Paradox

In the paradox of enrichment, an assumption that we took to be intuitively plausible turned out, along with other evidence, to lead to unexpected conclusions. For this reason, our intuitions about what is good for a population of animals had to be questioned and, as a result, progress was made in the form of new models of predator/prey relations (May 1972). So, by questioning our most basic intuitions, we are often led to new ways about thinking about our basic notions and, ironically, to further paradoxes.

Intuition has been variously defined as seeming truth, spontaneous mental judgment, what we would say in a given situation, and noninferential belief (Davis-Floyd and Arvidson 1997). And, more recently, it has been described by cognitive scientists as immediate judgments that arise from the recognition of familiar elements in a new situation (Kahneman 2011). It is intuitively obvious to me right now that I am typing on a laptop computer, that murder is immoral, that triangles have three sides, that the gray and tan object poking out from behind the computer screen is Coco the cat, and so on. In addition, my beliefs about these things were fairly immediate. I didn’t need to infer any of these beliefs from a set of assumptions. They came to me spontaneously and without forethought. But giving an account of intuitiveness is not an easy endeavor. If there were a way not only to explain the intuitiveness of the components of the paradox but also to quantify the degree to which a part of a paradox is intuitive, we would have a better understanding of the nature of paradox.

Recently, a new way of thinking about belief emerged in the philosophy of science that can give us a better understanding of the intuitive nature of paradox. A group of philosophers of science known as Bayesians started to use something called subjective probability to explain the degree to which a scientific hypothesis is confirmed or disconfirmed. Subjective probability is the degree to which a rational observer believes something, with 0 assigned to complete disbelief, 1 to complete certainty, 0.5 for neither belief nor disbelief, 0.7 for fairly strong belief, and so on (see table 1).

Table 1

Sample degrees of belief for one rational believer

Degree of belief	Sample belief
0.0	Belief that 2 + 2 = 5
0.1	Belief that meat consumption is morally acceptable
0.2	Belief that God exists
0.3	Belief that environmental damage can still be completely reversed
0.4	Belief that my dog will live to 18 years of age
0.5	Belief that the results of the next coin toss will be heads
0.6	Belief that the U.S. economy is improving
0.7	Belief that Hillary Clinton will run for president of the United States again
0.8	Belief that in five years most books will be digitized
0.9	Belief that smoking is bad for one’s health
1.0	Belief that 2 + 2 = 4

Thinking in terms of degrees of belief makes sense, especially when we are dealing with uncertainty about the future. Right now, for example, I believe that a gift I just ordered online will arrive in time for me to present it my friend on her birthday. I am, however, not completely certain about this. Because the online store usually delivers things in a timely fashion, I’m pretty sure it will arrive in time. But I did not opt for a guaranteed delivery date, so I’m taking my chances. Given what I know about the store, the mail service, the doormen who work in my friend’s building, and so on, my subjective probability that the gift will arrive on time is about 0.8. But this number could be revised up or down, depending on a number of factors. If, for example, a terrible storm hits the region, then my subjective probability would drop, and if I received an email saying the item was shipped yesterday, my subjective probability would rise. The degrees to which we believe things get raised or lowered, then, depending on the evidence we have available to us. And when the birthday comes, my degree of belief will be raised either to 1, complete belief, or lowered to 0, complete disbelief. I will know whether it has arrived in time.

Although it is called subjective probability, subjective probability is not a totally subjective measure, because it assumes that the believer is rational. As Richard Jeffrey has written (2004, 76), “Your ‘subjective’ probability is not something fetched out of the sky on a whim; it is what your judgment should be in view of your information to date, and of your sense of other people’s information, even if you do not regard it as a judgment that everyone must share on pain of being wrong in one sense or another.” Although there might be some leeway with regard to the degrees of rational belief, believers are constrained by rules of rationality and the information available to them. For example, a rational believer would not have a subjective probability of 1 for a contradictory proposition. Nor would such a believer assign a contradiction a probability of 0.6. Though I doubt people go through their daily lives assigning numerical probabilities to partial beliefs, when asked how sure they are about something, people can in general say whether they think something is more likely than not (greater than 0.5), almost certain (0.9), or similar estimates.

Using subjective probability, we can give an account of the intuitiveness of the parts of any paradox—even for the intuitiveness of a paradox as a whole. The more intuitive a statement in a paradox is, the higher its subjective probability is. And when we get contradictory conclusions, we know that that part of the paradox must have a subjective probability of 0. In the case of the deepest paradoxes, we have premises with extremely high subjective probabilities and a conclusion that is accorded 0—or just slightly above it. So, we can rate how deep a paradox is by using subjective probability. To see how this can happen, let’s examine how different subjective probabilities are combined.

