p2

VIII. THE COSTS OF POSTPONEMENT The second kind of cost that might be associated with gathering evidence is the cost of postponing the decision. In the most extreme case, looking for further evidence amounts to losing the option of doing A. For example, it might be that one’s friend is only available to listen to one’s secret today, so if one does not reveal it, one will lose that option forever. Or it might be that one is deciding whether to get married and one’s potential spouse has given one an ultimatum. Or it might be that one needs to choose a vehicle to drive critically injured patients to the hospital, and any delay will result in their certain death. In these cases, the overall value of gathering further evidence will be negative: it will be the difference between the expected value (on one’s current credences) of doing A and that of doing B, since deciding to gather further evidence is equivalent to deciding to do B.18 In a less extreme version, A might be an action that provides more good to the decision-maker the earlier he chooses it (in the event that X holds), so the utility of choosing A tomorrow might be slightly lower than the utility of choosing A today. If we imagine that the agent always prefers a day of marriage with a faithful spouse to a day of bachelorhood, or that in the event that God does exist, the agent prefers a day spent as a monk to a day spent as an ordinary citizen, then each day of postponing the decision is costly. For decisions where postponement is costly but does not prevent the agent from eventually choosing A, under what conditions do these costs outweigh the benefit of gathering additional evidence? To answer this question, let us assume that the only cost associated with postponing the decision occurs in the event that one eventually does A and that X obtains. Then let us fix the cost of postponing the decision while one does a particular experiment as c: specifically, c is the difference between doing A when X obtains without 18 Recall the supposition that one’s credence before gathering evidence is high enough to make A the rational choice. 22gathering more evidence (or while committing to A regardless of the evidence) and doing A when X obtains after gathering more evidence. We can now say something about when the value of performing the experiment is apt to be negative. It will be lower when c is higher; that is, when the costs of doing the experiment in the circumstances in which it is costly are higher. It will also be lower when M (the value of the ‘middling’ act B) is lower, and H and L (the value of A if X obtains and if ~X obtains, respectively) are higher; that is, when there is less of a risk involved in doing A. Next, it will be lower to the extent that one of the following situations is likely to result (that is, to the extent that these situations are probable, given the evidence). Both situations are situations in which X in fact holds. The first is when X in fact holds and the experiment yields some result that will make it rational for the individual to do A. In this situation, performing the experiment hurt the individual because he paid a cost c that he otherwise would not have paid. (For example, the third party says your friend is trustworthy and so you stick with your original plan of sharing your secret, but you incur the cost of the time it took to listen to the third party – perhaps by the time you ask the third party, the friend is no longer available today.) The second situation is when X in fact holds but the experiment yields some result that makes it no longer rational for the individual to do A. This is a situation in which the evidence is ‘misleading’ in the sense that it leads to the rational performance of an action that in fact has a lower payoff than its alternative. (For example, the third party says your friend is a gossip, and so you don’t reveal your secret, but your friend is in fact trustworthy – here you get a payoff of 1 rather than 10, because you have refrained from telling your secret rather than telling it to a trustworthy friend.)19 Both of these situations are more likely the higher the credence one already assigns to X. The second of these situations is more likely to the extent that evidence against X is non-conclusive: to the extent that it will lower one’s credence enough to make it rational to do B instead of A, but not enough to guarantee that B is in fact the better choice. The overall expected utility of gathering evidence depends on how likely each of the ‘beneficial’ situations is (the situation in which you get evidence that steers you away from your original choice when that would have in fact been the wrong choice, and, to a lesser extent, the situation in which you get evidence that confirms your original choice when that is in fact the right choice) as compared to how likely each of the ‘detrimental’ situations is (the situation in which you get evidence that doesn’t alter your choice but is costly, and the 19 See Buchak (2012) for specific mathematical results. 