A very good alternative discussion of causation and explanation can be found in Lewis 1986a. (Lewis's theory of causation is also eccentric. In fact, I guess every philosopher's theory of causation is eccentric; no two philosophers seem to agree. Lewis's discussion is compatible with a range of different views about causation, though.)
For unificationist theories of explanation, see Friedman 1974 and Kitcher 1989. These are fairly advanced papers.
Lewis discusses the Humean program in metaphysics in the preface to his 1986 Philosophical Papers, volume z. Armstrong 1983 is a clear introduction to the more purely philosophical side of the literature on laws of nature. Beebee z.ooo is a good discussion of the idea that laws "govern" things. Mitchell (zooo) defends an interesting position on laws.
14.1 New Hope
Through much of the twentieth century, the unsolved problem of confirmation hung over philosophy of science. What is it for an observation to provide evidence for, or confirm, a scientific theory? Back in chapter 3, I described how this problem was tackled by logical empiricism. The logical empiricists wanted to start from simple, obvious ideas-like the idea that seeing many black ravens confirms the hypothesis that all ravens are blackand build from there to an "inductive logic" that would help us understand testing in science. They failed, and we left the topic in a state of uncertainty and frustration.
Karl Popper was one person who could enjoy this situation, since he opposed the idea that confirmation is an essential part of science. After Popper, we launched into a discussion of historically oriented theories of science, like Kuhns. These theories were not focused on the problem of confirmation in the way logical empiricism was. But the problem did not go away. Some philosophers continued to work on it, and even when it was not being discussed, it lurked in the background. If someone had come up with a really convincing theory of confirmation, it would have been harder to argue for radical views of the kind discussed in chapters 7-9. The absence of such a theory put empiricist philosophers on the defensive.
The situation has now changed. Once again a large number of philosophers have real hope in a theory of confirmation and evidence. The new view is called Bayesianism. The core ideas of this approach developed slowly through the twentieth century, but eventually these ideas started to look like they might actually solve the problem. The attitude of many is summed up in the title of a recent book by John Earman: Bayes or Bust? (199z). This title refers to a widespread feeling that this approach had better work, or philosophy of science might really be in trouble again.
Although Bayesianism is the most popular approach to solving these problems today, I am not in the Bayesian camp. Some parts of Bayesianism are undeniably powerful, but I would cautiously put my money on some different ideas. These will be introduced at the end of the chapter.
And before setting out on these topics, I should stress that this is the hardest chapter in the book. Some readers might want to jump to chapter 15.
14.2 Understanding Evidence with Probability
At this point I will shift my terminology. The term "confirmation" was used by the logical empiricists, but more recent discussions tend to focus on the concept of evidence. From now on I will follow this usage.
Bayesianism tries to understand evidence using probability theory. This idea is not new. It has often seemed natural to express some claims about evidence in terms of probability. Rudolf Carnap spent decades trying to solve the problem in this way. And outside philosophy this idea is familiar; we say that seeing someone's car outside a party makes it very likely that he is at the party. The mathematical fields of statistics and data analysis use probability theory to describe the kinds of conclusions that can be drawn from surveys and samples. And in law courts, we have become familiar with the description of forensic evidence, like DNA evidence, in terms of probability.
Consequently, many philosophers have tried to understand evidence using probability. Here is an idea that lies behind many of these attempts: when there is uncertainty about a hypothesis, observational evidence can sometimes raise or lower the probability of the hypothesis.
Bayesianism is one version of this idea. For Bayesians, there is a formula that is like a magic bullet for the evidence problem: Bayes's theorem. Thomas Bayes, an English clergyman, proved his theorem in the eighteenth century. As a theorem-as a piece of mathematics-his idea is very simple. But the Bayesians believe that Thomas Bayes struck gold.
Here is the magic formula in its simplest form:
Here it is in a form that is more useful for showing how it works in philosophy of science:
Here is how to read formulas of this kind: P(X) is the probability of X. P(XIY) is the probability of X conditional upon Y, or the probability of X given Y.
How does this formula help us to understand confirmation of theories? Read "h" as a hypothesis and "e" as a piece of evidence. Then think of P(h) as the probability of h measured without regard for evidence e. P(hl e) is the probability of h given e, or the probability of the hypothesis in the light of e. Bayes's theorem tells us how to compute this latter number. As a consequence, we can measure what difference evidence e makes to the probability of h. So we can say that evidence e confirms h if P(hle) > P(h). That is, e confirms h if it makes h more probable than it would otherwise be.
Picture someone changing her beliefs as evidence comes in. She starts out with P(h) as her assessment of the probability of h. If she observes e, what should her new view be about the probability of h? It seems that her new view of the probability of h should be given by P(hle), which Bayes's theorem tells us how to compute. So Bayes's theorem tells us how to update probabilities in the light of evidence. (More on this updating later.)
Those are two central ideas in Bayesianism: the idea that e confirms h if e raises the probability of h, and the idea that probabilities should be updated in a way dictated by Bayes's theorem.