If I believe strongly that the gift I ordered for my friend will arrive in time for her birthday (0.8) and very strongly that my friend will like the gift (0.9), then the subjective probability that both of these things will happen is the combined subjectivity of both beliefs. Because it is less likely for both to happen than for one to happen individually, the combined subjective probability of the two beliefs would be slightly lower than each, and best calculated by multiplying the two degrees of probability.² If I multiply the two numbers (0.9 and 0.8), I get 0.72. So I think my chances of my gift being received and appreciated are pretty good. Why multiplication? We need to allow for how some uncertainty in individual beliefs combine to form even greater uncertainty in combined beliefs. If I believe something with a subjective probability of 0.2 and another with 0.9, then my subjective probability about both being the case should be quite low, given the low belief I have about one of them. And multiplication models this effect for us. Under this way of thinking about degrees of belief, the result would be 0.18—a small bit lower than 0.2, because when we combine a very low subjective probability (0.2) with something less than complete subjective certainty, such as a 0.9 belief, then the 0.2 belief should be lowered to account for the added amount of uncertainty.

Paradoxes, by the second definition mentioned in the first paragraph of this chapter, involve seemingly true assumptions and apparently correct reasoning. The intuitiveness of each part of a paradox and the depth of a paradox as a whole can be given using subjective probability.

To do this, let’s introduce what I’ll call a paradoxicality rating for paradoxes modeled, to start, on the definition of a paradox as an argument with seemingly true premises, seemingly valid ³ reasoning, and an obviously false or contradictory conclusion. The paradoxicality rating of an argument would then be determined by combining (a) the subjective probability of the conjunction of the premises Pr(p1, . . . , pn) being true, (b) the subjective probability of valid (Prv) reasoning, and (c) the subjective probability of a false conclusion c, or (1 – Prc). Thus we get the formula, which is really just one long multiplication of the subjective probabilities of the parts of a paradox, where Pr stands for subjective probability, p for the premises, v for the validity of the reasoning, and c for the conclusion:

Paradoxicality rating

Paradoxicality = Pr(p1, . . . , pn) × Prv × (1 – Prc)

Because paradoxes on this first definition are arguments, we need to factor in the subjective probabilities of all the parts of the paradoxical argument, including the subjective probabilities of the premises, that the reasoning is good, and that the conclusion is false. The higher the subjective probability of a premise, the more likely we are to bet that this premise is true, and the more “intuitive strength” the premise will have for us. With the probability of the conclusion and the argument’s validity held constant, the higher the subjective probability of a premise, the higher the degree to which an argument is paradoxical. For example, if we have an argument with a clearly false premise, it doesn’t pose much of a paradox for us, because we never found the premise to be intuitively plausible to begin with. If we never really thought that more food for prey would translate into more prey and hence a better situation for the predator, for example, we wouldn’t find the paradox of enrichment very deep or paradoxical at all. It is because we do think more food for prey means more prey and hence a better situation for predators that the conflicting evidence seems surprising and paradoxical, which is why, in the definition of paradoxicality, we do a straightforward multiplication of the subjective probabilities of the premises, Pr(p1, . . . , pn).

As for the reasoning involved, the more obvious it seems that the reasoning is good, the deeper the paradox becomes. For example, consider Rebecca Flint and her problems with the images of the future explosion. It does seem quite clear that if the assumption that the images presented to Flint are veridical is given, and we assume she has free will, we can use very straightforward reasoning to derive a contradiction. If the images are true, then the explosion will happen, but if Flint is free, she can prevent the explosion from happening. So, given the premises that (1) the images of the future explosion are true, which we assume and (2) Flint is free to prevent future events from happening, we are licensed to conclude that Flint cannot change the future (from 1) and yet can change the future (from 2). And this conclusion is a contradiction. It is an example of seemingly good reasoning. The more straightforward the reasoning is in an argument, the higher the degree to which the argument is paradoxical. If we knew, for example, that a logical fallacy (i.e., an error in reasoning) was used in the argument, then the argument would not be very deep or paradoxical. Real paradoxes, by this definition, are arguments for which the reasoning is straightforwardly correct. The more obviously correct the reasoning is, the more paradoxical the argument.

