2.1 Jamesian measures of epistemic utility

First, the measure of epistemic utility. In fact, in this paper, as in Joyce's, it will be most convenient to talk about measures of epistemic disutility. But such measures are easily obtained from measures of epistemic utility: the negative of an epistemic utility function is an epistemic disutility function; and vice versa. James' two commandments suggest a veritistic or accuracy-based account of epistemic utility. That is, it seems reasonable to interpret James as taking the sole fundamental source of epistemic utility to be the having of accurate belief states – Believe truth! – and the not having of inaccurate ones – Shun error!. This suggests that the epistemic disutility of a belief function b at a world w is determined as follows.Footnote ¹ First, there is a local inaccuracy measure ${\rm \frak s}$ : {0, 1} × {B, D, S} → [0, ∞]. The idea is this: ${\rm \frak s}$ (1, B) gives the inaccuracy of having a belief in a true proposition, while ${\rm \frak s}$ (0, B) gives the inaccuracy of having a belief in a false proposition. Similarly, ${\rm \frak s}$ (1, S) gives the inaccuracy of suspending judgment in a proposition that's true, and ${\rm \frak s}$ (0, S) gives the inaccuracy of suspending in a proposition that's false. And again ${\rm \frak s}$ (1, D) gives the inaccuracy of disbelieving a truth, while ${\rm \frak s}$ (0, D) gives the inaccuracy of disbelieving a falsehood. We make the following assumptions:

• • ${\rm \frak s}$ ${\rm \frak s}$ (1, B) =  ${\rm \frak s}$ (0, D). That is, a false disbelief is as inaccurate as a true belief. We denote this -R, where R > 0. ‘R’ for getting it right.

• • ${\rm \frak s}$ (1, D) =  ${\rm \frak s}$ (0, B). That is, a false belief is as inaccurate as a true disbelief. We denote this W, where W > 0. ‘W’ for getting it wrong.

• • ${\rm \frak s}$ (1, S) =  ${\rm \frak s}$ (0, S) = 0. That is, a suspension of judgment always has inaccuracy 0, regardless of the truth or falsity of the proposition.

Now we can use our local inaccuracy measure ${\rm \frak s}$ to define a global inaccuracy measure ${\rm \Im}$ for categorical doxastic states. This takes an entire belief function b defined on ${\rm \cal F}$ and a possible world w and returns a measure of the inaccuracy of b at w:

where v _w (X) = 1 if X is true and v _w (X) = 0 if X is false. That is, the inaccuracy of a belief function is simply the sum of the inaccuracies of the individual categorical attitudes that comprise it. Thus, for instance, a belief function that assigns belief (B) to X and disbelief (D) to $\overline X$ , evaluated at a world at which X is true, will have global inaccuracy ${\rm \frak s}$ (1, B) +  ${\rm \frak s}$ (0, D) = −2R.

Thus, a global inaccuracy measure is determined by a local inaccuracy measure; and a local inaccuracy measure is determined by the badness we assign to a true belief/false disbelief – that is, the value of R – and the badness we assign to a true disbelief/false belief – that is, the value of W. It is in this latter endeavour that we might think the balance between James' commandments makes a difference. If we “regard the chase for truth as paramount, and the avoidance of error as secondary”, then we will take R to be greater than W; if, on the other hand, we “treat the avoidance of error as more imperative, and let truth take its chance”, then we will take W to exceed R. And of course it's also possible to weight them equally and let W = R. Depending on what we choose in this case, quite different epistemic norms follow from decision-theoretic principles.

