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1 Most Counterfactuals are False Alan Hájek DRAFT DEAR READER, YOU MAY APPRECIATE SOME TIPS ON WHICH SECTIONS COULD BE SKIPPED ON A FIRST READING. IF YOU ARE LACKING IN TIME OR STAMINA, I RECOMMEND THAT

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1 Most Counterfactuals are False Alan Hájek DRAFT DEAR READER, YOU MAY APPRECIATE SOME TIPS ON WHICH SECTIONS COULD BE SKIPPED ON A FIRST READING. IF YOU ARE LACKING IN TIME OR STAMINA, I RECOMMEND THAT YOU SKIP OR SKIM THROUGH SECTIONS 6 AND 7, SPARING YOU OVER 20 PAGES. I VE PUT IN GRAY FONT THESE, AND OTHER BITS I SKIPPED IN MY APA TALK. FAST FORWARDING THROUGH SECTION 5 WOULD SPARE YOU ABOUT ANOTHER 10 PAGES, WITHOUT LOSING THE GIST OF THE PAPER. [THERE ARE ALSO VERY OCCASIONAL NOTES TO MYSELF, USUALLY IN SQUARE PARENTHESES.] A.H. 1. Introduction Even the crows on the rooftops are cawing about the question of which conditionals are true. So said Callimachus over 2000 years ago. 1 And my answer to the question is: when it comes to counterfactuals, relatively few of them. The crows have much to caw about. For counterfactuals are apparently implicated in much that we hold dear. They figure in influential analyses of causation, perception, knowledge, personal identity, laws of nature, rational decision, confirmation, dispositions, free action, explanation, and so on. Science freely traffics in counterfactuals, both explicitly (in drawing consequences of its theories 2 ) and implicitly (by trafficking in concepts that are themselves tacitly counterfactual). And counterfactuals are also earning their keep in the social sciences in psychology (e.g., in understanding emotions such as regret), in history (e.g., in studying the 1 Attributed by Sextus Empiricus, Against the Mathematicians. Quoted in B. Mates, Stoic Logic (Berkeley & Los Angeles: University of California Press, 1961), Consider this example from the classic physics textbook, Halliday and Resnick (19xx), p. 251: If we were to bore a hole through the earth, and drop a particle into the hole, it would move in simple harmonic motion. 2 incremental contribution by some commodity to economic growth), in the law (e.g., in apportioning responsibility), and so on. Yet it has long been recognized that counterfactuals are strange beasts. 3 They involve a modality that makes empiricists uncomfortable; they resist truth functional analysis, yet the best known possible worlds analyses of them make various philosophers uncomfortable; they putatively violate intuitive inference rules that material and strict conditionals obey, such as contraposition and transitivity and strengthening of the antecedent 4 ; and so on. I will argue that they are even stranger than has been generally recognized: while we use them nonchalantly in daily conversation, and while they are staples of numerous philosophical analyses, most counterfactuals are false. 2. Counterfactuals under indeterminism: most ordinary counterfactuals are probably false In what follows, by counterfactuals I will mean would counterfactuals of the form if X were the case, Y would be the case, denoted X Y. I will also speak of might counterfactuals (always so qualified) of the form if X were the case, Y might be the case, denoted X Y. I will focus on two strategies for showing a counterfactual to be false: appealing to indeterminism in particular, chanciness; and to indeterminacy in particular, imprecision. Both are strategies for securing the truth of might not counterfactuals that are, I will argue, incompatible with the corresponding would counterfactuals. I will find it convenient to argue via the truth of the might nots to the falsehood of the woulds, but I think that the strategies can also be deployed directly: a counterfactual cannot second guess 3 A fairly reliable source tells me that in Sweden they tried to pass a law to abolish the subjunctive conditional. 4 See Lewis 1973, 1.8. We will return to these rules in 7. 3 the outcome of a process that is chancy, or the resolution of an indeterminacy. The best way to understand what I mean is to see these strategies in action. I begin with indeterministic cases. Suppose for now that coin tossing is a chancy business: as a coin is tossed, it is a genuinely indeterministic matter whether it lands heads or tails; and for now assume for simplicity that these are the only possible outcomes. Here is a coin that in fact will never be tossed. Consider the counterfactual: If the coin were tossed, it would land heads, symbolically, Toss Heads. I submit that it is false. I think you can see that directly: focus on the chanciness of the outcome of the toss. Or consider the might counterfactual: If the coin were tossed, it might land tails, symbolically Toss Tails. This is true it is implied by our supposition of chanciness of the outcome. But I claim that the original would counterfactual and the might counterfactual are contraries: Toss Heads and Toss Tails cannot both be true. Since the might counterfactual is true, it s the would counterfactual that