23situation in which you get misleading evidence that steers you away from what would have in fact been the right choice). Holding fixed H, M, and L, in situations in which gathering the evidence proves costly in the event that X obtains and the agent does A, refraining from gathering further evidence is more likely to be rationally required (1) when this cost is high; (2) when one already has a high credence in X; or (3) when the experiment is likely to result in misleading evidence against X, that is, evidence that makes one ‘miss out’ on the possibility of doing A when X in fact holds. This third possibility holds to the extent that the potential evidence that could tell against X does not tell conclusively against X. The fact that costs associated with postponing a decision can make faith rational vindicates an observation made by William James, though he did not express it in these terms. James argued that when a decision about what to believe is momentous—in that it involves a once-in-a-lifetime opportunity, for example—then it must be made by the will, and that postponing the decision is a decision in itself. He used this observation to argue that it is rationally permissible to choose to believe in God even when one does not have conclusive evidence for God’s existence. I don’t think that it is rationally permissible to believe that God exists when one does not have conclusive evidence, if this means setting one’s credences differently from what one has evidence for (though I’m not saying that this is what James is suggesting). However, I do think that it is sometimes rationally permissible (and indeed, sometimes rationally required!) to have faith in God—as evidenced by doing some particular religious act without looking for further evidence—in circumstances in which postponing the decision to act is costly, provided one has the appropriate credences, and provided these are the correct credences to have given one’s evidence. The upshot of this discussion is that, if we accept expected utility theory as the correct theory of practical rationality, then faith can be rational—depending, of course, on one’s credences and the situation in which one finds oneself. We have seen three important results in this regard. First, if we think that faith requires only a weak preference for not gathering additional evidence—that is, if you count as having faith when you are indifferent between making the decision on current evidence and postponing the decision—then faith is rationally permissible, but not rationally required, in cases in which no piece of evidence that one could potentially gather would alter the agent’s decision about what to do. This will hold when no piece of evidence will tell conclusively enough against X such that doing A will no longer maximize expected utility. However, if we think that faith requires a strict preference for no 24additional evidence—that is, you must strictly prefer making the decision on current evidence—then faith will not be rationally permissible in these circumstances.20 Second, faith (under both the strict and weak reading of preference) will be rationally required in circumstances in which there is an interpersonal cost to looking for more evidence; that is, in which lacking faith is intrinsically worse than having faith. However, it is unclear whether such circumstances obtain. In my opinion, the right explanation for the fact that there are relational goods one can’t get unless one has faith isn’t that faith is in itself valuable, but rather that there are some goods that one can’t get if one is more suspicious of another person than the evidence warrants, or if one hesitates to act on a matter involving the relationship. Third, and most crucially, faith (under both readings) is rationally required in circumstances in which the costs of delaying the decision are high enough to outweigh the benefit of additional evidence. Holding fixed the costs of delay, whether these costs outweigh the benefits depends both on one’s credence in the proposition one has faith in and on the character of the potential evidence one might encounter: in particular, faith is more apt to be rational if potential evidence against X will be inconclusive. IX. RISK AVERSION AND THE POSSIBILITY OF MISLEADING EVIDENCE There are two reasons to think that our results so far are incomplete. First, one might think that faith requires more than a choice not to gather additional evidence—or more than a choice to commit to an action before the evidence comes in. It requires a choice to not gather additional evidence even when this evidence is cost-free. For example, we may think that the person who examines the private investigator’s envelope even when there are no ‘postponement costs’ lacks faith. Second, one might think that expected utility maximization is too strong a criterion of rationality, and that one can be practically rational without being an expected utility maximizer. 20 Perhaps we could argue that faith is rational in these circumstances by stipulating that every experiment has some cost? However, when we consider that faith requires not just a (strict) preference for avoiding evidence in the matter of X when deciding whether to do A or B, but more precisely a (strict) preference for committing to a decision before seeing the evidence, we realize that we would have to stipulate that not committing before performing the experiment always has a cost, and this is less plausible. 