For instance, consider the dominance principle to which Joyce appeals in his argument for probabilism. And consider an agent who entertains only two propositions, X and its negation . If R ≥ W, then the belief functions that are not accuracy dominated are these: believe X and disbelieve $\overline X$ ; believe $\overline X$ and disbelieve X; believe X and believe $\overline X$ ; disbelieve X and disbelieve $\overline X$ . That is, if we weigh Believe truth! more highly than Shun error!, then believing each of a pair of mutually inconsistent propositions is not ruled out as irrational by the dominance principle (nor is disbelieving both).Footnote ² On the other hand, if W > R, then the belief functions that are not accuracy dominated are these: believe X and disbelieve $\overline X$ ; believe $\overline X$ and disbelieve X; suspend judgment on X and $\overline X$ . That is, if we weigh Shun error! more highly than Believe truth!, then believing each of a pair of mutually inconsistent propositions is ruled out as irrational by the dominance principle (as is disbelieving each of them).

2.2 Jamesian decision principles

This gives us a taste of the consequences of different weightings of James' two imperatives when it comes to measuring the epistemic disutility of a categorical doxastic state. But these weightings do not only affect the way in which we measure epistemic disutility. They also affect the decision principle that we use in conjunction with that measure of epistemic disutility to assess the rationality of categorical doxastic states. After all, James' two commandments each encode opposite attitudes to epistemic risk. To believe or disbelieve a proposition gives you a shot at maximal epistemic utility (that is, minimal inaccuracy), however you set R and W: you will receive maximal epistemic utility if what you believe turns out to be true, or if what you disbelieve turns out to be false. But of course it also opens you up to the possibility of minimal epistemic utility (that is, maximal inaccuracy): you will receive minimal epistemic utility if what you believe turns out to be false, or what you disbelieve turns out to be true. Suspension, on the other hand, does not give you a shot at maximal epistemic utility, but nor does it open you up to the possibility of minimal epistemic utility. Thus, believing and disbelieving are risky doxastic attitudes in a way that suspending is not.

In ‘The Will to Believe’, James seems more concerned with this reading of his competing commandments than with the reading explicated in section 2.1 above. Thus, he talks of suspending judgment as being “attended with the […] risk of losing the truth” (James Reference James1896: 334) and the “awful risk of believing lies” (James Reference James1896: 338). However, at other points, where he also uses the language of risk, he is clearly imagining that this will be spelled out in terms of epistemic utility, such as when he writes: “worse things than being duped may happen to a man in this world” (James Reference James1896: 339).

We know from practical decision theory that there are two ways that we might encode an agent's sensitivity to risk. We might try to build some of the agent's attitudes to risk into her utility function; or we might capture it in the decision principle that she endorses (Buchak Reference Buchak2014). For instance, in practical decision theory, we might represent a risk-averse agent as assigning diminishing marginal utility to money: that is, we might say that her utility for an outcome is a concave function of the money she will receive in that outcome. But we might also take her to endorse a risk-averse decision principle. Such principles include: Minimax, which says that an agent should act to minimize her worst-case disutility; Minimax Regret, which says that an agent should act to minimize her worst-case regret; and certain versions of the Hurwicz Criterion, which takes the weighted sum of the best-case disutility of an option and its worst-case disutility and says that an agent should minimize that. In ${\rm \S}$ 2.1, we saw how we might encode an agent's attitudes to epistemic risk into her epistemic utility function: a risk-averse agent, who weighs Believe truth! more heavily than Shun error!, will assign a greater penalty to being wrong – believing a falsehood or disbelieving a truth – than she will assign a reward to being right – believing a truth or disbelieving a falsehood. A risk-seeking agent, who weighs the commandments in the opposite way, will do the opposite. In this section, we are interested in encoding those attitudes in a decision principle. The natural choice is the Hurwicz Criterion, since that allows us to weigh two competing attitudes to risk differently; and, as we will see below, those two attitudes correspond to James' two commandments.

Let us begin by stating the Hurwicz Criterion as a decision principle for general decision theory – that is, decision theory in which we make no assumptions about the nature of the options between which the agent is choosing. We will then adapt it to give a decision principle for epistemic utility theory, and we will describe the epistemic norm that we derive from it in conjunction with our account of epistemic utility from section 2.1.

