If Free Will Were Coherent, We Ought to Believe in It

Summary. I don't believe in libertarian free will because I don't understand what it could possibly mean. However, if the concept were coherent, I think causal decision theory would encourage us to believe that we have it, regardless of its apparent probability. Evidential decision theorists might disagree.

About This Essay

Libertarian free will, which as far as I'm concerned is the proper definition of the term free will, appears to me to be an incoherent concept -- at the very least, I am unable to understand what it could mean. If the future is not determined, then it's random. If the past does exert control over the future, then the future is not purely random, and as the amount of control of the future by the past increases, the amount of randomness decreases. Free will posits that the future can be controlled by the past (at the moment of decision) but that the future is not yet controlled by the past -- there's some moment at which many possibilities are reduced to one reality. But what's the mechanism of this collapse? If it's random, then our decisions are hardly controlled by us, while if it's not random, then it must have been based on some fact about our natures and dispositions. But although we have particular natures and dispositions, we did not choose them in the first place -- they were imposed on us when they came into existence. Galen Strawson has made this point.

Below I reproduce an essay I wrote a while back. The argument is premised on the coherence of free will and the correctness of causal decision theory. It argues that even if we are unsure about whether free will factually obtains, we ought to assign 100% probability to the belief that we have it, assuming our goal is to improve the correctness of our beliefs. The reasoning is rather straightforward: The set of possible worlds in which we have free will is precisely the set in which there exists a free we that can choose to cause changes in our beliefs. If we are in one of those worlds, then choosing to follow the maxim Be certain you have free will causes the epistemically optimal outcome. If we are in a possible world without free will, there is no we that chooses to cause any outcome, so in particular, there is no free we that will be mislead by the maxim. (Of course, there could be lots of non-free agents that are deterministically or randomly compelled to follow the maxim, possibly due to reading clever arguments like this one, but the we that free will refers to has no power to affect such cases.)

The preceding argument assumes that our goal is to improve the accuracy of our beliefs, but a similar idea can be expressed regarding more prudential (consequence-motivated) matters. For instance, suppose God either will or will not send you to hell, and you're not sure which. If you lack free will, then whether or not you burn for eternity is already either true or false and can't be changed (or else the outcome will be decided randomly, which is no more palatable). But if there is a free you, then you can potentially cause an improvement in the outcome by your actions. A Pascalian intuition here says, It's best to act as though we can change the outcome, just in case we actually can.

The intuition about futility in the absence of free will comes from the framework of causal decision theory. However, I think Newcomb's problem (assuming the scenario itself is possible) demonstrates rather convincingly that causal decision theory does not always give the right answers. If we're willing to allow evidential considerations to affect our choices, then the absence of libertarian free will may not be a problem. For instance, in the case of Calvinism, causal decision theory warns that your situation is hopeless: Whether you are damned or saved has already been decided, and there's nothing you can freely do to cause a change in the outcome. With evidential decision theory, you should do those things that will make you think it more likely that you are indeed among the elect, even if your action had no causal role in the process.

This idea applies more generally. Since we in fact have reason to assign high probability to the absence of free will, even if that concept were coherent, it might be a better outcome if we find our (probably deterministic) selves believing we don't have free will. Thus, from the evidential perspective, our believing we don't have free will implies that it's the case that we're (probably deterministically) holding a belief likely to be true -- which is a good outcome, regardless of whether there was any inner self that chose our correct belief. This is analogous to one-boxing in Newcomb's paradox: Even though the contents of the boxes are fixed, we will be happier to find ourselves taking only the box with the $1 million. And the free-will argument presented below is like two-boxing: It claims that, in spite of the compelling evidence against free will, we ought to believe we have it, because the would-be-$1-million box is already empty (i.e., if we're not free, our beliefs are already determined), so we ought to at least take the $1000 (i.e., we ought to believe in free will on the off chance that it's right). (We could make the Newcomb scenario slightly closer to that of uncertainty about the truth of free will by imagining that the $1,000,000 is replaced by (n-1) lottery tickets and the $1000 is replaced by 1 lottery ticket. The total number of tickets in the lottery is n, and exactly one ticket is guaranteed to win. Here, 1/n corresponds to your initial probability for free will in the correct belief lottery. The causal decision theorist claims that the contents of the zero-or-(n-1)-ticket box is already fixed, so that you may as well pick up the one ticket in the other box.)