25While I am sympathetic to the general aim of decision theory, and hold that expected utility theory is largely correct in its analysis of practical rationality, I nonetheless think that expected utility theory employs too narrow a criterion of rationality and should therefore be modified.21 In my view, expected utility theory dictates a too-narrow way in which riskconsiderations can play a role in an individual’s choices. It dictates that rational individuals cannot, for example, care proportionately more about what happens in the worst-case scenario than the best, or vice versa. But most people do pay special attention to these features of their decisions. Moreover, rather than being an example of human irrationality, I think this tendency can be rational. Above, I explained that decision theory formalizes means-ends reasoning: utility corresponds to how much an individual values particular outcomes, while credence corresponds to the likelihood with which the individual thinks some particular act will realize one of these outcomes. But even once we know an individual’s credence and utility function, there is an additional question of how to move from considerations about how a choice will turn out under various circumstances to an overall evaluation of that choice. For example, how should one trade off a small chance of something great against a high chance of something fairly good? We need to know the relative priority the individual places on the worst-case and best-case scenario (and all the scenarios in between). We might think of this additional factor, the individual’s risk-attitude, as corresponding to how he structures the potential realization of some of his aims. What is at issue here is, roughly speaking, the following: when an individual is making a single decision, ought he to care only about how a decision would turn out on average if it were to be repeated, or can he place some weight on ‘global’ features of a decision like how spread out the utility values are?22 In an earlier paper, I demonstrated that individuals who care proportionately more about what happens in the worse-case scenario will sometimes rationally reject cost-free evidence (see Buchak 2010). In particular, in scenarios like those outlined in the previous section, it will be rationally required for these individuals to commit to an action A before looking at additional evidence 21 See L. Buchak (2013). 22 I say ‘roughly speaking’ because on certain interpretations of what utility amounts to, this will be a controversial way of putting things. But the exact formulation is not important to the point here, which is that there are some ways of evaluating gambles that expected utility theory cannot capture, that appear to stem from individuals caring proportionately more about certain possible outcomes of a gamble and less about others, and that (I argue elsewhere) are rational. 26rather than to look at additional evidence and then decide. This will be the case in situations with the following properties, familiar from the previous section: (1) the individual already has a high credence in X (that is, p(X) is antecedently high); and (2) the experiment is likely to result in misleading evidence against X (again, evidence that makes him ‘miss out’ on the possibility of doing A when X in fact holds) because potential evidence that tells against X will not be conclusive. This is the case because if an individual already has a high credence that X, more evidence in favour of X won’t be very helpful; and while evidence against X could be helpful if ~X holds, it will be harmful if X holds – and to the extent that evidence against X is inconclusive these two situations will be closer in likelihood. In short, agents who care about global features of decisions are concerned with a particular risk involved in looking for additional evidence: the risk of coming across evidence that makes it rational to refrain from doing A even though X in fact holds. In other words, they are concerned, and rationally so, about the risk of getting evidence that is misleading. I already mentioned that the possibility of misleading evidence will make it rational for the EU maximizer to reject costly evidence. However, on the more permissive decision theory, the risk of misleading evidence makes faith rational even in cases in which there is no cost to looking for evidence. If one accepts the more permissive decision theory, then faith is rational in more cases than it is on standard decision theory, and precisely in cases in which the risks of getting misleading evidence outweigh the benefits of getting non-misleading evidence. X. CONCLUSION We have seen that whether faith that X, expressed by A, is rational depends on two important factors: (1) whether one has a high enough (rational) credence in X, and (2) the character of the available evidence. Specifically, faith in X is rational only if the available evidence is such that no potential piece of evidence would tell conclusively enough against X. There are two interesting practical upshots of this conclusion. First, notice that in a standard class of cases, when one has a high degree of belief in a proposition, the odds of any particular experiment being such that it could drastically lower one’s degree of belief decreases the larger the collection of evidence the agent already has.23 So, in a rough-and23 As James Joyce avers, it is ‘usually the case that the greater volume of data a person has for a hypothesis the more resilient her credence tends to be across a wide range of additional data’ (2005: 161). 27ready way, we might say that faith that X (expressed by some particular act A) is practically rational to the extent that the individual’s degree of belief in X is already based on a large body of evidence. Second, whether faith is rational depends on the kind of situation we find ourselves in. Faith will be rational to the extent that potential counterevidence wouldn’t be very conclusive with respect to the position in question, or to the extent that our decisions usually do have postponement costs. We won’t be able to vindicate the claim that faith is rational regardless of the circumstances. But we can explain why having faith is rational in certain circumstances, perhaps circumstances some of us find ourselves in some of the time. Individuals who lack faith because they insist on gathering all of the available evidence before making a decision stand to miss out on opportunities that could greatly benefit them.

