p6

As an analogy: we might think that giving $10 to one person and taking $20 from another preserves the overall value of a social distribution only if we are taking $20 from the best-off person and giving the $10 to the worst-off person, since we might think it’s okay to take more from the top in order to give something to the bottom, but not vice versa. If which tradeoffs an agent is willing to accept is sensitive to where in the structure of the gamble these tradeoffs occur (in the worst state or the best state, for example), then facts about which tradeoffs an agent considers appropriate will only reveal utility differences when we are dealing with gambles that put the events in the same order. By putting the events in the same order I mean: if event E is at least as good as event F for one gamble, then event E is at least as good as event F for another gamble. For example, the gambles in the first three pairs above are all such that the TAILS state is at least as good at the HEADS state, but this is not so in the gambles in the last pair. The technical term for gambles that order the events in the same way is comonotonic gambles, and a set of gambles that all order the events in a particular way is called a comoncone. 18 The Comonotonic Tradeoff Consistency Axiom, the axiom that the REU theorist accepts, dictates that tradeoffs reveal utility differences only when the gambles order events in the same way. It does this by restricting the Unrestricted Tradeoff Consistency Axiom to hold only when we are dealing with comonotonic gambles. In short: if two pairs ($10 rather than $0 and $110 rather than $100) are both appropriate tradeoffs in the same event in some pair of gambles, then the utility value difference between them must be the same unless the tradeoffs occur in different structural parts of the gamble. If $10 rather than $0 in HEADS and $110 rather than $100 in HEADS are both appropriate tradeoffs in the same event for the same pair, and they are tradeoffs in the same structural part of the gamble – if they are “comonotonic tradeoff equal” – then the utility value difference between them is the same: u($10) – u($0) = u($110) – u($100). Again, we can put this in terms of preferences without mentioning utility: if $10 rather than $0 plays the same compensatory role in some event in some pair of gambles as $110 rather than $100 plays in that event in that pair of gambles and all four of these gambles order the events in the same way as each other, then $10 rather than $0 plays the same compensatory role as $110 rather than $100 in any event and any gamble such that all four of the gambles order the events in the same way as each other. This is essentially what Comonotonic Tradeoff Consistency says (with variables instead of specific amounts of money, of course). 29 So Comonotonic Tradeoff Consistency limits the situations in which we can infer utility differences from which tradeoffs an agent is willing to accept. In sum, Unrestricted Tradeoff Consistency says that if one pair plays the same compensatory role in some event in some gamble as another pair plays, then the two pairs must play the same compensatory role in every event in every gamble. Comonotonic Tradeoff Consistency says that this holds only when the two compensating pairs occur in the same structural part of the gamble as each other, for example, when the tradeoffs both occur in the worst outcome or both occur in the best outcome. Here is another way to put this. Unrestricted Tradeoff Consistency entails that the utility contribution made by each outcome is separable from what happens in other states. In other words, y-in-E rather than x-in-E makes the same difference to the overall gamble (it exactly compensates for the same subgambles) regardless of what happens in E. Furthermore, y rather than x makes the same value difference regardless of which event the substitution occurs in – not in terms of absolute value, but in terms of which other tradeoffs it is equivalent to.30 Comonotonic Tradeoff Consistency entails that the utility contribution made by each outcome is separable from what happens in other states if and only if we stay within a comoncone. In other words, y-in-E rather than x-in-E makes the same difference to the overall gamble regardless of what happens in E, as long as E occupies the same position in the “event ordering” in each relevant gamble. But still, if we remain in the same comoncone, then which event E is will not matter, so the value difference a tradeoff makes will be relativized to a gamble, but not to an event. 19 Why might it make a difference which structural part of the gamble a tradeoff occurs in? For example, why does $10 rather than $0 in HEADS play the same compensatory role as $110 rather than $100 in HEADS when HEADS is the worst outcome in both cases, but $10 rather than $0 in HEADS when HEADS is the worst outcome doesn’t play the same compensatory role as $110 rather than $100 when HEADS is the best outcome? There are two possibilities. The first is that the agent considers HEADS more likely when it has a worse outcome associated with it, and less likely when it has a better outcome associated with it. He cares more about what happens in the worst possible state because the worst possible state is more likely. In this case, the agent would not have a fixed view of the likelihood of events but would instead be pessimistic: he would consider an event less likely simply because its obtaining would be good for him. But another axiom of REU theory, the axiom of Strong Comparative Probability, rules this interpretation out: in the presence of the other axioms, it entails that an agent has a stable probability function p of events. Aside from having the standard properties of a probability function, it is an important feature of p that it takes a higher value for one event than another just in case the agent would always rather put a better outcome on the first event than the second, holding fixed what happens in the rest of the gamble. This is reason to take the function p rather than the function r to reflect an agent’s beliefs. The second possibility is that what happens in E matters less to