I think the response of the causal decision theorist would be to claim that the stance of evidential decision theory is confused: Sure, you'll be happier if you find yourself taking just one box, but what free agent are you addressing your advice to? How is anyone going to freely choose to follow your reasoning in support of one-boxing? The evidential determinist can reply that his advice can be followed just fine by deterministic agents who read and understand it -- determinism doesn't imply stupidity after all. And even though the fact that he's giving this good advice was itself determined, it's still good advice, and he can be glad that he finds himself giving it.

I think this illustrates nicely that the difference between incompatibilists and compatibilists is, as far as I can tell, a sort of special case of the difference between causal decision theorists and evidential decision theorists (plus subjunctive decision theorists like Gary Drescher). Causal decision theory seems to me much more intuitive, and it's true that evidential decision theory gives terrible advice in some cases. But since utilitarians are playing to win, they certainly should take only one box in the Newcomb situation (so that they'll be able to donate much more toward relieving suffering). Of course, exactly what should means here is debated by the two camps.

In any event, below is the text of the original essay.

Summary. In this piece, I present a practical argument that, if sound, implies that we ought to believe in free will, if we can, regardless of what we currently think its probability is. This may have implications for other beliefs that we hold; for instance, it means we ought to reject scientific theories that imply hard determinism, as well as religious doctrines according to which our every action has been decided in advance.

Background

The idea of causal determinism maintains that all future events are in principle uniquely determined by initial conditions (e.g., the current positions and velocities of all particles). A famous proponent of this view was Pierre-Simon Laplace, who--writing during the heyday of Newtonian mechanics--advanced an idea that would later be known as Laplace's demon:

Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it -- an intelligence sufficiently vast to submit these data to analysis-it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes. The human mind offers, in the perfection which it has been able to give to astronomy, a feeble idea of this intelligence. (A Philosophical Essay on Probabilities, Chapter 2)

The "intelligence" to which Laplace referred could not be physical, or else it would itself be a part of the material world, which leads to contradictions. This is not important, however; the point is that in a deterministic universe, knowledge of all future outcomes exists in principle, even if it can't be computed.

In later years, it was recognized that Newtonian mechanics might come into conflict with Laplacian determinism in the realm of certain thought experiments involving unbounded spaces or velocities (source). In addition, certain interpretations of quantum mechanics, notably the Copenhagen interpretation with its notion of wavefunction collapse, posit fundamental randomness to the universe. Some physicists and philosophers, such as Arthur Stanley Eddington in his "The Decline of Determinism," have claimed that quantum randomness provides a path to free will. But this view is mistaken if quantum mechanics is viewed in a strictly materialist sense.

For instance, suppose I decide to get up and have a drink of water. Laplacian determinism claims that I did this as a result of the movements of material particles in my brain, and that those movements were in principle predetermined ever since the universe began. Quantum indeterminism claims that certain quantum particles in my brain randomly collapsed in such a way that, by chance, I decided to get a drink. But in neither case is there any sort of "me" that made a decision. As one anonymous quip puts it, "Either our actions are determined, in which case there is nothing we can do about them, or our actions are random, in which case there is nothing we can do about them."

Definitions

In what follows, I'll use the term "determinism" to refer to either the position that future events are predetermined or the position that they are not predetermined but are determined by arbitrary chance. A person is said to be "determined" to do something if he does something for either of these reasons.

By "free will," I refer to an ability to make decisions and beliefs that are not arbitrary but that are made by some sort of "self" or "spirit." Though different philosophers may define things differently, this definition is the one held by (philosophical) libertarians. Some have argued that materialism need not be inconsistent with this view of human consciousness and decision-making (see naturalistic libertarianism). Of course, many religious notions of "the spirit" and everyday common-sense notions of "freedom of action" are also consistent with free will (see metaphysical libertarianism).