FREE ACTS AND CHANCE: WHY THE ROLLBACK ARGUMENT FAILS BY LARA BUCHAK
The ‘rollback argument,’ pioneered by Peter van Inwagen, purports to show that indeterminism in any form is incompatible with free will. The argument has two major premises: the first claims that certain facts about chances obtain in a certain kind of hypothetical situation, and the second that these facts entail that some actual act is not free. Since the publication of the rollback argument, the second claim has been vehemently debated, but everyone seems to have taken the first claim for granted. Nevertheless, the first claim is totally unjustified. Even if we accept the second claim, therefore, the argument gives us no reason to think that free will and indeterminism are incompatible. Furthermore, seeing where the rollback argument goes wrong illuminates how a certain kind of incompatibilist, the ‘chance-incompatibilist,’ ought to think about free will and chance, and points to a possibility for free will that has remained largely unexplored. Libertarians hold that free will is incompatible with determinism, but that we nonetheless have free will. Of course, the truth of indeterminism is not enough to guarantee free will: for an act to be free, it must originate from the agent herself in some important sense. Whether an act is free thus depends on the source of the indeterminism. We might take for granted that there are sources of indeterminism conducive to free acts. Recently, however, Peter van Inwagen has introduced an argument that has come to be known as the ‘rollback argument,’ that challenges whether indeterminism in any form can leave room for freedom. This argument purports to show that if indeterminism holds, then regardless of what this indeterminism consists in, every act is a mere matter of chance in the sense incompatible with free will. If the rollback argument is sound, then libertarians must conclude that free will is compatible with neither determinism nor its denial, and so, in the words of van Inwagen, ‘free will remains a mystery.’ Determinism is the thesis that the state of the world at time t1 in conjunction with the laws of physics entail the state of the world at a later time t2. Libertarians hold that determinism is incompatible with free will, usually on the grounds that if there is only one physically possible future, © 2012 The Author The Philosophical Quarterly © 2012 The Editors of The Philosophical Quarterly Published by Blackwell Publishing, 9600 Garsington Road, Oxford ox4 2DQ, UK, and 350 Main Street, Malden, MA 02148, USA21 FREE ACTS AND CHANCE then an agent’s actions are not ‘up to’ him in the sense relevant for free will. Indeterminism is just the denial of determinism, though it is clear that not just any kind of indeterminism will do for free will. For example, if an agent’s actions at t2 are undetermined at t1 because they are to be determined by the flip of a coin between t1 and t2, then the agent’s actions are not up to her any more than if they are determined at t1. They are mere matters of chance. Free acts, according to libertarians, need to be not only undetermined, but undetermined in the right way: undetermined because they are ultimately up to the agent. Van Inwagen’s rollback argument challenges the idea that acts can ever be undetermined in the sense required for free will. The argument purports to show that regardless of what governs agent acts under indeterminism, all agent acts will have the same status as acts governed by coin-flips, which is to say, they will not be free. Van Inwagen specifically argues that agent-causation is not sufficient to make agent acts free, but his argument easily generalises to any way of spelling out what holds of an agent in an indeterministic world. Here is his argument.1 Consider an agent, Alice, who is deciding whether to lie. Let us assume her choice is undetermined by the state of the world at t1 and the laws of physics. And let us say she lies at t2. Can this have been a free act? To show that it cannot have been, van Inwagen asks us to consider what would have to be true if, hypothetically, God were to reset the universe to t1 and let events transpire as they may; and if God were to do this many times over. Since Alice’s lying is not determined, it would have to be the case that she would lie in some replays and not lie in others. Now, if God were to replay the event enough times, the proportion of replays in which Alice lies to replays in which she tells the truth would almost certainly converge to some definite number. For example, let’s say that after 100 replays, she has lied 35 times; after 1000 replays, she has lied 326 times, and after 10000 replays she has lied 3076 times. We would then be confident that the proportion of lies to total cases would settle