the agent when E is higher in the structural ordering not because she considers E itself less likely, but because this feature of the overall gamble plays a smaller role in the agent’s considerations. If an agent is more concerned with guaranteeing himself a higher minimum, for example, then tradeoffs that raise the minimum are going to matter more than tradeoffs that raise the maximum. I stressed that one thing an agent must determine in instrumental reasoning is the extent to which he is willing to trade off a guarantee of realizing some minimum value against the possibility of getting something of much higher value: that is, the extent to which he is willing to trade raising the minimum against raising the maximum. And, again, this is because agents must determine for themselves how to structure their goals. So we can now see that restricting Tradeoff Consistency to gambles with the same structural properties – gambles that order the events in the same way – captures the idea that agents who are riskaverse in the sense of caring about global or structural properties are structuring their goals differently than EU maximizers. Unrestricted Tradeoff Consistency says that substituting one outcome for another must make the same value difference to a gamble regardless of how these outcomes feature into the structural properties of the gamble. But Comonotonic Tradeoff Consistency says that the difference a substitution makes depends not just on the difference in value between the outcomes in the particular state, but on where in the structure of the gamble this substitution occurs. If the agent cares about these structural properties then he will only obey the comonotonic version of the axiom. Furthermore, if he has 20 stable beliefs about the state of the world, beliefs that don’t depend on how good various events are for him, then he will obey the comonotonic version of the axiom not because he is pessimistic but because he structures his goals so as to place more importance on what happens in the worst possible state. 6. Conclusion I have proposed a theory on which agents subjectively determine the three elements of instrumental rationality: their utilities, their credences, and the tradeoffs they are willing to make in the face of risk. In this paper I have explained how allowing agents to subjectively determine which sorts of tradeoffs they are willing to make corresponds to adopting a weaker set of axioms on preferences than those endorsed by the EU theorist. On EU theory, which tradeoffs an agent is willing to make must be determined solely by the outcomes and events those tradeoffs involve. This means that lowering the value of what happens in an event has the same effect on the value of a gamble regardless of what happens in the rest of the gamble. However, on REU theory, agents can care about where in the structure of the gamble the tradeoffs occur. Therefore, the effect on the value of the gamble can depend on whether it is the value of the minimum or maximum that is lowered. Furthermore, even if the agent assigns the same probability to two events E and F, she needn’t think that lowering the value of E in exchange for raising the value of F (by the same utility) is an acceptable tradeoff. In particular, if the worst-case scenario is proportionately more important to her than the best-case scenario, this may not be an acceptable tradeoff when the prize in E is already worse than the prize in F. What r represents, then, is the extent to which an agent privileges what happens in various structural parts of the gamble: whether she is prudent in making sure the minimum value is high, or venturesome in making sure the maximum value is high. Now that we’ve seen the difference between what EU theory requires of agents and what the more permissive REU theory requires of them, we can properly address the question of which theory captures the requirements of instrumental rationality. Since we can see what decision-makers who are supposedly irrational are doing in terms of taking the means to their ends, the burden will be on the defender of EU theory to show why individuals ought to adopt a very particular strategy for attaining their goals: averaging the utility values without regard to the spread of possibilities, or ignoring global considerations when deciding which tradeoffs to make. I contend that EU theory will not be able to meet this burden, and that it is rational to be sensitive to global properties of gambles in the way I suggest here. But that is a discussion for another time. 21 APPENDIX: REPRESENTATION THEOREM AND TECHINAL DISCUSSION The theorem here draws on two other results, one by Veronika Köbberling and Peter Wakker, and the other by Mark Machina and David Schmeidler.31 The set of axioms I use in the REU representation theorem are a combination of Köbberling and Wakker’s and Machina and Schmeidler’s axioms, strictly stronger than either set of axioms. I start by explaining the spaces and relations we are dealing with.32 The state space is a set of states SS = {…, s, …}, whose subsets are called events. The event space, EE, is the set of all subsets of SS. Since we want to represent agents who have preferences over not just monetary outcomes but discrete goods and, indeed, over outcomes described to include every feature of the world that the agent cares about, it is important that the outcome space be general. Thus, the outcome space is a set of outcomes XX = {…, x, …}. I follow Savage (1954/1972) in defining the entities an agent has preferences over as “acts” that yield a known outcome in each state. The act space AA = {…, f(.), g(.), …} is thus the set of all finite-valued functions from SS to XX, where the inverse of each outcome f-1(x) is the set of states that yields that outcome: f-1(x)∈EE. So for any act f∈AA, there is some partition of the state space SS into {E1, … En} and some finite set of outcomes Y ⊆ XX such that f can be thought of as a member of Yn. And as long as f(s) is the same for all s∈Ei, we can write f(Ei) as shorthand for “f(s) such that s∈Ei.” For any fixed finite partition of events M = {E1, …, En}, all the acts on those events will form a subset AM ⊆ AA. Thus, AM is defined to contain all the acts that yield for each event in the partition, the same act for all states in that event: AM = {f∈AA | (∀Ei∈M)(∃x∈XX)(∀s∈Ei)(f(s) = x)}. An upshot is that for all acts in AM, we can determine the outcome of the act by knowing which event in M obtains: we needn’t know the state of the world in a more fine-grained way. The preference relation ≥ is a two-place relation over the act space. This gives rise to the indifference relation and the strict preference relation: f ~ g iff f ≥ g and g ≥ f; and f > g if f ≥ g and ¬(g ≥ f). For all x∈XX, x denotes the constant act {f(s) = x for all s∈SS}. The relata of the preference relation must be acts, but it will be useful to talk about preferences between outcomes. Thus, we will define an auxiliary preference relation over outcomes: x ≥ y iff x ≥ y (for x, y∈XX) where indifference and strict preferences are defined as above. It will be useful to talk about preferences between outcomes of particular acts, so, following the above definition, f(s) ≥ g(s) holds iff f(s) ≥ g(s), the constant act that yields f(s) in every state is weakly preferred to the constant act that yields g(s) in every state. Furthermore, xEf denotes the act that agrees with f on all states not contained in E, and yields x on 22 any state contained in E: xEf(s) = {x if s∈E; f(s) if s∉E}. Likewise, for disjoint E1 and E2 in EE, xE1yE2f is the act that agrees with f on all states not contained in E1 and E2, and yields x on E1 and y on E2: xE1yE2f(s) = {x if s∈E1, y if s∈E2, f(s) if s∉E1∪ E2}. Similarly, gEf is the act that agrees with g on all states contained in E and agrees with f on all states not contained in E: gEf(s) = {g(s) if s∈E; f(s) if s∉E}. We say that an event E is null on F⊆AA just in case the agent is indifferent between any pair of acts which differ only on E: gEf ~ f for all gEf, f∈F.33 The concepts in this paragraph and the next are important in Köbberling and Wakker’s result. The first, comonotonicity, was introduced by Schmeidler (1989). Two acts f and g are comonotonic if there are no states s1,s2∈ SS such that f(s1) > f(s2) and g(s1) < g(s2). This is equivalent to the claim that for any partition AM of acts such that f,g∈AM, there are no events E1, E2∈M such that f(E1) > f(E2) and g(E1) < g(E2). The acts f and g order the states (and, consequently, the events) in the same way: if s1 leads to a strictly preferred outcome to that of s2 for act f, then s1 leads to a weakly preferred outcome to that of s2 for act g. We say that a subset C of some AM is a comoncone if all the acts in C order the events in the same way: for example, the set of all acts on coin-flips in which the heads outcome is as good as or better than the tails outcome forms a comoncone. Formally, as Köbberling and Wakker define it, take any fixed partition of events M = {E1, …, En}. A permutation ρ from {1, …, n} to {1, …, n} is a rank-ordering permutation of f if f(Eρ(1)) ≥ … ≥ f(Eρ(n)). So a comoncone is a subset Cρ of AM that is rank-ordered by a given permutation: Cρ = {f ∈ AM | f(Eρ(1)) ≥ … ≥ f(Eρ(n))} for some ρ. For each fixed partition of events of size n, there are n! comoncones.34 Here is an example to illustrate the idea of a comoncone. Consider the following gambles: f = {HEADS, $50; TAILS, $0} g = {HEADS, $100; TAILS, $99} h = {HEADS, $0; TAILS, $50} j = {HEADS or TAILS, $70} The set [f, g, j] forms a comoncone, because for each gamble in the set, the heads outcome is weakly preferred to the tails outcome. The set [h, j] forms a comoncone, because for each gamble in the set, the tails outcome is weakly preferred to the heads outcome. We say that outcomes x1, x2 , … form a standard sequence on F⊆AA if there exist an act f∈F, events Ei ≠ Ej that are non-null on F, and outcomes y, z with ¬(y ~ z) such that for all k, (xk+1)Ei(y)Ejf ~ (xk)Ei(z)Ejf, with all acts (xk)Ei(y)Ejf, (xk)Ei(z)Ejf ∈ F.35 The intended interpretation is that the set of outcomes x1, x2, x3, …, will be “equally spaced.” Since the agent is indifferent for each pair of gambles, and since each pair of gambles differs only in that the “left-hand” gamble offers y rather than z if Ej obtains, and offers xk+1 rather than xk if Ei obtains, the latter tradeoff must exactly make up for the former. And since the possibility of xk+1 rather than xk (if Ei) makes up for y rather than z (if Ej) for each k, the difference between each xk+1 and xk must be constant. Note that a standard sequence can be increasing or 23 decreasing, and will be increasing if z > y and decreasing if y > z. A standard sequence is bounded if there exist outcomes v and w such that ∀i(v ≥ xi ≥ w). We are now in a position to define a relation that is important for Köbberling and Wakker’s result and that also makes use of the idea that one tradeoff exactly makes up for another. For each partition M, we define the relation ~(F) for F ⊆ AM and outcomes x, y, z, w ∈ XX as follows: xy ~(F) zw iff ∃f,g∈F and ∃E∈EE that is non-null on F such that xEf ~ yEg and zEf ~ wEg, where all four acts are contained in F.36 Köbberling and Wakker explain the relation ~(F) as follows: “The interpretation is that receiving x instead of y apparently does the same as receiving z instead of w; i.e. it exactly offsets the receipt of the [f’s] instead of the [g’s] contingent on [E].”37 The idea here is that if one gamble offers f if E obtains, whereas another gamble offers g if E obtains, then this is a point in favor of (let’s say) the first gamble. So in order for an agent to be indifferent between the two gambles, there has to be some compensating point in favor of the second gamble: it has to offer a better outcome if E obtains. And it has to offer an outcome that is better by the right amount to exactly offset this point. Now let’s assume that offering y rather than x (on E), and offering w rather than z (on E) both have this feature: they both exactly offset the fact that a gamble offers f rather than g (on E). That is, if one gamble offers f if E, and a second gamble offers g if E, then this positive feature of the first gamble would be exactly offset if the first offered x if E and the second offered y if E – and it would be exactly offset if instead