Both hard determinists and libertarians fall into the category of incompatibilists: those who believe that free will and determinism cannot both be true (see this diagram). There are some, called compatibilists, who hold that this needn't be the case. Basically, compatibilists define free will differently than do libertarians, usually maintaining that actions which are caused by outside influences can still be considered free actions because the actor has an internal sense of having chosen them. The debate between incompatibilists and compatibilists, then, is over a definition: incompatibilists maintain that 2+2 cannot equal 6; compatibilists say that 2+2 does equal 6 if only we'd be willing to define "2" to mean what we normally think of as "3."

Thus, the compatibilist school presents no problem for the argument I advance in this piece. I'll show that we ought to believe in free will as I've defined it (i.e., libertarian free will) if we are able. Compatibilist definitions of free will fall under the category of "determinism," i.e., the absence of free will as I've defined it.

Finally, here are two types of beliefs/probabilities that I'll refer to:

The Argument

With competing views on either side of the libertarian-determinist debate, what should we believe about free will? Well, take a step back. What does it mean to ask "what should we believe?" This question assumes that there is a "we" that can choose freely to believe either that free will exists or that it doesn't. If there is no free will, then there is no "self" that can make this decision; things will just happen one way or the other. In that case, it's meaningless to talk about "what we should do."

On the basis of this idea, Richard Chappell articulated the following argument:

1) If I don't have free will, then I can't choose what to believe.
2) If I can choose what to believe, then I have free will [from 1]
3) If I have free will, then I ought to believe it.
4) If I can choose what to believe, then I ought to believe that I have free will. [from 2,3]
5) I ought, if I can, to choose to believe that I have free will. [restatement of 4]

[...]

Here's another way of putting it:

a) We can either choose what to believe (about free will), or we can't.

b) If we can't choose, then we can't make the wrong choice (since we can't make any choice).

c) If we can choose, then we make the wrong choice iff we choose not to believe in free will.

d) So we can make the wrong choice by not believing in free will, but we can't make the wrong choice by believing in it.

e) Therefore, anyone faced with the choice ought to choose to believe in free will.

We can imagine thought experiments in which Chappell's premise 3 is not true. An example would be if God commanded us to believe that we don't have free will, even if we do; in that case, we ought not to believe in free will if we are able. I'll ignore these cases below. If desired, the reader may substitute "am epistemically required" in place of "ought" in the argument; I've used the terminology of epistemic requirement, rather than normative obligation, in the remainder of this piece.

But there may be another problem with premise 3. As currently stated, it seems only to follow from something like this principle: "If X is true, then we are epistemically required to believe X." But in general, this principle is hard to defend. For instance, it might turn out to be true that little green men have lived on Mars for centuries, yet that would not mean that we right now were epistemically obligated to believe in those little green men.

It may be possible to restate premise 3 in a weaker, but still adequate, form. However, I prefer to formulate the argument in a different way. I think the following is intuitive:

Principle: Suppose a person lives either in world W1 or world W2, though she doesn't know which one. The person thinks of two actions, A1 and A2. If the person lives in W1, she will be unable to do either action. If she lives in W2, she must choose to do exactly one of the two actions, and in this case doing A2 is guaranteed to cause her beliefs to be more accurate than doing A1. If the person lives in W2, she will be unable to tell whether she has done one of the actions or not. The person knows all of this ahead of time. Then, if the person lives in W1, she has no obligation to do either action, because she can't do either action. If the person lives in W2, she has an epistemic obligation to do A2 over A1.

Now, replace W1 with "a world in which the person lacks free will," W2 with "a world in which the person has free will," A1 with "freely choosing to assign less than 100% epistemic probability to free will," and A2 with "freely choosing to assign 100% epistemic probability to free will."

Analogy

Suppose we find ourselves on a desert island that contains only a train car and two pieces of train track that are connected. Each piece of train track is labeled. The one that the car currently rests on says, "This car will not budge"; the other part of the track says, "This car can be moved." A message in a bottle floats our way and contains the following note:

Exactly one of the two pieces of track says a true statement.

Here ends the message that this bottle contains.