The final item to be factored in is the subjective probability that the conclusion is false. In the previous Flint paradox example, we derived a contradiction. Contradictions always get a subjective probability of 0. If the conclusion were probably true, then we wouldn’t think that the argument was very paradoxical. If we accepted the conclusion that a person with 1,000,000 hairs is bald, for example, we wouldn’t find the sorites paradox very deep. The problem, though, is that we do think the conclusion is false—and even if we didn’t, we could posit an even higher number of hairs as meeting the requirement for baldness. So, with other factors held constant, the lower the subjective probability of the conclusion, the more paradoxical the argument is. If you look at the formula in box 3, though, you’ll see that although I’m doing a straight multiplication of probabilities, the conclusion’s probability is subtracted from 1. The reason we have the (1 – Prc) at the end of the definition is that we don’t want to factor in how high the subjective probability of the conclusion is, but instead how low it is. And subtracting the subjective probability of the conclusion from 1 lets us do this.

So that you can learn how to use the formula for paradoxicality to determine how intuitively plausible a paradox is, let’s look at a few cases. For example, in an argument with two premises, each with a subjective probability of 0.5, the premises would have a combined probability of 0.25. If the reasoning is unquestionably valid, then we can give the argument a Prv value of 1, and if the conclusion is highly unlikely—say, 0.2—then 1 – Prc is 0.8. The total paradoxicality rating is 0.25 × 1 × 0.8 = 0.2, which is not very high, because the premises were not very likely.

If you have an argument like the sorites paradox discussed in the introduction, then you have an argument with extremely plausible premises. The first premise—that a person with no hair on his head is bald—is close to a conceptual truth and deserves a probability of 1. The second premise, which says that adding 1 hair won’t make a difference regarding whether someone is bald, is seemingly conceptual as well, but perhaps there is a bit more room for doubt with this premise. I would give this premise a 0.95. The reasoning is straightforward, and Prv would be 1. The conclusion that a person with 1,000,000 hairs is bald is almost conceptually false and can be increased indefinitely. Thus Prc should be 0, and 1 – 0 is 1. So we have 1 × 0.95 × 1 × (1 – 0) = 0.95. Taken together, we have an argument with a paradoxicality rating of 0.95, a very high paradoxicality rating. On this way of ranking paradoxes, the sorites turns out to be quite paradoxical.

Not all arguments, of course, get such a high paradoxicality rating. Consider an argument known to be sound. By definition, a sound argument’s premises are true, its reasoning valid, and its conclusion true. So the premises of such an argument have a combined subjective probability of 1, a subjective probability of its validity of 1, and a subjective probability of its conclusion of 1. A sound argument’s paradoxicality rating is always 0, because of the conclusion. The conclusion has a subjective probability of 1 and the paradoxicality rating is determined by the formula Pr(p1, . . . , pn) × Prv × (1 – Prc). Thus we have 1 × 1 × 0, which assigns to a sound argument a paradoxicality rating of 0. Not only sound arguments, but all arguments with clearly true conclusions receive paradoxicality ratings of 0. An argument with a clearly true conclusion is nonparadoxical on any plausible definition and would therefore receive a paradoxicality rating of 0. In addition, arguments must not have obviously false premises to count as paradoxes. If the subjective probability of a premise is 0, then—regardless of the subjective probabilities of the conclusion and validity—the argument has a paradoxicality rating of 0.

By using the paradoxicality rating, we can distinguish between obviously nonparadoxical arguments, such as sound arguments and arguments with obviously false premises, and clear cases of paradoxicality. However, as you’ll soon see, it would be unwise to give a set number for distinguishing the paradoxical from the nonparadoxical. Indeed, to do so would lead to further paradoxes. The paradoxicality rating determines what is more paradoxical than something else and identifies the very clear cases. As for the borderline cases of paradoxicality—for example, arguments in the 0.5 or 0.6 range—a paradoxicality rating merely points to the borderline nature of the paradox and distinguishes it from more clear cases.