Suppose ${\rm \cal O}$ is a set of options and ${\rm \cal W}$ is the set of possible worlds. Now suppose that ${\rm \frak U}$ is a utility function that takes each option and each possible world and returns the utility of the outcome of choosing that option in that world. Thus $- {\rm \frak U}$ is a disutility function. Then, as I said above, the Hurwicz Criterion asks an agent to minimize a weighted sum of the best-case disutility of an option and its worst-case disutility. Thus, different versions of the Hurwicz Criterion are given by different choices of weight. Let 0 ≤ λ ≤ 1. Then, for any option o in ${\rm \cal O}$ , define

So λ is the weight assigned to the best-case scenario; the scenario in which the disutility of the option is minimal. Then:

That is, all options o for which $H_\lambda ^{ - {\rm \frak U}} (o)$ is not minimal are ruled irrational. Thus, suppose our agent is faced with the following decision problem, where the numbers in the table represent the disutility of the relevant outcomes:

Then

• • ${H_\lambda} ({\kern 1pt} {\rm Umbrella}{\kern 1pt} ) = 6\lambda + 10(1 - \lambda ) = 10 - 4\lambda$

• • ${H_\lambda} ({\kern 1pt} {\rm No}\;{\rm Umbrella}{\kern 1pt} ) = 0\lambda + 12(1 - \lambda ) = 12 - 12\lambda$

Thus, for $\lambda \lt \displaystyle{1 \over 4}$ , Umbrella is the rational option; for $\lambda = \displaystyle{1 \over 4}$ , both are rationally permitted; and for $\lambda \gt \displaystyle{1 \over 4}$ , No Umbrella is the rational option. Thus, as λ increases and more weight is given to the best-case situation, the agent becomes more risk-seeking and less risk-averse.

Now suppose that ${\rm \Im}$ is a global inaccuracy measure – that is, an epistemic disutility function for belief functions. Then, for a belief function b,

Again, λ weights the best-case (or minimal) disutility. Then the Hurwicz _λ Criterion demands:

Thus, λ represents the weight given to the Jamesian commandment Believe truth!, since that is the risk-seeking commandment, which enjoins us to try to be accurate even though doing so inevitably exposes us to the risk of being inaccurate. On the other hand, 1 − λ represents the weight given to Shun error!, since that is the risk-averse commandment, which enjoins us to avoid exposing ourselves to the risk of inaccuracy, even if it is only by doing so that we open ourselves to the possibility of accuracy.

Now, suppose ${\rm \Im}$ is determined by the local inaccuracy measure ${\rm \frak s}$ , and that ${\rm \frak s}$ is in turn determined by R and W. Then our question is this: What epistemic norm follows from Hurwicz _λ Criterion when it is applied to ${\rm \Im}$ ? To state the norm, we first define what it means to be a Hurwicz _λ belief function. It turns out that these belief functions are precisely those permitted by Hurwicz _λ Criterion (categorical; epistemic).

Suppose we are considering belief functions defined on an algebra of propositions ${\rm \cal F}$ . Our first job is to say when a probability function defined on ${\rm \cal F}$ probabilifies a belief function on ${\rm \cal F}$ relative to R and W. There are three cases that we must consider: W > R, W = R, W < R. Let's take them in turn:

Thus, roughly speaking, p probabilifies b if p and b jointly satisfy the Lockean thesis for belief with respect to the belief threshold $\displaystyle{W \over {R + W}}$ . That thesis, again roughly, says that an agent believes X if she assigns sufficiently high probability to X; she disbelieves X if she assigns sufficiently low probability to X; and otherwise she suspends. On the threshold itself, she has some choice.

Thus, for instance, if ${\rm \cal F}$ contains only God exists and God does not exist, and R = 1 and W = 2, then p probabilifies a belief in God exists and disbelief in God does not exist if $p({\kern 1pt} {\rm God}\;{\rm exists}{\kern 1pt} ) \ge \displaystyle{2 \over 3} = \displaystyle{2 \over {2 + 1}}$ .