Bayes's theorem expresses P(hle) as a function of two different kinds of probability. Probabilities of hypotheses of the form P(h) are called prior probabilities. Looking at formula z, we see P(h) and P(not-h); these are the prior probability of h and the prior probability of the negation of h. These two numbers must add up to one. Probabilities of the form P(elh) are often called "likelihoods;' or the likelihoods of evidence on theory. In formula z we see two different likelihoods, P(elh) and P(elnot-h). (These need not add up to any particular value.) Finally, P(hle) is the "posterior probability" of h.
Suppose that all these probabilities make sense and can be known; let's see what Bayes's theorem can do. Imagine you are unsure about whether someone is at a party. The hypothesis that he is at the party is h. Then you see his car outside. This is evidence e. Suppose that before seeing the car, you think the probability of his going to the party is 0.5. And the probability of his car's being outside if he is at the party is o.8, because he usually drives to such events, while the probability of his car's being outside if he is not at the party is only o. i. Then we can work out the probability that he is at the party given that his car is outside. Plugging the numbers into Bayes's theorem, we get P(hle) = (.5)(.8)/[(.5)(.8) + (.5)(.1)1, which is almost o.9. So seeing the car raises the probability of h from o.5 to about o.9; seeing the car strongly confirms the hypothesis that the person is at the party.
This all seems to be working well. We can do a lot with Bayes's theorem, if it makes sense to talk about probabilities in these ways. It is common to think that it is not too hard to interpret probabilities of the form P(elh), the likelihoods. Scientific theories are supposed to tell us what we are likely to see. Some Bayesians underestimate the problems that can arise with this idea, but there is no need to pursue that yet. The probabilities that are more controversial are the prior probabilities of hypotheses, like P(h). What could this number possibly be measuring? And the posterior probability of h can only be computed if we have its prior probability. So although it would be good to use Bayes's theorem to discuss evidence, many interpretations of probability will not allow th
is because they cannot make sense of prior probabilities of theories. If we want to use Bayes's theorem, we need an interpretation of probability that will allow us to talk about prior probabilities. And that is what Bayesians have developed. This interpretation of probability is called the subjectivist interpretation.
14.3 The Subjectivist Interpretation of Probability
Most attempts to analyze probability have taken probabilities to measure some real and "objective" feature of events. A probability value is seen as measuring the chance of an event happening, where this chance is somehow a feature of the event itself and its location in the world. That is how we usually speak about the probabilities of horses winning races, for example. But according to the subjectivist interpretation, probabilities are degrees of belief. A probability measures a person's degree of confidence in the truth of some proposition. So if someone says that the probability the horse "Tom B" will win its race tomorrow is 0.4, he is saying something about his degree of confidence that the horse will win.
The subjectivist approach to probability was pioneered (independently) by two philosopher-mathematicians, Frank Ramsey and Bruno de Finetti, in the z9zos and 193 os. This interpretation of probability is not only important in philosophy; it is central to decision theory, which has great importance in the social sciences (especially economics). The majority of philosophers who want to use Bayes's theorem to understand evidence hold a subjectivist view of probability-at least for applications of probability theory to this set of problems, and sometimes more generally. Some are subjectivists because they feel they have to be in order to use Bayes's theorem; others think that subjectivism is the only interpretation of probability that makes sense anyway. The philosophical debates about Bayesianism also connect to debates about probability within mathematical statistics itself.
So let us look more closely at subjectivism about probability and how it relates to Bayesianism. The details of this topic are ferociously technical, but the main ideas are not too difficult.
Subjectivism sees probabilities as degrees of belief in propositions or hypotheses about the world. To find out what someone's degree of belief in a proposition is, we do not ask him or look inside his mind. Instead, we see his degrees of belief as revealed in his gambling behavior, both actual and possible. Your degrees of belief are revealed in which bets you would accept and which you would reject. Real people may be averse to gambling, even when they think the odds are good, or they may be prone to it even when the odds are bad. Here and in other places, Bayesianism seems to be treating not actual people but idealized people. But let us not worry too much about that. To read off a person's degrees of belief from his gambling behavior, we look for the odds on a given bet such that the person would be equally willing to take either side of the bet. Call these odds the person's subjectively fair odds for that bet. If we know a person's subjectively fair odds for a bet, we can read off his degree of belief in the proposition that the bet is about.
For example, suppose you think that 3:z is fair for a bet on the truth of h. That is, a person who bets that h is true wins $z if she is right and loses $3 if she is wrong. More generally, let us say that to bet on h at odds of X:i is to be willing to risk losing $X if h is false, in return for a gain of $z if h is true. So a large X corresponds to a lot of confidence in h. And if your subjectively fair odds for a bet on h are X:i, then your degree of belief in h is X/(X + I).
So far we have just considered one proposition, h. But your degree of belief for h will be related to your degrees of belief for other propositions as well. You will have a degree of belief for h & j as well, and for not-h, and so on. To find your subjective probability for h & j, we find your subjectively fair odds for a bet on h & j. So a person's belief system at a particular time can be described as a network of subjective probabilities. These subjective probabilities work in concert with the person's preferences ("utilities") to generate his or her behavior. From the Bayesian point of view, all of life is a series of gambles, in which our behavior manifests our bets about what the world is like.