So far, we have assumed the definition of paradox on which a paradox is an argument. If we think of a paradox as a set of mutually inconsistent statements, each of which seems true, we can give an alternative formula for determining how intuitive a paradox is. This formula combines the subjective probability of each of the statements, along with the subjective probability that the statements are inconsistent. We can symbolize this as follows, where Pr again represents subjective probability, each s refers to a statement in the set, and i represents the claim that the set of statements is inconsistent:

Paradoxicality for sets of statements

Paradoxicality = Pr(s1, s2, . . . , sn) × Pr(i)

In other words, paradoxicality is determined by how high the subjective probability of each statement is, along with the probability that the set of statements is inconsistent. For example, consider another paradox: the basic liar paradox. If we consider the statement (L), “This sentence is false,” we see that if the statement (L) is true, then it is true that the statement is false, because the sentence claims that it is false. And if (L) is false, then it is false that the statement (L) is false, so the statement is true. So, if (L) is true, then it is false. And if (L) is false, then it is true. But because statements are either true or false, then it seems to follow that the statement would have to be both true and false, which is a contradiction. So we have a set of statements:

Basic liar paradox

1. If L is true, then L is false.

2. If L is false, then L is true.

3. L is either true or false, but not both.

Each of these statements has a high subjective probability, given what L states. Yet all three cannot be true at the same time without contradiction, because the truth of L leads to its being false, and its falsity leads to its truth. And this contradiction violates statement 3, which claims that L cannot be both true and false. So we have three statements with high subjective probability and, taken as a set, they are inconsistent. When we plug very high values into the formula, Pr(1 × 1 × 0.9) × Pr(1), we get a paradox with a paradoxicality rating of 0.9, a very high rating. This result makes sense given that this paradox is considered one of deepest philosophical paradoxes.

As shown earlier, the paradoxicality rating can be used to assess the degree to which an argument is paradoxical. It also gives us a way of explaining why something is more paradoxical than something else. Consider the basic liar paradox and a strengthened version of the paradox. This next, stronger version of the paradox is more paradoxical because it makes an assumption that is harder to call into question. Because the basic liar sentence “This sentence is false” predicates falsity of itself, it is false in the event that it is true, and true in the event that it is false. Yet according to a basic rule called the principle of bivalence (the claim that every statement is either true or false), all propositions are either true or false. So regardless of which truth-value (true or false) is assigned to the sentence, the other one will automatically be assigned to it as well. Thus, it will be assigned both truth-values.

The only difference between the basic liar paradox just mentioned and the strengthened liar paradox is that the strengthened liar sentence is “This sentence is not true.” If you abandon bivalence, you can solve the simple liar paradox, holding that the sentence is neither true nor false. However, the strengthened liar paradox assumes only the law of excluded middle, which claims that every statement or its denial is true. The principle of bivalence can, with some explanation, be denied, whereas it is far harder to deny a logical principle such as the law of excluded middle without leading to contradiction. So although someone might argue that the basic liar sentence is neither true nor false but meaningless, it would be much harder to apply the same treatment to the strengthened liar sentence. Doing so entails denying that the sentence is neither true nor untrue. In the first case, we could say that denying the truth of “This sentence is false” doesn’t lead to the sentence being false, as the sentence could be meaningless or otherwise ill formed. In the second case, if we say that it’s not the case that “This sentence is not true” is true, we seemingly are saying that it is true.

Thus, the strengthened liar paradox must have a higher paradoxicality rating than the basic liar paradox. Assume that all other parts of the liar and strengthened liar paradoxes are unquestionable—that is, the other premises are assigned subjective probabilities of 1 and the validity of each paradox is assigned 1—and that both conclusions are obviously false; hence 1 – Prc is 1 – 0, or 1. In this case, the only difference between the two paradoxes in terms of their paradoxicality rating is due to the subjective probabilities of the basic liar sentence and the strengthened liar sentence. If, for example, the basic liar sentence is assigned a subjective probability of 0.90 and the strengthened liar sentence is assigned 0.95, then the strengthened liar sentence will have a paradoxicality rating of 0.95 and the basic liar sentence will have only a 0.90 rating. Thus, the strengthened liar statement will turn out to be just as paradoxical as the sorites paradox, but both the sorites and strengthened liar statements will be more paradoxical than the basic liar paradox.

Thus, when we make the idea of intuition more explicit by thinking of it in terms of subjective probability, we can explain why some things are more paradoxical than others. The paradoxicality rating gives us a way to do this. The liar paradox, in both its basic and strengthened versions, turns out to be more paradoxical than the paradox of enrichment, because both versions of the liar paradox have parts with high subjective probability. By contrast, the conclusion of the paradox of enrichment, though surprising, still has a lower probability than the parts of either paradox. So this paradox turns out to be less paradoxical (see table 2).