the first offered z if E and the second offered w if E. If this is the case, then there is some important relationship between x and y on the one hand and z and w on the other: there is a situation in which having the first member of each pair rather than the second both play the same compensatory role. This relationship ~ is called tradeoff equality. We write xy ~(C) zw if there exists a comoncone F ⊆ AM such that xy ~(F) zw: that is, if x and y play the same compensatory role as z and w in some gambles f and g where all of the modified gambles after x, y, z, and w have been substituted in are in the same comoncone. The relation ~(F), and particularly ~(C), features centrally in the representation theorem, because one important axiom places restrictions on when it can hold: on when pairs of outcomes can play the same compensatory role. This relation also plays a crucial role in determining the cardinal utility difference between outcomes using ordinal preferences. What we are interested in is the utility contribution each outcome makes to each gamble it is part of: this will help us determine the utility values of outcomes. More precisely, since utility differences are what matter, we are interested in the utility contribution that x rather than y makes to each gamble. And tradeoff equality gives us a way to begin to determine this: if getting y rather than x in event E and getting z rather than w in event E both exactly 24 compensate for getting f rather than g in event E, then y rather than x and z rather than w make the same difference in utility contribution in event E in those gamble pairs. In order to get from these differences in utility contributions to utility full stop, we need to fix when it is that two pairs making the same difference in utility contribution means that they have the same difference in utility. And to do this, we identify the conditions under which if two pairs have the same difference in utility (full stop), they must make the same difference in utility contribution; and we constrain the rational agent to treat a pair consistently in these situations – to consistently make tradeoffs. Tradeoff consistency axioms provide such a constraint. With the preliminaries out of the way, I can now present the axioms of REU theory, side-by-side with those of the analogous representation theorem for EU theory that Köbberling and Wakker spell out.38 EXPECTED UTILITY THEORY A1. Ordering: ≥ is complete, reflexive, and transitive. A2. State-wise dominance: If f(s) ≥ g(s) for all s∈SS, then f ≥ g. If f(s) ≥ g(s) for all s∈SS and f(s) > g(s) for all s∈E⊆SS, where E is non-null on AA, then f > g. A3. Preference Richness: (i) There exist outcomes x and y such that x > y. (ii) For any fixed partition of events E1, …, En, and for all acts f(E1, …, En), g(E1, …, En) on those events, outcomes x, y, and events Ei with xEif > g > yEif, there exists an “intermediate” outcome z such that zEif ~ g. A4. Small Event Continuity: For all acts f > g and any outcome x, there exists a finite partition of events {E1, …, En} such that for all i, f > xEig and xEif > g B5. Archimedean Axiom: Every bounded standard sequence on AA is finite. B6. Unrestricted Tradeoff Consistency: For all AM ⊆ AA, improving an outcome in any ~(AM) relationship breaks that relationship. In other words, xy ~(AM) zw and y’ > y entails ¬(xy’ ~(AM) zw). RISK-WEIGHTED EXPECTED UTILITY A1. Ordering: ≥ is complete, reflexive, and transitive. A2. State-wise dominance: If f(s) ≥ g(s) for all s∈SS, then f ≥ g. If f(s) ≥ g(s) for all s∈SS and f(s) > g(s) for all s∈E⊆SS, where E is non-null on AA, then f > g. A3. Preference Richness: (i) There exist outcomes x and y such that x > y. (ii) For any fixed partition of events E1, …, En, and for all acts f(E1, …, En), g(E1, …, En) on those events, outcomes x, y, and events Ei with xEif > g > yEif, there exists an “intermediate” outcome z such that zEif ~ g. A4. Small Event Continuity: For all acts f > g and any outcome x, there exists a finite partition of events {E1, …, En} such that for all i, f > xEig and xEif > g A5. Comonotonic Archimedean Axiom: For each comoncone F ⊆ AM ⊆ AA, every bounded standard sequence on F is finite. A6. Comonotonic Tradeoff Consistency: Improving an outcome in any ~(C) relationship breaks that relationship. In other words, xy ~(C) zw and y’ > y entails ¬(xy’ ~(C) zw). A7. Strong Comparative Probability: For all pairs of disjoint events E1 and E2, all outcomes x’ > x and y’ > y, and all acts g,h∈AA, x’E1xE2g ≥ xE1x’E2g => y’E1yE2h ≥ yE1y’E2 26 Any agent whose preferences obey the axioms in the left-hand column maximizes expected utility relative to a unique probability function and a utility function unique up to positive affine transformation. Furthermore, in the presence of (A3), any agent who maximizes expected utility will satisfy the remaining axioms. Analogously, if a preference relation ≥ on AA satisfies (A1) through (A7), then there exist (i) a unique finitely additive, non-atomic probability function p: EE [0, 1]; (ii) a unique risk function r: [0, 1] [0, 1]; and (iii) a utility function unique up to positive affine transformation such that REU represents the preference relation ≥. If there are three such functions so that REU(f) represents the preference relation, we say that REU holds. Thus, if ≥ satisfies (A1) through (A7), then REU holds. Furthermore, in the presence of (A3), if REU holds with a continuous r-function, then the remaining axioms are satisfied. Therefore, if we assume preference richness (A3), we have: (A1), (A2), (A4), (A5), (A6), (A7) are sufficient conditions for REU. (A1), (A2), (A4), (A5), (A6), (A7) are necessary conditions for REU with continuous r-function. The proof of this theorem, with references to details found in Köbberling and Wakker and in Machina and Schmeidler, can be found in Buchak (2013).