out to 0.3: she lies in 30% of the cases. But to say that she lies in 30% of the cases is just to say that there is a 30% chance of her lying in any particular case, including some hypothetical next case. And including, indeed, the actual case at hand. Furthermore, if there is a definite objective probability to her lying, then whether she lies in the case at hand is a mere matter of chance: it is as if whether she lies is determined by the flip of a biased coin which has a 30% 1 P. van Inwagen, ‘Free Will Remains a Mystery’, Philosophical Perspectives, 14 (2000), pp. 1–20, at pp. 13–18. © 2012 The Author The Philosophical Quarterly © 2012 The Editors of The Philosophical Quarterly22 LARA BUCHAK chance of landing heads. Finally, notice that to reach this conclusion we did not rely on a particular assumption about the source of the indeterminism or the source of its resolution between t1 and t2: regardless of the mechanics of choice, says the argument, an undetermined choice is relevantly like flipping a coin. To see the crucial steps of the argument, here it is in premiseconclusion form: (P1) If indeterminism holds, then if God replayed the universe numerous times in the above scenario, it would become increasingly likely, as the number of replays increased, that the ratio of lies to truths would converge to some definite real number. (P2) If the ratio of lies to truths would converge to a definite real number in the above scenario, then Alice’s lying in the case at hand and Alice’s telling the truth in the case at hand each have a definite objective probability at t1, namely the ratio of lies to total cases and the ratio of truths to total cases. (C1) If indeterminism holds, then Alice’s lying and Alice’s telling the truth each have a definite objective probability at t1. (P3) If an act has a definite objective probability at a time, then it cannot be a free act at that time. (C2) If indeterminism holds, then whatever Alice does, it won’t be a free act. Discussion of the rollback argument has centered around (P3): denials of this claim are articulated by Mark Balaguer, Michael Almeida and Mark Bernstein, Timothy O’Connor, Laura Eckstrom, and Christopher Evans Franklin; and Seth Shabo provides an additional argument in its favour.2 However, to my knowledge, everyone who discusses the argument has taken (P1) and (P2) for granted.3 Denying (P2) is not a 2 M. Balaguer, Free Will as an Open Scientific Problem, (MIT Press, 2010), chapter 3; M. Almeida and M. Bernstein, ‘Rollbacks, Endorsements, and Indeterminism’, in R. Kane (ed.), The Oxford Handbook of Free Will, (Oxford UP, 2011), pp. 484–95; T. O’Connor, ‘Agent-Causal Theories of Freedom’, in R. Kane (ed.), The Oxford Handbook of Free Will, (Oxford UP, 2011), pp. 309–28; L. Eckstrom, ‘Free Will, Chance, and Mystery’, Philosophical Studies, 113 (2003), pp. 153–80; L. Eckstrom, ‘Free Will Is Not a Mystery’, in R. Kane (ed.), The Oxford Handbook of Free Will, (Oxford UP, 2011), pp. 366–80; C. Evans Franklin, ‘Farewell to the luck (and Mind) argument’, Philosophical Studies, 156 (2011), pp. 199–230; S. Shabo, ‘Why Free Will Remains a Mystery’, Pacific Philosophical Quarterly, 92 (2011), pp. 105–25. 3 In their discussion of the rollback argument, Almeida and Bernstein (pp. 485) and Franklin (pp. 216) explicitly endorse (C1) without argument, so we may assume they endorse (P1) and (P2). Eckstrom does note in passing that if an agent’s choices are ungoverned by laws, they will have no probability (a denial of (C1)), though she doesn’t detail where the rollback argument goes wrong if this holds or what ‘ungoverned by laws’ means in this context, since her focus is a denial of (P3). © 2012 The Author The Philosophical Quarterly © 2012 The Editors of The Philosophical Quarterly23 FREE ACTS AND CHANCE very attractive option, since it seems to be an unproblematic instance of inference to the best explanation. But I claim that we have no reason to accept (P1). It is worth looking in detail at why van Inwagen thinks that the ratio of lies will converge to some definite real number. Here is what he says: ‘Now let us suppose that God a thousand times caused the universe to revert to exactly the state it was in at t1 (and let us suppose that we are somehow suitably placed, metaphysically speaking, to observe the whole sequence of “replays”). What would have happened? What should we expect to observe? Well, again, we can’t say what would have happened, but we can say what would probably have happened: sometimes Alice would have lied and sometimes she would have told the truth. As the number of “replays” increases, we observers shall—almost certainly— observe the ratio of the outcome “truth” to the outcome “lie” settling down to, converging on, some value…”Almost certainly” because it is possible that the ratio not converge. Possible but most unlikely: as the number of replays increases, the probability of “no convergence” tends to 0.’4 Van Inwagen’s reason for thinking that the convergence will occur is clearly the law of large numbers, which says roughly that if we repeat an event with two possible outcomes many times over, the ratio of each outcome to the number of trials will, with increasing likelihood, tend to the (objective) probability of each outcome. For example, if we flip a biased coin long enough, the proportion of heads to total flips will almost certainly converge to the coin’s bias towards heads. However, van Inwagen fails to notice