To tell the truth about conditionals

Transcription

1 To tell the truth about conditionals Vann McGee If two people are arguing If p, will q? and both are in doubt as to p, Ramsey tells us, 1 they are adding p hypothetically to their stock of knowledge, and arguing on that basis about q; so that, in a sense If p, q and If p, ~q are contradictories. We can say they are fixing their degrees of belief in q given p. Ramsey s observation has proven so accurate that it has become the standard account of when English speakers are willing to assert or accept an indicative conditional; but it is not, I am sorry to report, unfailingly reliable. The counterexample, developed at length below, is a conditional If p, then q that we are willing to assert on the basis of the testimony of someone we regard as a highly reliable authority. Learning p would, however, sufficiently undermine our confidence in the authority that we would no longer have grounds for believing either the conditional or its consequent. So even though If p then q is assertable, the posterior probability that q would have once we learned that p, and hence our present conditional probability of q given p, is low. Unfortunately, the details are elaborate, so I must ask your indulgence as I set the example up. On the old TV game show To Tell the Truth, on the last game of each show, a well-known celebrity would appear as a contestant. Along with him were two other contestants, who pretended to be the celebrity. The celebrity guest was sworn, to the best of his ability, always to tell the truth, whereas the other two contestants were free to answer questions any way they liked. The aim of the game was to determine, by skilful questioning, which of the three contestants was the celebrity Let me tell you about a particular episode of To Tell the Truth in which the celebrity guest was none other than the illustrious detective Sherlock Holmes. Holmes appearance on the show happened at a time when there was widespread public discussion of a criminal case with which Holmes was intimately involved. A certain Rutherford Murdoch drowned under circumstances that somewhat suggested foul play. Suspicion fell upon Murdoch s business partner, Gingrich Brown, because, while there were others who might have had the opportunity to kill Murdoch, no one else appeared to have a motive to do so. Now imagine that, having followed the case closely in the newspapers, 1 P. 155n of General Propositions and Causality in F. P. Ramsey s Philosophical Papers (Cambridge and New York: Cambridge University Press, 1990), Analysis 60.1, January 2000, pp Vann McGee

2 108 vann mcgee you have convinced yourself, although without great confidence, that, in spite of the public uproar, Murdoch s death was an accident. You regard it as rather unlikely that Brown killed Murdoch, and even less likely that someone other than Brown killed Murdoch. Thus, you are inclined to disbelieve the conditional: If Brown didn t kill Murdoch, someone else did. Such, at least, was your state of belief before Holmes appearance on the game show. On the basis of questions that have been answered before the Murdoch case is discussed, you have convinced yourself, with near certainty, that Player Number One is Sherlock Holmes. When Player Number One is asked about the Murdoch case, he replies: Examining this case thoroughly, at the behest of Scotland Yard, I found utterly conclusive evidence that Murdoch was murdered. Investigating further, I found convincing evidence that the culprit was Brown. Thus I am virtually certain that Brown killed Murdoch. Moreover, independent of the evidence that points to Brown, the evidence that Murdoch s death was a murder is so compelling that I can state with complete assurance that, if Brown didn t kill Murdoch, someone else did. After you have heard Player Number One s answer, what should your new beliefs be? Holmes expertise in forensic investigations is legendary, and it would be foolhardy of you to oppose your own superficial opinion, based on what you read in the papers, to his expert view. Since you are convinced that the master detective Sherlock Holmes has told you that, if Brown didn t kill Murdoch, someone else did, you now believe: If Brown didn t kill Murdoch, someone else did. The greatest detective in the world, or, at least, someone you have good reason to believe to be the greatest detective in the world, has told you that he has incontrovertible evidence that Murdoch s death was a murder. Who are you to say he is wrong? Ramsey s account handily explains Holmes assertion of the conditional, If Brown didn t kill Murdoch, someone else did. The evidence that Murdoch s death was a homicide is so strong that it would hold up even if Holmes were to learn, much to his surprise, that Brown wasn t the malefactor. According to the new subjective probability measure Holmes would have after he learned that Brown didn t kill Murdoch, the probability that someone other than Brown killed Murdoch would be close to one. That implies, according to the Bayesian theory of belief revision, that his present conditional probability that someone other than Brown killed Murdoch,