Suppose we have reason to trust absolutely what the bottle has told us. We would prefer for it to be the case that the car can be moved, because then we could move it and guarantee being rescued. On the other hand, we may think to ourselves, "That train car looks really big and heavy. There's no way I'd be able to push it. I'm almost certain that the statement 'This car will not budge' is true. Given that, my best bet is to keep the car where it is, because that will at least give me a 50% of being rescued."

But there's a better solution. If we try to push the car, and it doesn't move, then we've done no harm. On the other hand, in the highly improbable event that the car does move, we will have pushed it onto the other piece of track, in which case we will be saved from the island instead of struck by lightning. So we should push the car. But how hard? Should we push it with only half our strength, since it probably won't move anyway? Of course not! We should push with full force, because that will guarantee that the car lies over the right piece of track if it can be moved. If we had pushed with only half our might, when in fact our full might was required, we would have done the wrong thing. But if we push as hard as possible, we maximize our chance of being saved from the island, regardless how small we think the probability is that the car will move, as long as that probability is greater than zero.

In this example, "being saved from the island" represents having a correct belief about free will, "being struck by lightning" represents having an incorrect belief about free will, "pushing the car" represents trying, if we can, to believe in free will, and "pushing with all our might" represents trying to assign free will 100% epistemic probability. A "movable train car" represents free will being true. I assumed a 50% chance of being rescued if the car won't budge because there's some chance that, if free will isn't true, we'll be determined to correctly believe that it isn't true, in which case we'll hold an accurate belief.

Discussion

It's easy to be fooled by the simplicity of the free-will argument into underestimating its robustness. For instance, we might be tempted to think that the argument is merely prudential, like positive self-delusions. Consider this example. Ted thinks he's a really good baseball player, when in fact he's pretty much average. But if Ted were to realize that he wasn't really good, he would become unsettled and lose self-confidence; this would turn him from an average player into a poor player. So it's best for Ted to think he's really good. If, however, Ted needed to make an epistemic assessment of his abilities, things might be different. For instance, suppose Ted enters a $10 billion bet on who will win the next baseball game in which he plays. Clearly, it's no longer best for Ted to believe that he's really good; instead, it's in his best interest to believe the more objective assessment of his abilities.

Importantly, the free-will argument does not lead to a separation of our prudential beliefs from our epistemic beliefs. It does not tell us to "fool ourselves into thinking that we have free will, even though we secretly continue to realize that we don't. " Instead, it tells us to assign 100% epistemic probability to the belief that we have free will. To see this, suppose it were otherwise. Suppose that after hearing the above argument, along with other evidence about free will and determinism, Bill assigned only a 10% probability to free will. Suppose Bill felt that, on prudential grounds, it would be best to act as though he had free will. "However," Bill said to himself, "for the really big questions--such as what type of God exists or what types of physical theories are true--I should use my real, objective assessment of the probability of free will. I shouldn't naively assume that free will is true just because it's generally best to act as though it is." A few years later, Bill enters a bet with his friend on whether there is free will. The bet is to be arbited by an omniscient being who will reveal the correct answer after the bets have been placed. Bill, recalling the example of baseball player Ted, bets that there is no free will.

Was this action better than betting that free will did exist? It depends. The outcome might turn out to be better. If free will doesn't exist, then Bill was determined to make the bet he did, and he happened to be lucky enough to be rewarded for it. But the decision, if it was a decision, was a bad one. If Bill decided to bet against free will, he was guaranteed to lose, and if he decided to bet for free will, he would always win. In other words, the only way to make a bad decision is to bet against free will. More generally, by giving epistemic credence to the absence of free will in the process of what we think is decision-making, we can only make our decisions worse. If free will is false, we haven't done any harm by giving 100% probability to free will because "we" haven't done anything. In that case, "we" had no control over the matter of assigning 100% epistemic probability.