that there is an important difference between the coin case and Alice’s case. In the case of the coin, we apply the law of large numbers because we assume the coin does have some definite objective probability of landing heads. That there is some definite probability involved is a presupposition of the law of large numbers. For example, here is a typical statement of the law: ‘In repeated, independent trials with the same probability p of success in each trial, the percentage of successes is increasingly likely to be close to the chance of success as the number of trials increases. More precisely, the chance that the percentage of successes differs from the probability p by more than a fixed positive amount, 4 Van Inwagen (p. 14 and footnote 16). The passage I have quoted is from van Inwagen’s argument that undetermined acts aren’t free, without allowing for the possibility of agent-causation. He goes on to claim that the argument works the same way for agentcaused undetermined acts, since it is nowhere mentioned whether or not Alice’s acts result from agent-causation. We may also assume that it is supposed to work against any kind of act under indeterminism, since it nowhere relies on the mechanics of action or of indeterminism. © 2012 The Author The Philosophical Quarterly © 2012 The Editors of The Philosophical Quarterly24 LARA BUCHAK e > 0, converges to zero as the number of trials n goes to infinity, for every number e > 0.’5 We can only apply the law at all if its antecedent is satisfied: i.e., if the event in question has some probability p, and has this probability in each of the trials. But this is precisely what van Inwagen is trying to argue for in this step of the argument: he is trying to argue that we can assign a probability to the event that Alice lies.6 The rollback argument directly begs the question of whether Alice’s lying has an objective probability. Without the assumption that it does, there is nothing at all in the setup of the rollback scenario itself to guarantee the truth of (P1). There is nothing at all to rule out, for example, the following series of choices: the first time God reruns the situation, Alice lies; the next 9 times, she tells the truth; the next 90 times, she lies; the next 900 times, she tells the truth; and so forth. In this example, the proportion of lies never converges (it will alternate between roughly 1/11 and 10/11, after each 10n trials). Contra van Inwagen, there is nothing in his setup even to make this unlikely. Unlike in the coin-flipping case, there may not be a chancy mechanism– or a mechanism that behaves as if it is governed by chance– grounding Alice’s actions. Since (P1) and (P2) are supposed to supply an argument for (C1), van Inwagen can’t support (P1) using the law of large numbers, because to do so assumes (C1), the very thing at issue. The truth of (P1) is an empirical question, and one we are incapable of testing in principle. It is now clear that we have no reason to be convinced by the rollback argument as it stands. But the insight here goes beyond a refutation of the rollback argument. That one can accept (P3) without concluding indeterminism and free will are incompatible points to an unexplored possibil5 P.B. Stark, ‘Glossary of Statistical Terms,’ http://www.stat.berkeley.edu/~stark/SticiGui/ Text/gloss.htm. Accessed on 2/09/2011. 6 After establishing that the replays would converge to a definite proportion, van Inwagen imagines for illustration that the proportion of lies to truths is roughly even, and then writes (p. 15): ‘A sheaf of possible futures (possible in the sense of being consistent with the laws) leads ‘away’ from [t1], and, if the sheaf is assigned a measure of 1, surely, we must assign a measure of 0.5 to the largest sub-sheaf in all of whose members Alice tells the truth and the same measure to the largest sub-sheaf in all of whose members she lies. We must make this assignment because it is the only reasonable explanation of the observed approximate equality of the ‘truth’ and ‘lie’ outcomes in the series of replays. And if we accept this general conclusion, what other conclusion can we accept about the seven-hundredand-twenty-seventh replay (which is about to commence) than this: each of the two possible outcomes of this replay has an objective, ‘ground-floor’ probability of 0.5—and there’s nothing more to be said? And this, surely, means that, in the strictest sense imaginable, the outcome of the replay will be a matter of chance.’ Thus, it is clear that the rollback scenario is supposed to establish that lying has a definite probability, namely a probability equal to the (convergent) proportion of cases in which the agent lies. © 2012 The Author The Philosophical Quarterly © 2012 The Editors of The Philosophical Quarterly25 FREE ACTS AND CHANCE ity for ‘chance-incompatibilists,’ i.e., incompatibilists who think that an act cannot have been free at a time if its occurrence had a definite chance at that time. In particular, it is open to chance-incompatibilists to deny that a free act has a definite objective chance of occurring before the agent exercises her free will. The thought that there is a difference between agent acts and ordinary goings-on in the world– in this case a difference in whether we can assign objective probabilities to their occurrence ahead of time– naturally calls to mind the original target of van Inwagen’s argument: agent-causation. Agent-causation views say that