1. (ii) Suppose W = R. Then p probabilities b if, for each X in ${\rm \cal F}$ , one of the following holds:

   • • $\displaystyle{1 \over 2} \lt p(X)$ and b(X) = B

   • • $\displaystyle{1 \over 2} = p(X)$ and b(X) = S or b(X) = D or b(X) = B

   • • $p(X) \lt \displaystyle{1 \over 2}$ and b(X) = D

2. (iii) Suppose W < R. Then p probabilifies b if, for each X in ${\rm \cal F}$ , one of the following holds:

   • • $\displaystyle{1 \over 2} \lt p(X)$ and b(X) = B

   • • $\displaystyle{1 \over 2} = p(X)$ and b(X) = D or b(X) = B

   • • $p(X) \lt \displaystyle{1 \over 2}$ and b(X) = D

The crucial result is the following, due to Hempel (Reference Hempel, Feigl and Maxwell1962) and Easwaran (Reference Easwaranms):

That's the first stage in our definition of the Hurwicz _λ belief functions. In the second stage, we will define them as the belief functions that are probabilified by a particular sort of probability function.

Suppose ${\rm \cal F}$ is a finite algebra. Then let ${{\rm \cal W}_{\rm \cal F}} = \{ {w_1}, \ldots, {w_n}\}$ be the atoms of that algebra: one might think of these as the maximally specific possibilities relative to ${\rm \cal F}$ . Now, for each 1 ≤ k ≤ n, we need to define the k ^th Hurwicz _λ probability function on ${\rm \cal F}$ . We denote it $p_k^\lambda$ and we define it as follows:

Thus, suppose you are thinking about the colour of my newest tie. You partition the space of possibilities into three: red, yellow, blue. Thus, you entertain only the propositions in the algebra generated by the following three possibilities: w ₁ = Red, w ₂ = Blue, w ₃ = Yellow. And suppose $\lambda = \displaystyle{1 \over 2}$ . Then $p_1^\lambda (Red) = \lambda = \displaystyle{1 \over 2}$ , while $p_1^\lambda (Blue) = \displaystyle{{1 - \lambda} \over {n - 1}} = \displaystyle{1 \over 4}$ and $p_1^\lambda (Yellow) = \displaystyle{{1 - \lambda} \over {n - 1}} = \displaystyle{1 \over 4}$ . If, on the other hand, $\lambda = \displaystyle{1 \over 3}$ , then $p_1^\lambda (Red) = p_1^\lambda (Blue) = p_1^\lambda (Yellow) = \displaystyle{1 \over 3}$ .

As we will see, these probability functions play an equally important role when we turn to ask what the consequences of the Hurwicz _λ Criterion are in the case of credences. Here is the crucial theorem that states the consequences of the Hurwicz _λ Criterion in the case of categorical doxastic states:

The proof is given in the Appendix. Let's see this result in action. I ask you to pick a number from 1 to 9. Let X _k be the proposition that says that you pick k. Thus, I entertain the propositions X ₁, …, X ₉.Footnote ³ Now suppose that R = 1 and W = 3. Thus, when I pick my epistemic disutility function, I assign three times greater weight to James' commandment to shun error than to his commandment to believe truth. Now let us see what happens when we change the relative weights that I assign to these commandments when I pick the version of the Hurwicz Criterion that I will apply in the presence of that epistemic disutility function.

First, note that, for any $\displaystyle{1 \over 9} \lt \lambda \le 1$ , $p_k^\lambda ({X_j}) = \displaystyle{{1 - \lambda} \over 8} \lt \displaystyle{1 \over 4} = \displaystyle{R \over {R + W}}$ (for j ≠ k). Thus, since Hurwicz _λ Criterion (categorical; epistemic) demands that I have belief function that is probabilified by $p_k^\lambda$ for some 1 ≤ k ≤ 9, and since any such belief function will assign disbelief to eight out of the nine propositions X ₁, …X ₉, I must disbelieve eight out of the nine propositions. What Hurwicz _λ Criterion (categorical; epistemic) demands in the case of the remaining proposition depends on λ. If $\displaystyle{3 \over 4} \lt \lambda$ , then it demands that I believe the remaining proposition; if $\lambda = \displaystyle{3 \over 4}$ , it permits me to believe or suspend; if $\displaystyle{1 \over 4} \le \lambda \lt \displaystyle{3 \over 4}$ , it demands that I suspend; and if $\lambda \lt \displaystyle{1 \over 4}$ , it demands that I disbelieve.