Bayesians claim to give a theory of when a person's total network of degrees of belief is "coherent," or rational. They argue that a coherent set of degrees of belief has to follow the standard rules of the mathematics of probability.
Here is a quick sketch of these more technical ideas. Modern treatments of probability start from a set of axioms (most basic principles) first devel oped by the Russian mathematician Kolmogorov. Here is a version of those axioms that is used by subjectivists.
Axiom z: All probabilities are numbers between o and i (inclusive).
Axiom z: If a proposition is a tautology (trivially or analytically true), then it has a probability of i.
Axiom 3: If h and h" are exclusive alternatives (they cannot both be true), then P(h-or-h") = P(h) + P(h").
Axiom q: P(hlj) = P(h&j)IP(j), provided that P(j) > o.
(Bayes's theorem is a consequence of axiom 4. P(h & j) can be broken down both as P(hlj)P(j) and as P(jlh)P(h). So these are equal to each other and Bayes's theorem follows trivially.)
Why should your degrees of belief follow these rules? Subjectivists argue for this with a famous form of argument called a "Dutch book." (My apologies to any readers who are Dutch.)
The argument is as follows: if your degrees of belief do not conform to the principles of the probability calculus, there are possible gambling situations in which you are guaranteed to lose money, no matter how things turn out. How can there be a guarantee? Because these situations are ones in which you are betting on both sides of a proposition (or betting on all the horses in the race), at various different odds. In these situations, if the degrees of belief you have do not conform to the probability calculus, and you are willing to accept any bet that fits with your degrees of belief, then you will be willing to accept combinations of bets that guarantee you a loss.
Here is a simple example involving a coin toss. Suppose your degree of belief that the toss will come out heads is o.6 and your degree of belief that the toss will come out tails is o.6. Then you have violated the probability calculus because, by axiom 3, P(heads or tails) = i.z, which axiom i says is impossible. But suppose you persist with these degrees of belief and are willing to bet on them. Now suppose someone (a "Dutch bookie") offers you the following bets: (z) You are to bet $io at 1.5:1 that the result will be heads, and (z) you are to bet $io at I.5:1 that the result will be tails.
You should accept both bets, because your subjectively fair odds for heads and for tails are both I.5:1. (To go from a degree of belief p to odds of X:i, use X = p/(z - p).) But now you have accepted two bets that each pay worse than even money on the only two possible outcomes. So you are guaranteed to lose. If the coin lands heads, you win $io on the heads bet but lose $15 on the tails bet, so you are $5 behind. The same applies-a net loss of $ 5-if the coin lands tails. You have fallen victim to a Dutch book. If you want to ensure that no one could possibly make a Dutch book against you, you must ensure that your degrees of belief follow the rules of probability theory. This is a simple case, but more complex arguments of the same kind can be used to show that any violation of the mathematical rules of probability makes a person vulnerable to a Dutch book.
Of course, there are not very many Dutch bookies out there, and one can avoid the threat by refusing to gamble at all. That is not the point. The point is supposed to be that the Dutch book argument shows that anyone who does not keep his degrees of belief in line with the probability calculus is irrational in an important sense.
Let us now connect these ideas to the problem of evidence. The ideas about belief and probability discussed in this section so far apply to a person's beliefs at a specific time. But we can use these ideas to give a theory of the rational updating of beliefs as evidence comes in. Bayes's theorem tells us about the relations between P(h) and P(hl e). Both those assignments of probability are made before e is observed. Then suppose e is actually observed. According to Bayesianism, the rational agent will
update her degrees of belief so that her new overall confidence in h is derived from her old value of P(hle). So the key relationship in this updating process is
The probability PneW(h) then becomes the agent's new prior probability for h, for use in assessing how to react to the next piece of evidence. So "today's posteriors are tomorrow's priors." A different set of Dutch book arguments is used to argue that a rational agent should do her updating of beliefs in accordance with formula 3. (Bayesianism has to make special moves to deal with "old evidence," evidence known before its relation to a hypothesis is assessed, and it also has a different formula to use when evidence e is itself uncertain.)
To finish this section (sigh of relief), note again that according to a subjectivist interpretation of probability, there is no way that one set of degrees of belief can be "closer to the real facts about probability" than another, so long as both sets of degrees of belief each follow the basic rules (axioms) of probability. At least, that is how things work according to strict subjectivism. Some Bayesians would like to recognize an objective sense of probability as well as a subjective sense, but this requires extra arguments.
14.4 Assessing Bayesianism
Bayesianism is an impressive set of ideas. There is a big literature on these topics, and I will not try to predict whether Bayesianism will work in the end. But many of the debates have to do with the role of prior probabilities, so they are worth further discussion.
In standard presentations of Bayesianism, a person is imagined to start out with an initial set of prior probabilities for various hypotheses, which are updated as evidence comes in. This initial set of prior probabilities is a sort of free choice; no initial set of prior probabilities is better than another so long as the axioms of probability are followed. This feature of Bayesianism is sometimes seen as a strength and sometimes as a weakness.
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