an act is free just in case the agent in question is a ‘substance’ that acts rather than a mere locus for physical events in the causal chain of that act: this is to say, if we list only the physical events leading up to a free act, then we have left out a member of the causal chain.7 The metaphysics of agent-causation are notoriously tricky, but the discussion here points us to one concrete metaphysical difference that the proponent of agent causation could postulate: agentcaused events lack objective probabilities. Of course, introducing agentcausation is not the only way for the chance-incompatibilist to deny that agent acts have objective probabilities. There may be other theories about the metaphysics of free will that can plausibly deny this. The point is that there are avenues open to the chance-incompatibilist to resist the conclusion that we lack free will: as long as one analyses free will in such a way that free acts lack objective probability, van Inwagen’s argument will have no purchase. Is maintaining that free acts lack objective probability inconsistent with what current physics tells us? While a full discussion of this question goes beyond my knowledge of physics, here is a reason to think that it is not. This reason originates in an argument for a seemingly unrelated point: specifically, in Alan Ha ´jek’s argument that conditional probability rather than unconditional probability ought to be thought of as primitive.8 In the course of his argument, he notes that quantum mechanics primarily tells us about certain objective conditional probabilities. For example, he says that the ‘Born rule’ tells us about probabilities of the form p(Ok | M), where M is the proposition that a particular measurement takes place (according to Ha´jek, the act of some agent) and Ok is the proposition that 7 The classic statement of this view can be found in R.M. Chisholm, ‘Human Freedom and the Self’, University of Kansas Lindley Lecture, Department of Philosophy, University of Kansas (1964), pp. 3–15. Reprinted in G. Watson (ed.), Free Will, (Oxford UP, 2003), pp. 26–37. 8 A. Ha ´jek, ‘What Conditional Probability Could Not Be’, Synthese, 137 (2003), pp. 273323. © 2012 The Author The Philosophical Quarterly © 2012 The Editors of The Philosophical Quarterly26 LARA BUCHAK a particular outcome eventuates.9 Ha ´jek argues that quantum mechanics itself (QM uninterpreted) does not assign an unconditional probability to the proposition M: it is silent on p(M). He argues further that quantum mechanics cannot in principle deliver probabilities of the form p(M). I will not rehash his arguments here. And while Ha ´jek’s conclusion is not uncontroversial (and he states as much),10 the point for present purposes is that physics hasn’t made up its mind about whether all events– in particular events involving the actions of agents– have objective unconditional probability. Indeed, Ha´jek cautions us against inferring from the fact that the micro-level events which are the central subject of physics have probabilities relative to the measurements of observers to the claim that all events have unconditional probabilities: ‘It seems to me that the intuition that chances must always exist, even for free acts, parallels the intuition that values for observables (such as position and momentum) must always exist. But the latter intuition has been challenged since Bohr, and has hit particularly hard times since the Kochen-Specker theorem.’ (307) We shouldn’t be too quick to assume that our current physical theories will assign objective chance to acts, nor that they will say the same things about the behaviour of agents that they do about the behaviour of particles. They might or might not, but it is an empirical question we are not currently in a position to answer. If Ha ´jek’s argument is right, then physics is not committed to assigning unconditional chances to free acts– and there may be additional reasons to think that we cannot assign them. However, we typically will be able to assign conditional chances to propositions. So the important question will be what sorts of conditional chances we can assign to agent acts at the time when they are purportedly free, and whether being able to assign these is incompatible with free will. For example, we might ask which conditional chances of Alice lying at t2 get assignments at t1: if A is the proposition that Alice lies at t2, for what conditions {C} does p(A | C) have 9 Interestingly enough, earlier in the article and in quite a different context than the discussion in this paper, Ha ´jek (p. 304) uses what is essentially a modus tollens version of van Inwagen’s argument to show that not all propositions have unconditional relative frequency (relative frequency being a candidate interpretation for objective probability). Ha ´jek asks us to consider an agent freely deciding whether to toss a coin, and points out that the agent could decide to deliberately make choices so that the frequency with which he decides in the affirmative fluctuates wildly over time, i.e., so that the sequence has no limiting frequency. Of course, this isn’t an argument that we do have free will (and Ha´jek certainly doesn’t intend it to be!), but his use of the modus tollens argument further shows that the truth of (P1) ought not be considered settled in contexts outside of this debate. 