Second, note that, if $\lambda \le \displaystyle{1 \over 9}$ , then $p_k^\lambda ({X_j}) = \displaystyle{1 \over 9}$ for all 1 ≤ j ≤ 9. Thus, since $\displaystyle{1 \over 9} \lt \displaystyle{1 \over 4}$ , $p_k^\lambda$ probabilifies a belief function that assigns disbelief to each of the nine propositions. Of course, this gives rise to a situation akin to the Lottery Paradox, where we disbelieve each of a set of mutually exclusive and exhaustive propositions. Recall that we saw above that, if R > W, there were inconsistent sets of propositions such that believing each is not accuracy dominated. The implications of the Hurwicz _λ Criterion (categorical; epistemic) just sketched show that such attitudes may be permitted even if R < W.

3.1 Jamesian measures of epistemic disutility

We are now in a position to ask how our attitudes to the Jamesian commandments might be reflected in the way that we measure the inaccuracy of credence functions.Footnote ⁴ Recall that, in the case of belief functions, we said that the agent who weighs Believe truth! more heavily than Shun error! will assign a greater reward to getting it right than she will assign a penalty to getting it wrong. Thus, for such agents – whom we might think of as epistemic risk-seekers – moving from the neutral position of suspension to a belief introduces the possibility of penalty and reward, but the potential reward has more goodness than the penalty has badness. And the opposite is true of agents – whom we might think of epistemic risk-avoiders – who weigh Shun error! more heavily than Believe truth!. For them, moving from the neutral position of suspension introduces the possibility of penalty and reward, but the potential reward has less goodness than the penalty has badness. Translating this to the credal case, we get the following: an epistemic risk-seeker should have a local inaccuracy measure where a move away from neutral ground – which we might take to be the point at which ${\rm \frak s}$ (1, x) =  ${\rm \frak s}$ (x, 0) – will introduce the possibility of penalty and reward, but the potential reward – that is, the decrease in inaccuracy if the move is in the right direction – will have more goodness than the penalty – that is the increase in inaccuracy if the move is in the wrong direction – will have badness. And the opposite for the epistemic risk-avoider. More precisely:

• • If ${\rm \frak s}$ is a risk-seeking local inaccuracy measure and x* is the neutral point for ${\rm \frak s}$ – that is, ${\rm \frak s}$ (1, x*) =  ${\rm \frak s}$ (0, x*) – then, for all x and ε:

  • – If x > x* and x + ε ≤ 1, then

    $${\rm \frak s}(1,x) - {\rm \frak s}(1,x + \varepsilon ) \gt {\rm \frak s}(0,x + \varepsilon ) - {\rm \frak s}(0,x)$$

  • – If x < x* and x − ε ≥ 0, then

    $${\rm \frak s}(0,x) - {\rm \frak s}(0,x - \varepsilon ) \gt {\rm \frak s}(1,x - \varepsilon ) - {\rm \frak s}(1,x)$$

• • If ${\rm \frak s}$ is a risk-avoiding local inaccuracy measure and x* is the neutral point for ${\rm \frak s}$ – that is, ${\rm \frak s}$ (1, x*) =  ${\rm \frak s}$ (0, x*) – then, for all x and ε:

  • – If x > x* and x + ε ≤ 1, then

    $${\rm \frak s}(1,x) - {\rm \frak s}(1,x + \varepsilon ) \lt {\rm \frak s}(0,x + \varepsilon ) - {\rm \frak s}(0,x)$$

  • – If x < x* and x − ε ≥ 0, then

    $${\rm \frak s}(0,x) - {\rm \frak s}(0,x - \varepsilon ) \lt {\rm \frak s}(1,x - \varepsilon ) - {\rm \frak s}(1,x)$$

Here is an example of a risk-avoiding local inaccuracy measure. It is called the quadratic scoring rule and it is defined as follows: ${\rm \frak q}$ (1, x) = (1 − x)² and ${\rm \frak q}$ (0, x) = x ². And here is an example of a risk-seeking local inaccuracy measure. It is called the square root scoring rule and it is defined as follows: ${\rm \frak r}(1,x) = \sqrt {1 - x}$ and ${\rm \frak r}(0,x) = \sqrt x$ .

