguments together entails the overall conclusion that taking both boxes is the most rational thing and also not the most rational thing. This is unacceptable, yet the arguments from which it derives are apparently acceptable.

This chapter concerns problems about knowledge or rational belief. The first main section, called “Paradoxes of confirmation,” is about two paradoxes that might be called “philosophers' paradoxes.” Let me explain.

Most of the paradoxes in this book are quite straightforward to state. Seeing what is paradoxical about them does not require any special knowledge – you do not have to be a games theorist or a statistician to see what is paradoxical about Newcomb's paradox or the Prisoner's Dilemma, nor do you have to be a physicist or sportsman to see what is paradoxical about Zeno's paradoxes. By contrast, the paradoxes of confirmation arise, and can only be understood, in the context of a specifically philosophical project. Therefore these paradoxes need some background (section 5.1.1) before being introduced (in sections 5.1.2 and 5.1.3). The background section sets out the nature of the project within which the paradoxes arise.

The last three main sections of the chapter (5.2–5.4) concern the paradox of the Unexpected Examination. Although it is hard to resolve, one form of it is easy enough to state. (More complex forms, discussed in sections 5.3–5.4, involve technicalities: these sections can be omitted without loss of continuity.) This paradox has been used to cast doubt on intuitively natural principles about rational belief and knowledge.

Paradoxes of confirmation

Background

We all believe that there is a firm distinction between strong, good, or reliable evidence on the one hand, and weak, bad, or unreliable evidence on the other.

Paradoxes are fun. In most cases, they are easy to state and immediately provoke one into trying to “solve” them.

One of the hardest paradoxes to handle is also one of the easiest to state: the Liar paradox. One version of it asks you to consider the man who simply says, “What I am now saying is false.” Is what he says true or false? The problem is that if he speaks truly, he is truly saying that what he says is false, so he is speaking falsely; but if he is speaking falsely, then, since this is just what he says he is doing, he must be speaking truly. So if what he says is false, it is true; and if it is true, it is false. This paradox is said to have “tormented many ancient logicians and caused the premature death of at least one of them, Philetas of Cos.” Fun can go too far.

Paradoxes are serious. Unlike party puzzles and teasers, which are also fun, paradoxes raise serious problems. Historically, they are associated with crises in thought and with revolutionary advances. To grapple with them is not merely to engage in an intellectual game, but is to come to grips with key issues. In this book, I report some famous (and some less famous) paradoxes and indicate how one might respond to them. These responses lead into some rather deep waters.

(An asterisk before a title indicates that there is an observation on the entry in Appendix II.)

THE GALLOWS

The law of a certain land is that all who wish to enter the city are asked to state their business there. Those who reply truly are allowed to enter and depart in peace. Those who reply falsely are hanged. What should happen to the traveler who, when asked his business, replies, “I have come to be hanged”?

٭BURIDAN'S EIGHTH SOPHISM

Socrates in Troy says, “What Plato is now saying in Athens is false.” At the same time, Plato in Athens says, “What Socrates is now saying in Troy is false.” (Cf. Buridan, in Hughes 1982, pp. 73–9).

THE LAWYER

Protagoras, teacher of lawyers, has this contract with pupils: “Pay me a fee if and only if you win your first case.” One of his pupils, Euathlus, sues him for free tuition, arguing as follows: “If I win the case, then I win free tuition, as that is what I am suing for. If I lose, then my tuition is free anyway, since this is my first case.”

Protagoras, in court, responds as follows: “If you give judgment for Euathlus, then he will owe me a fee, since it is his first case and that was our agreement; if you give judgment for me, then he will owe me a fee, since that is the content of the judgment.

ACTING RATIONALLY

Newcomb's Paradox

You are confronted with a choice. There are two boxes before you, A and B. You may either open both boxes, or else just open B. You may keep what is inside any box you open, but you may not keep what is inside any box you do not open. The background is this.

A very powerful being, who has been invariably accurate in his predictions about your behavior in the past, has already acted in the following way:

1. He has put $1,000 in box A.

2. If he has predicted that you will open just box B, he has in addition put $1,000,000 in box B.

3. If he has predicted that you will open both boxes, he has put nothing in box B.

The paradox consists in the fact that there appears to be a decisive argument for the view that the most rational thing to do is to open both boxes; and also a decisive argument for the view that the most rational thing to do is to open just box B. The arguments commend incompatible courses of action: If you take both boxes, you cannot also take just box B. Putting the arguments together entails the overall conclusion that taking both boxes is the most rational thing and also not the most rational thing. This is unacceptable, yet the arguments from which it derives are apparently acceptable.

Logic ought to guide our thinking. It is better, more rational, more intelligent to think logically than to think illogically. Illogical thought leads to bad judgment and error. In any case, if logic had no role to play as a guide to thought, why should we bother with it?

The somewhat naive opinions of the previous paragraph are subject to attack from many sides. It may be objected that an activity does not count as thinking at all unless it is at least minimally logical, so logic is constitutive of thought rather than a guide to it. Or it may be objected that whereas logic describes a system of timeless relations between propositions, thinking is a dynamic process involving revisions, and so could not use a merely static guide. Or again the objection may be that there is no such thing as logic, only a whole variety of different logics, not all of which could possibly be good guides.

I aim to disarm the last two objections to the initial idea that logic should be a guide to thought.

Logic and belief revision

How could logic guide our thought? Anything we can believe has an infinite number of logical consequences, but no sensible guide would tell us to take steps to believe all the logical consequences of anything we believe, for there is not enough time to obey and most of the added beliefs would be trivial.

There is something to the idea that we should consider our beliefs not just one by one but in larger groups, to see if we can logically extract some useful further information.