Table 2

Sample list of paradoxes and approximate paradoxicality ratings

Paradox	Paradoxicality rating
Enrichment	0.07
Liar, basic	0.90
Liar, strengthened	0.95
Sorites	0.95

Solutions can also be explained in terms of the subjective probability of the parts of the paradox. There are many ways to solve paradoxes, but most involve pointing to one part of the paradox and lowering its subjective probability. For example, many philosophers have tried to solve the sorites paradox by showing that the premise that states that adding one hair won’t make the difference in someone’s being bald or non-bald (for any number n, if a person with n hairs on his head is bald, then a person with n + 1 hairs on his head is bald) is either false or very misleading.

Would-be solvers of the sorites paradox often try to show that this premise (which is sometimes called the inductive premise) is not true, and that one hair does make this difference. Timothy Williamson (1994), for example, argues that the principle of bivalence—which states that every statement is either true or false—is a true principle, and that because of this, in a series of hairs ranging from zero to a million, there must be a point in which one hair marks a shift from being bald to being non-bald. Once we reach that number—let’s call it n—the statement “A person with n hairs is bald” would be true, and the statement “A person with n + 1 hairs is bald” false. This claim follows, Williamson thinks, from the premise that every statement is either true or false. Williamson’s project is to show how this could be, and why our intuitions about the truth of the claim that the addition of one hair won’t change anyone’s status as bald are mistaken. To try to lower our subjective probability about this part of the paradox, Williamson says that it only seems like one hair wouldn’t make a difference in whether someone is bald because we cannot know that point. Just because we are unable, due to our own limited abilities, to tell where the shift in numbers of hairs is, it doesn’t follow that shift isn’t there. In fact, the principle of bivalence shows that this shift must be there. In other words, we confuse our ignorance of the cutoff between baldness and non-baldness with the fact that there is no cutoff. If Williamson succeeded in this project, then the subjective probability of the second premise in the sorites paradox would be lowered.

Whether Williamson was successful is another matter. The point, though, is that his treatment of the paradox involved pointing to part of it and then trying to lower this part’s subjective probability, showing how we were mistaken in giving that part of the paradox a high subjective probability.

For another example, consider Flint’s problem of preventing the future explosion that was known to be coming. We saw that there was a conflict regarding how it is possible both for the images of the future explosion to be accurate and for it to be possible for Flint to stop the future explosion. One way to attempt to solve this paradox would be to claim that although it looks plausible that Flint has the power to prevent the future explosion, if in fact the images are accurate, then the future explosion must happen, despite Flint’s best efforts to prevent it. The future is predetermined, and nothing can prevent the explosion from happening. Flint’s actions may actually bring about the explosion, if the images are truly accurate. So the paradox is solved by getting rid of the conflict between Flint’s freedom to prevent the future explosion and the accuracy of the images, by denying the first statement—namely, that Flint has the freedom to prevent the explosion. One might argue that it only looks as if she is free to prevent the event, but her supposed freedom is an illusion. Thus the subjective probability of the part of the paradox that claims that Flint can prevent the explosion is lowered.

One interesting result of treating paradoxes the way we are presently doing (i.e., using subjective probability) is that the very nature of paradox can be shown to be paradoxical. The following argument, modeled on the sorites paradox, is a paradox about paradoxes themselves—that is, a higher-order paradox:

The paradoxicality paradox

1. An argument with a paradoxicality rating of 0 is nonparadoxical.

2. For any number n, if an argument with a paradoxicality rating of n is nonparadoxical, then an argument with a paradoxicality rating of (n + 0.001) is nonparadoxical.

3. An argument with a paradoxicality rating of 1 is nonparadoxical.

The paradoxicality of the previous argument is quite high. As in the sorites, both premises seem true; the argument’s reasoning is straightforward; and the conclusion is, according to the definition of paradoxicality, conceptually false. Such an argument shows that the very concept of paradox is not immune from paradoxicality. Because paradoxes themselves admit of degrees, they too engender paradoxes. And because there are myriad paradoxes about a wide array of concepts, it makes sense that the concept of the paradox is itself paradoxical.

Taking intuition as the basis for paradoxes, and then analyzing our intuitions using subjective probability, we were able to create the paradoxicality rating, which in turn gave us a way of explaining why one paradox is deeper than another. We also have a way of making clearer the intuitions that are necessary in order for something to be a paradox. Solutions to paradoxes, we also saw, can be thought of as pointing to a part of the paradox and then showing that the subjective probability of the paradox should be lowered. Now that you have a fairly thorough understanding of the intuitive basis of paradoxes, we can turn to the various solutions offered for them.