A Clarification

Readers may be thinking to themselves, "It's absurd to suggest that free will must be true! What about all the evidence, scientific and otherwise, against it? And even if we had no such evidence, it would be rash to assign 100% probability to something that might be true and might be false. What if we turn out to be wrong?" Let me clarify my position. I am not suggesting that it somehow must be the case that free will is true or that the evidence will somehow convince us that free will is true. Rather, I am asserting that regardless of what the evidence is--regardless of how low our current probability assessment is--we ought to force ourselves to assign 100% probability to free will, if we can. But this is true in a stronger sense than in a case of prudential belief. Ted should force himself to think he's really good at baseball in order to keep him from becoming a bad player; if he succeeds in doing this, he will improve his utility but will sacrifice the epistemic correctness of his beliefs. In the case of free will, however, forcing ourselves to assign 100% probability, if we can, can only keep constant or improve the epistemic correctness of our beliefs, quite apart from considerations of utility.

Consequences

This result has important practical consequences. If we assign zero probability to the absence of free will, then we should reject all physical theories that imply hard determinism. We would also assign probability 0 to religious scenarios according to which everything that will happen to us has been decided ahead of time without consideration of our own volitions. (This does not mean, however, that we should reject all types of religious predestination. As an example, the doctrine of conditional election held by some Christians maintains that while God has "predestined" us to salvation or damnation, he has done so with our consent and with a foreknowledge of what we would have done freely. In this case, our decisions to act one way or another do matter, even though God happens to already know how they will turn out.)

Qualifications

The possibility this argument is wrong.

If the free-will argument is sound and if we ignore the above possibility, then we ought to assign zero probability to the absence of free will. However, there's a nonzero chance that the argument itself, for whatever reason, is not sound. So in practice, we shouldn't deny the possibility that there is no free will; we should only give it reduced credence, with the amount of reduction depending on how strongly we believe the free-will argument.

Which things do we have free will over?

We may have free will over a large number of decisions or only a limited number. Intuitively one may be tempted to say, based on the above, that in any case where we aren't sure whether we have free will to do Y, it's better if we assume that we do have free will to do Y. However, this may not always be true, as the following example illustrates. Suppose Alice knows for certain that she has free will about her beliefs; that is, she can freely decide to believe anything she wants. However, Alice doesn't know whether she can freely decide what color socks she will wear. Alice's mother tells her, "If you believe that your sock-color choice has been predetermined since the beginning of time, then I'll give you a $100/week allowance." Clearly, Alice should (freely) choose to believe that she doesn't have free will about sock choice. Unlike the previous free-will argument, this one is purely prudential, because Alice already knows that she has free will about beliefs. Similar arguments can be made to justify why we shouldn't believe that we have free will over, say, how quickly Pluto orbits the sun. It's when we're uncertain about our freedom to choose our beliefs that we should always choose, if we can, to believe that we do have free choice.

Free will might turn on and off.

It may be that our ability to exercise free will over our beliefs alternates. For instance, we might imagine a God who usually grants humans free choice in their decisions, but at times when he wants to accomplish a particular task through people, he temporarily disables their free will. In these cases, believing that we always have free will would lead us to an incorrect epistemic belief. For instance, if we have free will for the next five minutes, after which our beliefs and actions will be completely determined, then we ought, at least from an epistemic perspective, to make ourselves believe we don't have free will, so that we'll give ourselves the belief that will be true five minutes later and onward. Of course, if we find ourselves in such a situation, it would be even better to believe exactly what's true: that we have free will now but will no longer have free will after five minutes. But right now, we don't know whether we're in such a situation, so we should assign it some probability between 0 and 1; to that extent, we ought to believe that we won't always have free will. Note, though, that there is never a case in which we are epistemically required to believe that we never have had free will, because in those cases, we were never free to believe anything. The point I raise in here seems relatively unimportant, because most people (determinists and libertarians) assign very low probabilities to the idea that free will oscillates on and off.

A Second Analogy

Lack of free will is highly counterintuitive, and it's sometimes easy to reason mistakenly about it. It might be easier to think about something more distanced, and for this purpose I give the following analogy.

