Figure 1
Boyle’s self-flowing flask
Image taken from Wikimedia commons. Also on http://www.lhup .edu/~dsimanek/museum/people/people.htm.
Introduction: Is There Trouble in Paradox?
In a BBC television series aptly called Paradox, a British astrophysicist claims to have images of a future explosion in which many people are going to die. After seeing these images of the future, poor harried detective Rebecca Flint must try to prevent this vision from coming to be. But if Flint is successful, then wouldn’t the images of the “future” thus be false? If all the destruction predicted in the images of the future never came to be, even if it were thanks to Flint, in what sense could these images be thought reliable? The puzzling nature of time has given rise to many philosophical paradoxes. In addition to problems like Flint’s, which are about altering the future based on our foreknowledge of it, there are also problems associated with the past, such as the grandfather paradox: the paradox associated with the question of whether it is conceptually possible to go back in time and kill your own grandfather before he met your grandmother. If it is indeed possible, then because you would not have existed in this scenario, it seems to be impossible. Because one of your parents would not have been born, you would not have been born, and hence you could not go back in time to kill your grandfather before he met your grandmother.
On a very broad construal, a paradox can be anything from a tough problem or a counterintuitive opinion or conclusion to a visual sleight of hand. An Internet search on the word paradox, for example, turns up the intricate and surprisingly beautiful prints of M. C. Escher; a picture of a glass ashtray with a “no smoking” symbol imprinted on it; a picture of a self-flowing flask, attributed to Robert Boyle, that is constantly refilling itself with the water that pours out from its base (figure 1); and a Wikipedia article listing more than two hundred paradoxes, including the grandfather paradox mentioned earlier. The paradoxes listed come from such diverse areas as statistics, thermodynamics, economics, biology, and logic. What, then, makes them all paradoxes?
Figure 1
Boyle’s self-flowing flask
Image taken from Wikimedia commons. Also on http://www.lhup .edu/~dsimanek/museum/people/people.htm.
Philosophers are by no means in complete agreement about the correct way to define paradox, but each of the prominent definitions points to an important feature of paradox. One common definition (Rescher 2001) holds that a paradox is a set of mutually inconsistent propositions, each of which seems true. Consider Flint’s problem of trying to prevent an event in the future from happening. There are a number of propositions—let’s call them statements—associated with her situation. First, assume that the images of the future explosion are reliable predictors of what will happen. Second, if this is the case, then nothing Flint does would seem to be able to alter what would happen, because we’ve assumed that they are correct. But, third, Flint seems to have the freedom to act in ways that would prevent the future explosion from happening. So, fourth, if Flint does succeed in preventing the explosion from happening, then the images of the future explosion were not accurate. Notice that the fourth statement contradicts the first one, which was taken as an assumption. If the pictures are reliable, then nothing can be done about the explosion. But because Flint is free to act as she chooses, she can do things that would prevent the explosion. However, if this is the case, then the pictures did not accurately predict the future. Each statement, in and of itself, looks acceptable, yet when put together, we have a contradiction:
Box 1
Flint’s paradox
1. The pictures of the future explosion are accurate (assumption).
2. Nothing Flint can do can prevent the explosion (follows from 1).
3. Flint has the freedom to behave in ways that would prevent the explosion.
4. The pictures of the future explosion may turn out to not be accurate (from 3; contradicts 1).
This example illustrates that paradoxes involve some type of contradiction among claims that, at least on the surface, have nothing wrong with them. Perhaps this is why an Internet search for paradox turns up a picture of an ashtray with a “no smoking” symbol inscribed on it. Individually, the ashtray and the symbol are perfectly common items in our environment. Yet by putting them together in one object, a tension arises between an object that was created with the idea that smoking would happen and a sign that prohibits smoking from happening. In both the ashtray and in the grandfather paradox, the inconsistency is highlighted, along with the fact that no one member of the inconsistent set of assumptions is obviously wrong. An inconsistency among seemingly innocuous elements is thus central to the idea of paradox.