Recall from above: If R < W – that is, the agent has a risk-averse epistemic utility function – then only the consistent categorical doxastic states are not dominated, but if R > W – that is, the agent's epistemic utility function is risk-seeking – then inconsistent states are undominated. Similarly, it turns out, the credence functions that are undominated relative to the inaccuracy measure generated by the quadratic scoring rule are precisely the coherent ones – that is, the ones that satisfy the axioms of the probability calculus – whereas there are incoherent credence functions that are undominated relative to the inaccuracy measure generated by the square root scoring rule. The former fact is a theorem due to de Finetti (Reference de Finetti1974) and generalised by Joyce (Reference Joyce1998), Predd et al. (Reference Predd, Seiringer, Lieb, Osherson, Poor and Kulkarni2009), and Pettigrew (Reference Pettigrewms); the latter is easily seen by considering the credence function that assigns 0 to X and 0 to $\overline X$ .

3.2 Jamesian decision principles

Finally, we turn to the question of decision principles that reflect different attitudes to the Jamesian commandments. In fact, we have very little work to do here, since the natural decision principle was introduced above. It is the Hurwicz Criterion. Here is the version that pertains to credences. Given a global inaccuracy measure ${\rm \Im}$ for credence functions – that is, an epistemic disutility function for those states – let:

So our question is: Which credence functions are ruled irrational by the Hurwicz _λ Criterion (graded; epistemic) and which are permitted? Suppose the credence functions between which we are choosing are all defined on a finite algebra ${\rm \cal F}$ with ${{\rm \cal W}_{\rm \cal F}} = \{ {w_1}, \ldots, {w_n}\}$ , as above. Then, under a natural assumption, it turns out to be only the Hurwicz _λ probability functions on ${\rm \cal F}$ , defined above, that are permitted (not ruled irrational) by Hurwicz _λ Criterion (graded; epistemic). The natural assumption is this: our local inaccuracy measure is strictly proper.

That is, ${\rm \frak s}$ is strictly proper if a given credence p in a proposition X will expect itself to be the least inaccurate amongst all possible credences in X. A number of epistemic utility theorists have made this assumption or something similar (Greaves and Wallace Reference Greaves and Wallace2006; Joyce Reference Joyce, Huber and Schmidt-Petri2009; Pettigrew, Reference Pettigrewms). However, it is worth noting that while the quadratic scoring rule ${\rm \frak q}$ defined above is strictly proper, the other scoring rule ${\rm \frak r}$ is not. Nonetheless, if we make this assumption, we can identify the credence functions permitted by Hurwicz _λ Criterion (graded; epistemic) by appealing to the following theorem:

Thus, if we consider again the situation in which I ask you to pick a number from 1 to 9, and proposition X _i says that you pick number i, then we have the following. If I am very risk averse and assign $\lambda \le \displaystyle{1 \over 9}$ , then my credence should be simply the uniform distribution over the nine possibilities: $c_k^\lambda ({X_k}) = \displaystyle{1 \over 9}$ . But, on the other hand, if I am more willing to take epistemic risks, and I give more weight to the best-case disutility by setting $\displaystyle{1 \over 9} \lt \lambda$ , then I should plump for one of the possible numbers k (with 1 ≤ k ≤ 9) and have credence λ that you picked k, while dividing my credences equally over the other eight options, thus assigning $\displaystyle{{1 - \lambda} \over 8}$ to each.