Think of a switch that can be flipped toward one of two positions. The "on" position of the switch is labeled "I have free will right now" and the off position says "I don't have free will right now." Connected to the switch is a plastic cover that can be clamped over the switch to prevent anyone from pressing it. "We" are sitting next to the switch. This "we" represents the "self" or "spirit" that would decide our free choices if indeed we did make free choices. If the switch cover is clamped, then we are unable to flip the switch. This represents the nonexistence of free will. (In reality, lack of free will means that the "self" sitting near the switch doesn't exist, but for purposes of this analogy, let a clamped switch cover represent the nonexistence of free will, instead.) The choice of whether to believe in free will is then analogous to the question of which way we ought to flip the switch.

One obvious question that arises is whether we want the switch to point toward the correct label. If we would prefer for the switch to say "I don't have free will right now" even though the switch cover is not blocking us from changing the switch, then we would be in a position similar to someone who wants to believe that he doesn't have free will even when he actually does. Epistemically, though, we are required to flip the switch toward the correct label if the cover is not blocking us.

Other complications arise. For instance, we might be unsure that we're reading the labels correctly, or we might be afraid that we don't fully understand what's going on, in which case we might have some reason to want to keep the switch toward "I don't have free will right now." This is analogous to the first qualification above. Similarly, we might be afraid that, if the cover is off and we flip the switch toward "I have free will right now," someone might come along later and close the switch cover before we have the chance to flip it back. This is like the third qualification above.

A Generalization--Varying Degrees of Control over Belief

Those maintaining libertarian free will have to decide how easy they think it is for people to make various decisions. Our actions cannot be totally divorced from caused events in the external world. When someone slaps you in the face, it's harder to make the free decision not to get angry than when someone pats you on the back. Drug users have a harder time making the free choice not to abuse drugs than do people who haven't become addicted. Of course, libertarians maintain that it's possible for people to overcome these external influences on their behavior, but libertarians don't deny that this can be easier or harder depending on the situation. An interesting question is, What does the free-will argument tell us about what we should believe about how hard it is to make free decisions?

Actually, the free-will argument will inform a slightly narrower question: What should we believe about how hard it is to make free decisions specifically about our beliefs about free will? The following examples should clarify this.

Example 1

First, I'll restate the free-will argument from above in a new way. Suppose we're considering only two possibilities:

  1. Free will doesn't exist, or
  2. Free will exists, and it's guaranteed that we'll change our beliefs about free will if our "inner self" wants us to.

If (1) is true, then we have no "inner self" that makes choices, but it does no harm, for the present purpose, to pretend that in this case we have such an inner self but that it has absolutely no power to change our actions. So we could restate the two possibilities as follows:

  1. The probability is 0 that our inner self will succeed in changing our beliefs about free will if it tries to do so.
  2. The probability is 1 that our inner self will succeed in changing our beliefs about free will if it tries to do so.

Suppose we assign subjective prior probabilities to these possible states of the world: say 95% probability to (1) and 5% probability to (2). Suppose further that if our inner self doesn't succeed in changing our beliefs, we have an 80% chance of being determined eventually to believe (1) and a 20% chance of being determined eventually to believe (2).

Since there are only two options, our inner self (which is assumed for the purpose of this example always to exist) is epistemically required to maximize the probability that we will hold the correct belief. Suppose it decides to try to make us believe that (1) is true. 95% of the time it will fail, and in 80% of these cases, we will end up correctly believing that (1) is true. In the 5% of cases where (2) is true, our inner self will succeed, and in doing so it will guarantee wrong belief. So the probability that this strategy will bring about correct belief is 0.95*0.8 = 0.76.

Now consider the strategy in which our inner self tries to make us believe (2). Again, it will fail 95% of the time, and there will again be an 80% chance that we'll end up correctly believing (1). But now, if (2) is true, we're guaranteed in that case to hold a correct belief. So the probability for this strategy is 0.95*0.8 + 0.05*1 = 0.81. It's clear that, as long as the prior probability of (2) is greater than 0, the strategy of trying to believe in (2) will always be epistemically better.