Other definitions highlight the reasoning involved in paradoxes. For example, some claim that a paradox is an argument with seemingly valid reasoning and true premises, but an obviously false conclusion (Mackie 1973) and still others claim that paradoxes are unacceptable conclusions drawn from seemingly true premises and correct reasoning (Sainsbury 2009). Arguments are pieces of reasoning in which one claim (the conclusion) is supported by other claims (the premises). When the reasoning is correct, true premises will always lead to true conclusions. But, in the case of paradoxes, it seems that something has gone wrong, in that true premises and correct reasoning lead to an obviously false or contradictory conclusion. In the Flint case, for example, we assumed that images of the future explosion were correct, but then concluded that if she had prevented the explosion from happening, then the images could not have been correct. So, we’ve concluded something that contradicts what we took as a given. Also, consider the sorites paradox, an early and famous paradox about vagueness that shows, it seems, that when there are no sharp boundaries between concepts such as bald and non-bald or rich and not rich, we can conclude some obviously false things. The sorites can be put in the form of the following argument:
Box 2
Sorites paradox
1. A person with 0 hairs is bald.
2. For any number n, if a person with n hairs is bald, then a person with (n + 1) hairs is bald.
3. Therefore, a person with 1,000,000 hairs is bald.
In the sorites paradox, (1) and (2) are premises and (3) is the conclusion of the argument. The first premise, which claims that a person with zero hairs is bald, describes the paradigm case of baldness. Such a premise looks obviously true, because if any person were to be bald, the person with the fewest possible hairs (0) would be. The second premise, though perhaps difficult to read at first, is very intuitive as well. It claims that the difference of one hair is not enough to warrant the change in classification from being bald to being non-bald. If you add one hair to any person’s head, in other words, it won’t change whether that person is bald. Given that the difference of one hair is hardly noticeable by the human eye, it is hard to imagine how any kind of principled distinction between baldness and non-baldness could be made based on one hair. With regard to its reasoning, the sorites is straightforward. The first premise claims that a person with a specific number of hairs (0) is bald. The second premise makes a claim about all numbers of hairs, saying that for any arbitrary number, one more would not make enough of a difference to warrant a change in classification from someone being bald to non-bald. The number used in the first premise (0) is plugged into the generalization in the second premise, repeatedly, to get the conclusion that a person with a million hairs is bald. So, phrased in this way, the sorites paradox is an argument with intuitively plausible premises, apparently correct reasoning, and an obviously false or contradictory conclusion. It can also be thought of, using Mark Sainsbury’s definition, as an unacceptable conclusion (“A person with 1,000,000 hairs is bald”) that is derived from apparently acceptable premises (“A person with 0 hairs is bald,” and “For any number n, if a person with n hairs is bald, a person with (n + 1) hairs is bald”), using apparently correct reasoning. These two definitions therefore highlight another important feature of paradoxes,1 namely, how reasoning from seemingly fine premises, using good reasoning, sometimes turns up unexpectedly strange conclusions.
Paradoxes expose some kind of trouble with our reasoning, or the statements we take as premises, or the basic concepts that underlie the paradox in question. In Flint’s case, the trouble the paradox exposes is whether foreknowledge of the future (that the images of the future explosion gave) means that the future is already predetermined. In the sorites paradox, it is troubling that vague terms like bald, strong, and rich have to apply to some things and not others, but saying precisely how few hairs makes someone bald, how much weight lifted makes something strong, or how many pennies one must have to count as rich doesn’t make much sense. We have to admit that there is a difference between being bald and not bald, but no specific number of hairs would be a good boundary to mark that difference. And the same goes for weight and strength, and pennies and wealth. It is this type of trouble that leads us to want to figure out what has gone awry in the paradox.
Presenting solutions to paradoxes, generating new paradoxes, and criticizing proposed solutions to paradoxes are all part of the workaday life of philosophers, theoretical physicists, economists, and other theoreticians. From the very brief introduction to the Flint, the grandfather, and the sorites paradoxes, possible solutions may have begun spontaneously emerging in your mind. For example, in the case of Flint’s paradox, you might think it shows that there can be no correct images of the future. Or perhaps you thought that the paradox showed that the future must be predetermined, and there was nothing Flint could do to stop the explosion. Or both. And perhaps you started thinking that the grandfather paradox shows that there can be no time travel.