3 to tell the truth about conditionals 109 given that Brown did not, should be close to one, and so, on the standard account of indicative conditionals, If Brown didn t kill Murdoch, someone else did should be highly assertable. The standard account does not do as good a job accounting for your assertion of the conditional, If Brown didn t kill Murdoch, someone else did. How would your beliefs change if you were to learn that Brown did not kill Murdoch? Player Number One has expressed virtual certainty that Brown killed Murdoch, and it is unlikely that Sherlock Holmes would be so badly mistaken about a case he has studied closely. The most likely explanation for Player Number One s assertion would be that Player Number One is not Holmes at all. But if Player Number One is not Holmes, his pronouncements about the Murdoch case don t mean a damn thing, so you fall back to the old belief you had before the contestant spoke, namely, Most likely, Murdoch s death was an accident, so that nobody killed Murdoch. Surely nobody other than Brown killed Murdoch. Were you, after having heard Player Number One s answer, to learn to your surprise that Brown didn t kill Murdoch, your conditional probability that someone other than Brown killed Murdoch given that Player Number One is Holmes would be, in deference to Holmes expertise, close to one. On the other hand, your conditional probability that someone other than Brown killed Murdoch given that Player Number One is someone other than Holmes would revert to the same small value the thesis that someone other than Murdoch killed Brown had before you turned on the television. Your subjective probability that Player Number One is Holmes would decrease drastically. The cumulative effect of all these changes is that your degree of belief that someone other than Brown killed Murdoch, while slightly greater than it was before you turned on the TV, would still be quite low. According to the Bayesian doctrine about belief revision, the fact that learning that Brown didn t kill Murdoch would result in your assigning a low probability to the thesis that someone other then Brown killed Murdoch means that your current conditional probability that someone other than Brown killed Murdoch given that Brown didn t kill Murdoch should be low. The doctrine in upheld in this case, inasmuch as the probability of Brown didn t kill Murdoch but someone else did is noticeably lower than that of Brown didn t kill Murdoch and no one else did. We find ourselves in a situation in which the conditional

4 110 vann mcgee If Brown didn t kill Murdoch, someone else did is highly assertable (and acceptable and probable) even though the conditional probability of the consequent given the antecedent is low. This is contrary to what the standard theory predicts, so the standard theory is wrong. Or so, at least, I want to claim. Admittedly, the data that support this drastic conclusion aren t many. The only evidence to support the contention that, in the circumstances described, we would be willing to assert the conditional is my own intuitions, and the intuitions of a few friends to whom I have shown the example. But though the data are few in number, they are high in quality. The intuition that, under the circumstances described, I would be willing to assert and accept the conditional is firm and unmistakable. The contestant I take to be Holmes has said, If Brown didn t kill Murdoch, someone else did, and, by golly, if he said it, I believe it. When Holmes tells us that, if Brown didn t kill Murdoch, someone else did, he does so as a way of reporting to us that there is further evidence that supports the thesis that Murdoch s death was a homicide in addition to the evidence that points to Brown as the malefactor. On the basis of the putative Holmes testimony, I too have evidence, albeit indirect evidence, that supports the thesis that Murdoch s death was a homicide, in addition to my reasons for believing that Brown was the malefactor. I would presume that it is this further evidence that I report when I say, If Brown didn t kill Murdoch, someone else did. Were I to learn, however, that Brown didn t kill Murdoch, Player Number One s testimony would lose most of its evidentiary value, so that I would no longer have good reason to suppose that Murdoch s death was a homicide. Notice that, throughout all the actual and hypothetical changes of belief, our disbelief in the subjunctive conditional If Brown hadn t killed Murdoch, someone else would have remains undisturbed. Thus, whatever else may be going on in this example, it s not a case of an indicative mood conditional acting like a counterfactual. The empirical hypothesis that we accept and are willing to assert an indicative conditional when and only when the conditional probability of the consequent given the antecedent is close to one has been upheld through a vast array of examples, both commonplace and exotic, both simple and sophisticated. The present example appears to be an exception to the pattern. Frankly, I don t know what to make of it. The probabilistic analysis of conditionals seems to be stymied here, and the possible-worlds account doesn t do any better. Stalnaker 2 extended his 2 Indicative Conditionals in W. Harper, R. Stalnaker, and G. Pearce, eds., Ifs (Dordrecht, Holland: D. Seidel, 1981),

5 to tell the truth about conditionals 111 theory, originally devised to describe subjunctive conditionals, to encompass indicative conditionals by requiring that, in evaluating an indicative conditional, the only worlds we look at are epistemically possible ones. In the present situation, the actual world, so we have reason to believe, is a world in which Player Number One is Holmes and in which Brown killed Murdoch. The nearest world in which Brown didn t kill Murdoch is a world in which Player Number One is Holmes and in which nobody killed Murdoch; that s why we regard the counterfactual If Brown hadn t killed Murdoch, someone else would have as false. But that world isn t epistemically possible, since it s a world in which Holmes is mistaken. Moreover, worlds in which Murdoch is still alive are epistemically impossible. Among the worlds that remain, the world in which Brown does not kill Murdoch that is most similar to the actual world is a world in which Murdoch simply lost his footing on the dock, and so a world in which no one killed Murdoch. Thus, on Stalnaker s account, the conditional If Brown didn t kill Murdoch, someone else did is false; yet we accept it. What I would like to do now is to provide an adapted version of the standard theory that corrects the deficiency the example illustrates. Unfortunately, I don t have one to offer. 3 Department of Linguistics and Philosophy Massachusetts Institute of Technology Cambridge, MA 02139, USA vmcgee@mit.edu 3 I would like to thank Ernest Adams, Aldo Antonelli, Horatio Arlo-Costa, Ned Hall, Richard Jeffrey, Krister Segerberg, and Brian Skyrms for helpful discussions.