Example 2

Suppose now there are three possibilities, to which we assign the subjective probabilities given:

  1. The probability is 0 that our inner self will succeed in changing our beliefs about free will if it tries to do so. (initial subjective probability = 85%)
  2. The probability is 0.5 that our inner self will succeed in changing our beliefs about free will if it tries to do so. (initial subjective probability = 9%)
  3. The probability is 1 that our inner self will succeed in changing our beliefs about free will if it tries to do so. (initial subjective probability = 6%)

Assume that if our inner self fails to persuade us, our probabilities of eventually believing these options are 75%, 15%, and 10%, respectively. Our inner self has three strategies:

  1. Try to believe (A). Probability of correct belief = Pr(A)*[Pr(inner self succeeds given A is true)*Pr(correct belief if inner self succeeds given A is true) + Pr(inner self fails given A is true)*Pr(correct belief if inner self fails given A is true)] + Pr(B)*[Pr(inner self succeeds given B is true)*Pr(correct belief if inner self succeeds given B is true) + Pr(inner self fails given B is true)*Pr(correct belief if inner self fails given B is true)] + Pr(C)*[Pr(inner self succeeds given C is true)*Pr(correct belief if inner self succeeds given C is true) + Pr(inner self fails given C is true)*Pr(correct belief if inner self fails given C is true)] = 0.85*[0*1+1*0.75] + 0.09*[0.5*0+0.5*0.15] + 0.06*[1*0+0*0.1] = 0.64425.
  2. Try to believe (B). Probability of correct belief = 0.85*[0*0+1*0.75] + 0.09*[0.5*1+0.5*0.15] + 0.06*[1*0+0*0.1] = 0.68925.
  3. Try to believe (C). Probability of correct belief = 0.85*[0*0+1*0.75] + 0.09*[0.5*0+0.5*0.15] + 0.06*[1*1+0*0.1] = 0.70425.

It's not hard to see that the second term in each pair of [ ] above is the same for each strategy; this is because those terms represent what happens if the inner self fails. So in comparing the three strategies, it suffices to look at the first terms in each pair of [ ]; the 75%, 15%, 10% assumptions turned out to be irrelevant. It suffices to look at these numbers:

  1. 0.85*0*1 + 0.09*0.5*0 + 0.06*1*0 = 0.
  2. 0.85*0*0 + 0.09*0.5*1 + 0.06*1*0 = 0.045.
  3. 0.85*0*0 + 0.09*0.5*0 + 0.06*1*1 = 0.06.

These represent the probabilities that the inner self will bring about correct belief by adopting a particular strategy.

If we try to maximize our probability of correct belief, (C) wins. But what is interesting is that, had our initial subjective probabilities been slightly different, say 85%, 11%, and 4%, respectively, it would have been epistemically better to believe in (B) than (C), because 0.11*0.5*1 > 0.04*1*1.

We could also consider "mixed strategies" in which our inner self doesn't try to believe a single thing but rather tries to make us assign epistemic probabilities to various things. For instance, the inner self might decide to try to make us believe (B) with 40% probability and (C) with 60% probability. Suppose our goal is now to maximize the expected value of the probability that we place on the correct belief. We can ignore what happens if our inner self fails, because as we saw above, those terms will be the same for any strategy. If (B) is true and our inner self succeeds, we'll place 40% probability on the correct belief. The probability of this happening is 0.09*0.05. If (C) is true and our inner self succeeds, we'll place 60% probability on the correct belief. The probability that this happens is 0.06*1. The expected value of the probability we assign to the correct belief is then 0.09*0.5*0.4 + 0.06*1*0.6 + stuff, where stuff represents other terms that will be the same regardless of what strategy we pick.

It's not hard to see that, in this case at least, our expected value increases as we assign more weight to (C) than (B). For instance, if we assign 70% probability to (C) and 30% to (B), we have 0.09*0.5*0.3 + 0.06*1*0.7 + stuff > 0.09*0.5*0.4 + 0.06*1*0.6 + stuff. Our best strategy is to assign 100% probability to (C). But had the probabilities of (B) and (C) been 11% and 4%, rather than 9% and 6%, our best strategy would have been to assign 100% probability to (B).

It is not hard to generalize the reasoning above to a greater number of possible options, including a continuous number of them, as I do in this Note. There I also consider an alternate decision procedure other than attempting to maximize the expected value of the probability we assign to the correct belief.


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