Rational Faith and Justified Belief Lara Buchak, UC Berkeley 1. Introduction In “Can It be Rational to Have Faith?”, I argued that to have faith in some proposition consists, roughly speaking, in stopping one’s search for evidence and committing to act on that proposition without further evidence. In that paper, I also outlined when and why stopping the search for evidence and acting is rationally required. Because the framework of that paper was that of formal decision theory, it primarily considered the relationship between faith and degrees of belief, rather than between faith and belief full stop (hereafter, “belief”). The purpose of this paper is to explore the relationship between rational faith and justified belief. Before rehearsing my account of faith, let me briefly say something about the overall epistemological picture that I am working with, which rests on two assumptions. The first is that beliefs come in degrees. This idea can be motivated by noticing that among the propositions I believe, the strength of my belief is not uniform. For example, while I believe that the sun will rise tomorrow and I believe that it will be a cloudless day, I believe the former much more strongly. I would be willing to bet more on the former: I would be willing, for example, to pay 99 cents for a bet that pays me a dollar if the sun rises tomorrow, but I would not pay 99 cents for a bet that pays me a dollar if it is a cloudless day. This distinction holds not only among propositions I believe, but also among propositions I fail to believe: I can rank these according to how likely I think they are to be true. While I don’t believe that the Giants will win the World Series and I don’t believe that the Cubs will win the World Series, I would be willing to pay 10 cents for a bet that pays a dollar if the Giants win, but I would be unwilling to accept such odds on a bet that pays off if the Cubs win. According to the epistemological picture here, my degree of belief that the sun will rise tomorrow is higher than my degree of belief that today will be cloudless, and my degree of belief that the Giants will win is higher than my degree of belief that the Cubs will win. On this picture, rational degrees of belief (or credences) are best thought of as subjective probabilities: subjective in the sense that they are the individual believer’s response to her own evidence, and probabilities in the sense that they obey the axioms of the probability calculus. Rational degrees of belief are also updated in response to new evidence: so, for example, on my current evidence I might have degree of belief 0.1 that the Giants will win the World Series, but if I learn that their star shortstop is injured, my degree of belief drops to 0.05. We can represent this by writing p(Giants win) = 0.1 and p(Giants win | star shortstop is injured) = 0.05, where p(X | Y) is read as “the probability of X conditional on Y” or “the probability of X 2 given Y. Finally, we can think of an individual’s degree of belief in a proposition as an estimate of the proposition’s truth-value, given her evidence.1 The second assumption is that rational believers ought to proportion their degrees of belief to the evidence. This thesis is known as evidentialism. Evidentialism rules out taking into account non-truthconducive reasons in deciding what degrees of belief to hold. For example, one cannot adopt a high degree of belief in a proposition simply because one wants it to be true. Nor can one adopt a high degree of belief in a proposition for moral reasons, for example, because it is a proposition about a friend, and one has a moral duty to think well of a friend. I won’t take a stand on whether a given body of evidence always recommends a unique set of credences or whether, on the contrary, two different individuals could have the same evidence and each rationally adopt different credences – but if this is possible, the difference must be explained by something epistemic rather than their non-epistemic values. Furthermore, I am only assuming evidentialism about degrees of belief: as we will see, one can be an evidentialist about degrees of belief while thinking that “on-off” belief ought to be sensitive to non-truthconducive factors. Epistemic rationality concerns which credences one ought to have and what one ought to believe. Instrumental rationality concerns what one ought to do. We saw above that there is a link between credence and betting behavior: of two propositions, a rational individual would rather take a bet on the proposition to which she assigns a higher credence. More generally, credences figure into a precise theory of rational action. A maxim that guides action is that one ought to take the means to one’s ends: one ought to choose the act that brings about the outcome one values most. But this maxim cannot always be followed as stated. In the typical case, one has many competing ends which one values to different degrees; furthermore, one is typically not certain of what the state of the world is, but instead assigns credence to several possible states. For example, consider the choice about whether to bring one’s umbrella to work: one values staying dry while carrying an umbrella more than getting wet, but one also values staying dry while not carrying an umbrella more than staying dry while carrying one; and one assigns some credence to the hypothesis that it is raining and some credence to the hypothesis that it is not raining. Decision theory makes precise the components of this decision, and how they interact to produce a recommendation about what one should do. An act is thought of as a gamble whose payoffs depend on the state of the world. For example, not carrying an umbrella is the gamble that results in getting wet if it rains and staying dry while not carrying an umbrella if it doesn’t: and if one thinks there is a 70% chance of rain, then this is the lottery {70% chance of getting wet, 30% chance of staying dry while not carrying 1See Joyce (2005). It is controversial how exactly to give content to what degree of belief represents, but the differences between the various proposals won’t matter here. 3 an umbrella}. Also introduced is a utility function, a function which measures how much one values particular outcomes. The standard view is that an instrumentally rational individual ought to choose the act with the highest average utility value (the highest expected utility), given the probabilities she assigns to the various possible states. We write