must take the fall. There are several ways to arrive at this conclusion. The would/might duality The first way regards would and might counterfactuals as duals, à la Lewis (1973): X Y is equivalent to (X Y). Then Toss Heads and Toss Heads are contradictories they necessarily have opposite truth values. The duality of the would and might counterfactuals has been defended by a number of authors (e.g. Bigelow and Pargetter (1990, 103), and Bennett (2003, 192)), and assumed by others (e.g. Hawthorne 2005, and 4 Williams 2007). It has also appealed to some authors who have written on counterfactuals in connection with the debate over whether God has middle knowledge of the truth of counterfactuals concerning free actions see, e.g., Adams (1977) and van Inwagen. But my argument does not need the full strength of the would/might duality it suffices that Toss Heads and Toss Heads are contraries. The elimination of possibilities Just try saying out loud: If the coin were tossed, it MIGHT land Tails, and it WOULD land Heads if it were tossed (*) There is surely something defective about (*). More generally, there is serious tension between assertions of 'might and corresponding would not counterfactuals, or between would and corresponding might not counterfactuals. DeRose (1999) calls this the phenomenon of inescapable clashes. Here is an argument that the clash is semantic. The might counterfactual recognizes Tails possibilities, while the would counterfactual eliminates them. Of course, the would counterfactual leaves open various possibilities, corresponding to various ways in which Heads could be realized the coin landing Heads at noon, the coin landing Heads with a soft metallic tinkle, the coin landing Heads and then immediately afterwards vanishing, the coin landing Heads while John Howard impersonates a chicken, and so on. But they are all Heads possibilities; Tails does not occur in any possibility left open by Toss Heads. On the other hand, the might counterfactual is committed to at least one Tails possibility remaining live. And rightly so: clearly, Toss Tails is true. Thus, Toss Heads is false. 5 Redundancy Here is another way to see the point. I say: if I were to toss the coin, it MIGHT not land heads. I then add: And furthermore, it s not true that it WOULD land heads if I were to toss it. You ought to be puzzled. My utterance of furthermore primed you to expect more information, but none was forthcoming. Instead, what came next was redundant. Disagreement and retraction You claim that if the coin were tossed, it would land heads. I disagree. One way of stating my disagreement is to remind you that if the coin were tossed, it might not land heads. We can t both be right. (In fact, how else can I merely deny what you said, rather than committing myself to something stronger than the denial? That is another consideration in favor of the would/might duality.) Similarly, I may assert Toss Heads and later come to retract it, so that I have temporal stages of myself disagreeing with each other. One way that my later self could give grounds for his retraction would be to note that if the coin were tossed, it might land tails. (The earlier self may have had misleading evidence regarding the coin e.g. being told by an unreliable source that it was two headed.) Evidence Think of the evidence that I could muster in support of my claim that if I were to toss the coin, it might land tails for example, the fact that it has landed tails in the past, or that another, similar coin that I just tossed landed tails. But to the same extent, these facts countersupport the would land heads counterfactual. 6 I assumed for the sake of simplicity that Heads and Tails were the only possible outcomes. If the assumption is false, for example, because edge is another possible outcome, no matter. In that case, Toss Heads and Toss Tails are clearly not contradictories, but they are still contraries. The truth of Toss Tails still implies the falsehood of Toss Heads. Or we may bypass considerations of Tails altogether: with some other outcome possible, a fortiori Toss Heads is true (there being a further way that Heads could fail to happen), and it is contrary to Toss Heads. Our foot is in the door; now let s kick it open. I deliberately did not specify the chance of Heads. It is fine if you assumed that the coin was fair, but I did not. The argument would go through equally well, whatever the chance of Tails, as long as it is a possible outcome (and our supposition of chanciness assures us that it is). Then Toss Tails is still true, and that is all I need to establish that Toss Heads is false. For example, let the chance of Tails be It remains true that the coin might land Tails if it were tossed, undermining the corresponding would counterfactual concerning Heads. The point generalizes. Whenever we have a true might counterfactual of the form X Y, any corresponding would counterfactual X Z, where Z is a contrary of Y, is false. If I were to play the lottery, I would lose is false, because I might win, no matter how many