10 Ha´jek (p. 307) notes that Bohm’s interpretation, collapse interpretations, and the many worlds interpretation all imply that probabilities of the form p(M) are well-defined. © 2012 The Author The Philosophical Quarterly © 2012 The Editors of The Philosophical Quarterly27 FREE ACTS AND CHANCE a determinate value at t1? And we can then ask whether the existence of any of these conditional chances ought to worry us. Simply showing that there are some conditional chances of the form p(A | C) won’t reveal a problem for the claim that A is a free act. If C is merely a physical description of Alice’s lying at t2, then p(A | C) will equal 1, but this is surely not troublesome. For in this case, the physical description merely is the free act, and since p(A) is not determinate, p(C) is not determinate. More generally, if C is a description of some act whose objective unconditional probability is determinate, then the indeterminateness of p(A) implies that at least one of p(A | C) and p(A | ~C) is indeterminate. This is to say, conditional on at least one of C or ~C the act does not have a determinate objective probability– which I take it is all the chance-incompatibilist needs. So we should expect not to be able to pick a C such that p(C), p(A | C), and p(A | ~C) are all determinate at t1. Therefore, conditional chances of the form p(A | C) where C is some physical event that has a determinate objective probability should not ordinarily pose a problem for the claim that A can be a free act. The chance-incompatibilist cannot, however, conclude that there won’t be any conditional chances that will undermine freedom. For the lack of a determinate p(A | C) for determinate p(C) does not imply that Alice does have free will: if Alice’s lying is determined by the free act of some other agent (if C is ‘Mary forces Alice to lie’), it is surely not free. This observation draws attention to the fact that there are two ways in which probabilistic facts can entail that an agent-act A is not free, according to the chance-incompatibilist. The first is if p(A) is determinate, which we already saw is not compelled by current physics (at least on some stillopen interpretations). The second is if there is a determinate p(A | C) where C is the free act of some other agent. If, at t1, there is some definite probability of Alice lying conditional on an act of Mary’s, chance-incompatibilists will presumably think that Alice is not free at t1, or at least won’t be free if Mary does perform the act: conditional on what Mary does, it is a mere matter of chance whether Alice will lie. It is open to all chance-incompatibilists– proponents of agent-causation and otherwise– to deny that agent-acts have determinate probabilities. However, the second way in which probabilistic facts can threaten freedom sheds light on which kinds of chance-incompatibilists can claim that there are free acts without departing too radically from current physics. Current physics says that many conditional probabilities of the form p(B | C) do exist: namely, conditional probabilities where C is the proposition that a particular measurement takes place and B is a description of a micro-level event of the type studied by physics. Therefore, if free acts are © 2012 The Author The Philosophical Quarterly © 2012 The Editors of The Philosophical Quarterly28 LARA BUCHAK just micro-level events of the type studied by physics, then there should be a determinate p(A | C) where, for example, A is the proposition that Alice lies and C is a proposition describing some measurement process. Given this, the chance-incompatibilist has two ways to make room for freedom. First, she can deny that p(A | ~C) is determinate, and argue that whether an act is free depends on whether the agent is part of a system for which a measurement is in fact not taken; but to take this route she will have to spell out why not taking a measurement should make a difference to freedom. Second, she can deny that free acts are micro-level events of the type studied by physics. This is what the proponent of agent-causation denies. There may be other ways to deny this, but denying this without departing too radically from current physics depends on finding some way to distinguish between agent acts and other kinds of events such that the objective probabilities conditional on measurements won’t always be determinate for agent acts even though they are for micro-events that don’t involve agents. And this may be a difficult task for the theorist who thinks that the decisions of free agents have ordinary micro-level descriptions. I have shown that libertarian freedom is not in as bad a spot as we might have thought. In particular, the rollback argument does not show, even for chance-incompatibilists, that free will is incompatible with indeterminism. If chance-incompatibilism is true, then the question of whether free will is compatible with determinism depends on what exactly agent acts are, and on what our best physical theory ultimately says about whether agent acts have objective chance. The discussion here points the way forward in two respects. First, it reminds us that taking physics seriously may be consistent with thinking there really is a difference between events involving free acts and other kinds of events. Second, it suggests that we ought to turn our attention to the question of what physics is actually committed to as regards the objective chances of acts involving agents, and whether what physics is committed to in this regard is incompatible with free will.