Proposing solutions is a natural response to paradoxes and is probably as old as philosophy itself. Very early in the Western philosophical tradition, Aristotle—the philosopher often called the “father” of systematic logic—studied and attempted to solve paradoxes. For him, paradoxes were flawed arguments, and to solve a paradox was to point to the flaw in the argument (Kneale and Kneale 1962). Some of his main targets were the paradoxes of Zeno of Elea, whose arguments on space, time, and motion are thought to be the earliest paradoxes in the Western philosophical tradition. And the generation of paradoxes and solutions continues until the present day. As shown in the final chapter of this book, the process of paradox generation and solution proposal is an interesting and important one. New sciences often stem from attempts to solve paradoxes, and the concepts used in the new sciences lead to further paradoxes.
One common misconception that I hope will be shown to be mistaken is that paradoxes are puzzles that—although they are interesting—remain removed from everyday life. Nothing could be further from the truth. Paradoxes emerge in everyday sources, in the newspapers, in religious texts, in conversations, and in practical dilemmas that must be faced in one’s life. To give an example, during the writing of this book, an article in the Wall Street Journal discussed Santon Bridge, a small English town that hosts an annual World’s Biggest Liar Competition (MacDonald 2011). The town must be awash in paradox.
Also, it will soon become apparent that the solutions to paradoxes have implications in other aspects of our lives. Fuzzy logic, for example, with its use of degrees of truth, underlies such mundane but useful things as signature recognition programs. Without a method of dealing with the borderline cases between, say, a handwritten a and a handwritten u, such programs would not be possible. And without decision theory, public policy would be handicapped. Solutions and their paradoxes, then, are important parts of the world in which we live.
This idea was brought home to me when I was in the British Library about ten years ago and found a dusty volume that contained the Conway Memorial Lecture given by J. C. Flugel to a London ethical society in 1941 at the height of World War II. The address began, “This is the sixth Conway Memorial Lecture (out of a total series of thirty-two) to be delivered during war; and since a year ago . . . war has come very appreciably nearer to our doors, so near indeed, that we may count ourselves fortunate if Conway Hall still stands, and we ourselves still free to meet within it; for we know that at any moment actual combat, with its noisy and destructive clamor, may break out around us, threatening our lives, the lives of those who are near and dear to us, and the works and monuments of those who lived here before us and established the traditions that we are seeking to maintain” (Flugel 1941, 1). Upon reading this, I thought, “Why on Earth were these people gathered in downtown London to listen to a lecture while bombs might fall down upon them?” The lecture must have been about something very important to the members of the audience at that time. Flugel continued, “At such times it is difficult or impossible to divert our thoughts for long from the tremendous conflict going on around us, and I have made no attempt to do so in this lecture” (1).
The subject that had brought out the attendees and speaker was the paradoxical nature of war—in particular, how something as obviously immoral as war could induce in the belligerent groups such impressive moral qualities as self-sacrifice and generosity toward one’s fellows. In the lecture, which was titled “The Moral Paradox of Peace and War,” Flugel questioned why war, though terribly destructive, induces a higher moral state within particular groups. This issue was a paradox that those in attendance were experiencing directly, and one that must have inspired the audience to attend a lecture at such a time. A number of responses to the paradox suggest themselves, such as that the supposed conflict between war being a horrible thing and it being the inspiration for benevolent and even heroic acts is nonexistent. Both can exist without much conflict at all. This is what I describe later as an It’s-All-Good response. But the point brought home by those people gathered at Conway Hall in 1941 London is even more profound than the paradox they discussed—namely, that paradoxes are important to humans, because they highlight conflicts between some of the beliefs we hold most dear. By bringing to light conflicts among our firmly held beliefs, paradoxes demand answers from us. If we are to lead lives guided by reason, we must respond to the paradoxes that arise among the beliefs we take to be true. This is why, I believe, it was perfectly understandable to forego other obligations to attend such a lecture at that very tense time. And it is why, in our pursuit of paradoxes and solutions, we are engaged in more than mere puzzle solving. We are committing ourselves to lives that are more thoughtful and better guided by reason.
In the following chapters, we will look at new ways of thinking about paradoxes (chapter 1), how they are generated (chapter 2), and also how they are best solved (chapter 3). The glossary of key terms provides further explanation for the general reader. There is, I hope, insight into the nature of paradoxes and enjoyment in considering their proposed solutions in much of what follows.