tickets there are in the lottery. Indeed, we can drive the chance of the consequent all the way to 1 and still have a false counterfactual. Now suppose that the coin is to be tossed repeatedly infinitely many times. It might land Tails on every toss, even though the chance of this is 0. (Again, I deliberately did not specify the chance of Heads: as long as tails is a possible outcome of any toss, it is a possible outcome of every toss, assuming that the trials are independent.) Thus, I cannot truly say if I were to toss this coin forever, it would land Heads eventually. It might not. 7 This infinite coin toss experiment corresponds to an infinite, highly biased lottery: ticket #i wins iff the coin lands heads for the first time on the i th toss. In this lottery, it is possible that no ticket will win (even though this can only happen in one way, and it can fail to happen in infinitely many ways to which, moreover, the probability distribution is heavily biased if the coin is fair). So I cannot even truly say if I were to hold all the tickets in the lottery, I would win. I might not. In an indeterministic world such as ours appears to be, lotteries in a broad sense abound. I say appears to be, because we are not certain that the world we live in is indeterministic, but the evidence from quantum mechanics certainly seems to point that way. 5 And it isn t just the canonical quantum mechanical examples radioactive decay, spin measurements on a particle in a Stern Gerlach apparatus, and so on that are indeterministic. The indeterminism reaches medium sized dry goods (and even oversized wet ones), just less obviously so. Two billiard balls colliding may approximate a deterministic system, but even they are not immune from quantum mechanical indeterminism. One ball might spontaneously tunnel through the other, or to China, or to the North Star incredibly unlikely, to be sure, but possible. Thus, I cannot truly say if the cue ball were to hit the 8 ball, the 8 ball would begin rolling. Or again, whenever I jump in the air, there is a minute chance that I will not come down I might vaporize instead, for the chance of that happening is non zero. Thus, I cannot truly say if I were to jump, I would come down. With indeterminism reaching so far, a surprisingly large array of ordinary counterfactuals really have chancy consequents. Thus, they are as bad as the coin tossing counterfactual with which I began, as bad as counterfactuals about lottery outcomes that are not guaranteed. Indeed, these ordinary counterfactuals are worse than the earlier counterfactual about the infinite sequence of coin tosses, because 5 To be sure, Bohmian mechanics is deterministic. In the next section I argue that determinism will not save most counterfactuals from being false. 8 unlike that counterfactual, the ordinary counterfactuals have consequents whose chances are less than 1. Ordinary the counterfactuals may be, but that doesn t save them from being false. At this point you may want to protest. PROTEST: Granted, anomalous results such as a billiard ball quantum tunneling to China or a person vaporizing have positive chance of occurring. But in the nearest possible worlds in which the relevant antecedents are true, the results do not occur. A person s vaporizing mid jump, for example, is such a bizarre event that a world in which it happens is rather remote from ours, and in particular more remote than worlds in which he falls normally. Thus, the counterfactuals come out true after all. I believe that you have effectively denied that the might counterfactuals are true. For example, in claiming that all the nearest worlds in which I jump are worlds in which I fall normally, you have ruled out my vaporizing from being among them, thus apparently ruling out that I might vaporize were I to jump. But what entitles you to do that? (We will revisit this point later if you are not already convinced of it.) Not this: the fact that the chance of my vaporizing is small. That would be reminiscent of a methodological precept a bad one adopted variously by Cournot, Kolmogorov, and Popper to regard events with small probabilities as impossible. You might think of it as a maxim to round off small probabilities. I think of it, at least in Popper s case, as an attempt to fit the round peg of inductivism into the square hole of deductivism to square the circle, if you will. The precept might make our lives easier, but it is bad for probability s philosophical foundations. Far from being impossible, events with small probabilities happen all the time indeed, the perfectly normal jump that I just performed was one of them. Your reading these words, exactly as you are, is another one of them. In any case, having a highly probable consequent is not sufficient for a counterfactual s being true, for reasons akin to the lottery paradox. Wherever you set the threshold for highly probable (assuming that it is below 1), it will not be high enough. Suppose, for example, that 9 you set the threshold at you claim, for example, that if a coin is biased at least % towards Heads, then Toss Heads is true. But imagine a non actual lottery with a 1,000,001 tickets. Consider the following list of a million and one counterfactuals, each of which has a consequent with probability above the threshold: lottery is played ticket #1 loses; lottery is played ticket #2 loses; lottery is played ticket # loses. They can t all be true, because the counterfactuals are jointly inconsistent (there is no possible world in which the lottery is played and in which all of the tickets lose). So at least one is false a counterexample to sufficiency of the threshold. And so it goes for any putative threshold (below 1); I will just imagine a lottery with enough tickets to thwart it. Notice that the analogue of a popular response to the lottery paradox denying that belief is closed under conjunction is not available here. For denying that truth is closed under conjunction would be a far more radical thesis than my thesis that most counterfactuals are false! Or you may instead insist that the threshold is context dependent unusually high in lottery contexts (so that all of the million and one counterfactuals above come out false), lower in various other contexts. It is unclear exactly what counts as a context, but there is a concern that on any reasonable understanding of it, the threshold account will deliver unintuitive results of the kind that it was intended to avoid. For example, counterfactuals like lottery is played the owner of the winning ticket is happy and perhaps even lottery is played the sun rises the following day 10 will come out false, since if the lottery is large enough, it will set up a context in which the threshold is so high that the probabilities of the consequents will not achieve it. And as I say, in a broad sense lotteries abound. I challenge the context dependent threshold theorist to come up with an account that correctly adjudicates as false the million and one counterfactuals above (and likewise for similar counterfactuals, however large the lottery), while adjudicating as true the ordinary counterfactuals about cup fallings and so on. 6 Indeed, even setting the probability threshold at 1 will not suffice. We have already seen a probability zero event that is possible: the coin landing Tails forever. Such events are forced upon us if we have uncountable probability spaces. For example, assuming that the radioactive decay laws involve continuous rather than step functions, just as physics says they do, then probability zero events abound. So in this sense even zero probabilities cannot be rounded off. They certainly are not equivalent to impossibilities and given our assumptions, that s a genuine certainty, not merely the probability 1 kind! I put the word bizarre in your mouth during your imagined Protest, and presumably that goes beyond merely having small probability. After all, your hearing these words, exactly as you are, was antecedently improbable, but it is not (I hope!) a bizarre event. Fleshing out what bizarre really means will be no mean feat. Still, I agree that in some good, intuitive sense, there is something strange about the events I am imagining; I used the word anomalous myself. My ill fated jump s positive probability does not save it from that epithet. But I am asking very little of that ill fate: all I need to be true is the modal claim that I might vaporize, were I to jump. If you want to take issue with that, you have not only me but also quantum mechanics to contend with. 6 A detailed discussion of all the moves and countermoves that might be considered would derail this already lengthy paper. Many of them will parallel moves that Hawthorne (200x) considers in his discussion of how knowledge claims are incompatible with their chanciness. My skepticism about the truth of counterfactuals in the face of chanciness parallels his skepticism about knowledge claims in the face of chanciness. 11 So if nothing else, quantum mechanics is there to guarantee the truth of the might counterfactuals that undermine the corresponding would counterfactuals. But often less esoteric facts will equally do the job. Holding my cup tantalizingly over a hard floor, you say: If I were to let go of the cup, it would fall. And if it were to fall and hit the floor, it would break. Well, no, and no it might not, and it might not. If I were to let go of the cup, a sudden gust of wind might lift it higher; and if it were to fall and hit the floor, another gust of wind might slow down its fall sufficiently to spare it a damaging impact. Or even less esoterically, I might catch the cup, sparing it an impact altogether. Quantum mechanics is just a handy, cover all way for me to secure the truth of a huge raft of undermining might counterfactuals in one fell swoop. But other anomalous happenings could do the job just as well on a case by case basis. You may grant me that that there would be a chance of these anomalous happenings, but still deny that they might happen if the relevant antecedents were realized. Y

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