u(O) to stand for the utility value of some outcome O, and p(X) to stand for the probability of some possible state of the world X. If u(getting wet) = -3, u(staying dry while not carrying an umbrella) = 3, u(staying dry and carrying an umbrella) = 1, p(rain) = 0.3, and p(notrain) = 0.7, then EU(don’t bring umbrella) = (0.3)(-3) + (0.7)(3) = 1.2 and EU(bring umbrella) = (0.7)(1) + (0.3)(1) = 1; therefore, one ought not to bring one’s umbrella. On this picture, one’s values and beliefs are subjective, and from them we can arrive at a recommendation about what to do. I hold – although this is controversial – that we ought additionally to allow an individual to determine for herself how to take risk into account. For example, some people care proportionally more about what happens in the worst-case scenario than what happens in the best-case scenario. Although the chance of rain is 0.3, when contemplating an act where “rain” leads to the worst outcome, the possibility of rain may weigh more heavily in deliberation. To determine which act is instrumentally rational for these risk-avoidant individuals, we can still calculate values according to a mathematical formula, but one which weights the minimum more heavily than its “probability share” of the state space.2 For example, one might shift the “decision weights” – the weights the various possibilities get for decision-making purposes – so that the risk-weighted expected utility (REU) of not bringing an umbrella is (0.6)(-3) + (0.4)(3) = -0.6, and as a result forgoing one’s umbrella is not recommended. (A similar point holds of “risk-inclined” individuals, who weight the maximum more heavily than the minimum.) Since this thesis is still controversial, I will make sure to say how things go both on standard decision theory (expected utility theory) and on the alternative I favor (risk-weighted expected utility theory). This is the basic framework of “formal” epistemology: credences are the epistemological entity; the norm of epistemic rationality is to have credences that obey the probability calculus and (according to evidentialism) to proportion them to the evidence; and the norm of instrumental rationality is to maximize expected utility (or, on my view, to maximize risk-weighted expected utility). You’ll notice that I haven’t yet said anything about the relationship between credence and belief. That is because there is currently no agreed upon view about how the two frameworks fit together. The bulk of this paper will consider what can be said about faith and belief according to several plausible but competing answers to this question. First, however, let me briefly outline the account of faith I offered in Buchak (2012), and how faith fits into the formal epistemological picture. 2For details, see Buchak (2013). 4 2. The Nature of Faith An account of the nature of faith should satisfy several criteria. First, it should capture what we take to be paradigm cases of faith, both intuitively and within the context of interpersonal relationships and religious practices. I assume throughout that religious faith is a special case of a general, unified attitude that encompasses “secular” cases of faith as well, such as faith in a friend. Thus, we are interested in a minimal core notion of faith that is consistent with distinct, thicker notions of faith such as Christian faith, faith in one’s spouse, and so forth. Second, an account of the nature of faith should be able to distinguish between what we take to be good cases of faith and what we take to be bad cases of faith. If we don’t agree about all cases, it should capture the cases we do agree about and yield a verdict on those we don’t. Finally, those who endorse faith as a virtue think of it as a central component within some sphere of activity, for example, religious practice or interpersonal relationships. Thus, the final desideratum for an account of faith is that it explains why faith can be a virtue (intellectual or otherwise), and what the attitude of faith can add to human life. To set these criteria out isn’t to prejudge the question of whether they can be satisfied (of whether there really is a notion of faith that is common to both religious and mundane contexts and according to which faith serves a positive role) but rather to begin a search for whether there is an account of faith that can meet them. Additionally, there may be no single sense of faith that explains all usages of the term: what I am after is the concept in the neighborhood that is normative – the concept according to which it is true of some people that they ought to have faith that a friend will come through for them or faith that God exists – if there is such a concept. Even given these caveats, there are at least two important senses of the term “faith”: propositional faith (faith that X, where X is some proposition) and interpersonal faith (faith in I, where I is some individual). My account is an account of propositional faith. My hope is that the correct account of interpersonal faith will ultimately rest on an account of propositional faith: for example, to have faith in a person is to have faith that some facts about her obtain. But even if these two senses of faith are largely independent, I take it there is still an important question about what propositional faith consists in. (The only substantive thesis I am ruling out is that an account of propositional faith rests on an account of interpersonal faith.) An account of propositional faith has two parts. The first delineates the set of propositions that are potential candidates for faith, and the second what it is to have faith in one of these propositions. While all propositions are potentially the objects of credence and of belief, not all propositions are even candidates for faith. Thus, I introduce three criteria that a proposition must meet in order to be a potential object of faith for a particular individual. First, in order for a proposition to be a potential object of faith, the individual must care whether or not the proposition is true. Faith that X is incompatible with indifference about whether X. 5 Second, the individual must have a positive attitude towards the truth of the proposition. This can be seen by noting that while I can be said to have or lack faith that you will quit smoking, I can’t appropriately be said to have or lack faith that you will continue smoking. The exact attitude one must have towards the proposition needs some spelling out, though. In my earlier account I said that the sense in which one must have a positive attitude is that one is basing some act on the proposition. However, this threatens to let in too many propositions as potential objects of faith. For example, it implies that my betting on your continuing to smoke is enough to make “you continue to smoke” an appropriate object of faith. One might instead claim that in order for X to be an appropriate object of faith, one must prefer that X, aside from any act one takes. However, this threatens to rule out too many propositions as potential objects of faith. For example, consider an individual whose friend brings her news that the individual’s child has been kidnapped and that the individual must pay a ransom to rescue him: it is felicitous to say that the individual pays in part because she has faith that her friend is telling the truth, but she of course prefers that the friend be lying.3 I don’t think this example reveals that a positive attitude is not a necessary condition of faith – there is something that one has a positive attitude towards in this example, namely the friend – but that we need a more nuanced account of what sort of attitude is required. I leave this aside for future work. Third, in order for a proposition X to be an appropriate object of faith for a particular individual, she must not take her evidence on its own to support her being certain that X: her evidence must leave open the possibility that not-X.4 For example, while it is felicitous to say, before you know the results of a friend’s exam, that you have faith that your friend passed the exam, it is infelicitous to say this once she shows you her passing grade. There are certain kinds of propositions for which evidence cannot generally produce certainty, because there is yet no fact of the matter: for those who hold that free actions must not be determined in advance, an example of such a proposition is a proposition concerning the future free act of another individual. Thus, these kinds of propositions will often be candidates for faith. Now that we have delineated the set propositions that are candidates for faith, we can ask what having faith involves. A key component is that faith that X involves a willingness to act on X in situations in which doing so constitutes taking a risk. When we have faith that a particular individual will act in a certain way – keep our secret, pick us up from the airport, do what is in our best interests – we take a risk that the individual will let us down. We are vulnerable to the individual not acting as we have faith that she will act. However, not every case of risk-taking will count as an act of faith. Faith involves a willingness to commit to acting on the proposition one has faith in without first looking for further evidence for or 3The general form this example takes is due to Alex Pruss. 4“Not-X” is hereafter represented as X. 6 against that proposition. An individual with faith in her friend’s ability to keep a secret must be willing to confide in her friend without first verifying with a third party that the friend isn’t a gossip. A man who has faith that his wife is constant must commit to his marriage without first hiring a private detective to observe how his wife behaves when he is not there. An individual who has faith that a particular bridge will hold his weight doesn’t test the bridge before stepping onto it. Not only do individuals with faith not need further evidence, they will choose not to obtain it if it is offered to them, when their only interest in obtaining it is in how it bears on the decision about the act. For example, I must decline if a third party offers to tell me about her experiences with my friend’s secret-keeping abilities. More specifically, individuals with faith will commit to the risky act without looking for further evidence. I want to be clear that having faith doesn’t mean generally avoiding all evidence for or against the proposition in question; rather, it means not looking for further evidence for the specific purpose of deciding whether to act on the proposition, or not basing one’s decision on how the evidence turns out. I also want to make clear that in many cases, a decision to eschew further evidence will be based on evidence one already has: faith need not be “blind” faith. (And, as we will see, faith tends to be rational to the extent that one has already amassed evidence: one must base one’s faith on evidence, even though faith involves eschewing further evidence – one must first climb the ladder before kicking it away, so to speak.) So faith involves two key components: taking a risk and doing so without the need for further evidence. Let us make this analysis of faith explicit. First, we will say that an act A constitutes an individual’s taking a risk on X just in case there is some alternative available act B such that A is preferred to B under the supposition that X, and B is preferred to A under the supposition that X. For example, my revealing a secret is a risk on my friend’s keeping the secret because I prefer to tell her on the supposition that she will keep it and not tell her on the supposition that she won’t. Whether an act is a risk on X will be relative to the individual performing the act. So we have: For an individual I, A is an act of faith that X if and only if X is a candidate proposition for faith and: (1) A constitutes I taking a risk on X. (2) I chooses to commit to A before examining additional evidence rather than to postpone her decision about A until she examines additional evidence.5 So to perform an act of faith in a proposition is to take a risk on that proposition, and to refrain from gathering further evidence before committing to taking that risk. 5In Buchak (2012), these conditions were formulated in terms of preference rather than choice because decision theory is primarily a theory about preference. I think the view is more intuitive when formulated in terms of choice, and since choice and preference are linked, I see no harm in doing so, although there may still be questions about whether the requirement is ultimately about choice or preferences, if the two do come apart. 7 Several points of clarification are in order. The first was already mentioned: to have faith that X does not require that one in general ignore future evidence in the matter of X. What it requires is that one choose to commit to the relevant act without first gathering additional evidence. For example, that the theologian has faith that God exists is compatible with her continuing to study theology, because she does not base her commitment to the Christian life on the results of her study. Indeed, gathering evidence can itself sometimes be a faithful act if the evidence is gathered for purposes other than committing to a further act (though of course the question of how such evidence will then bear on one’s faith arises). For the theologian, devoting herself to theological study is itself an act of faith because doing so constitutes taking a risk that God exists (if God does not exist, to study theology is a waste of her time, but if God does exist, then theological study will lead to a deeper understanding of God), and because she is willing to devote herself to study without first verifying through other means that God exists (praying for a sign, for example). Faith requires not engaging in an inquiry for the purpose of figuring out whether to take the risk on the claim in question. The second point of clarification concerns the move from acts of faith to faith itself. Whether one has faith that X is a matter of which acts of faith that X one is willing to perform. Just as belief comes in degrees, so too does faith. And one’s degree of faith will be a matter of which risks one is willing to take on X without looking for further evidence. I might have enough faith that God exists to attend a house of worship (a low-stakes risk) without gathering further evidence, but I might not have enough faith that God exists to donate all my money to charity and take up a life of poverty (a high-stakes risk) without gathering further evidence.6 Faith simpliciter, then, is a matter of one’s dispositional profile. Given that faith simpliciter is determined from the acts of faith one is willing to perform, one point to note about the requirements for a proposition’s being a candidate for faith in combination with the second condition in the above account of faith is that the account can distinguish between propositions that a risky act reveals faith in and propositions that are presuppositions of the act. When we say that an act constitutes a risk on X, in the sense that it is the preferred act if X holds but not the preferred act if X holds, this preference is determined against the individual’s background credences. Donating all of one’s money to Oxfam only constitutes taking a risk on {God exists} if one assumes that if God exists, God commands extreme charitable giving and that Oxfam is the most efficient charity. Therefore, this act also constitutes taking a risk on {if God exists, then God commands poverty} and {Oxfam is the most efficient charity}. Refraining from a prenuptial agreement only constitutes taking a risk on {my spouse will continue to be committed to me} if one assumes that it will be financially beneficial to separate one’s 6I don’t have a formal definition of how precisely one’s degree of faith is determined from which risks one is willing to take without gathering further evidence, though at the very least being willing to take higher-stakes risks indicates having more faith. I also don’t mean to suggest anything about the structure of degrees of faith, e.g., that they can be represented cardinally rather than ordinally. 8 assets from his in the event of a divorce. Therefore, refraining from a prenuptial agreement also constitutes taking a risk on {a prenuptial agreement will be financially beneficial in the event of divorce}. But we wouldn’t want to say that one has faith in all of these propositions. And, on my account, we don’t: the proposition that a prenuptial agreement will be financially beneficial is not an object of faith, because one is indifferent to the truth of that claim. One does not have faith in {Oxfam is the most efficient charity} even though one is not indifferent to the truth of this claim, if one is willing to research charities further before donating. Similarly, whether one has faith that {if God exists, then God commands poverty} is a matter of whether one is willing to read the relevant religious texts to get a better idea of what God commands. 3. When and Why Faith is Rational What, then, is the relationship between rational faith and degrees of belief? (Another way of putting this question is: under what evidential conditions is it rational to have faith?) Recall that for a proposition to be an object of faith, one must not have p(X) = 1: one must not be certain, on the basis of the evidence alone, that X holds. Assuming that X meets the other conditions for being an appropriate object of faith (one cares whether X and has a positive attitude towards the truth of X), we can characterize when it is rational to have faith that X by considering when it is rational to perform risky acts on X without more evidence. Again, every act can be thought of as a lottery which yields various results in various states. To commit to act A is just to take the gamble that yields A&X if X obtains and A&X if X obtains. This is to say: committing to A can be thought of as holding a lottery ticket that yields A&X with probability p(X) and A&X with probability p(X). Committing to tell one’s secret to the friend, without looking for further evidence, is a lottery which results in telling one’s secret and having it kept, with whatever probability one now assigns to the proposition that the friend will keep the secret; and which results in telling one’s secret and having it spilled, with whatever probability one now assigns to the proposition that the friend won’t keep the secret.