2. Preliminaries

As this is a technical note, I will be quick with the formal preliminaries. Further discussion can be found in Pettigrew’s (Reference Pettigrew2022) paper as well as his book (Pettigrew Reference Pettigrew2016).

Let $W$ be a finite set of worlds. A credence function $c:{2^W} \to [0,1]$ is a function from the powerset of $W$ into the unit interval. Let ${\cal C}$ be the set of all credence functions. If $({c_n})$ is a sequence of credence functions and $c \in {\cal C}$ , we write ${c_n} \to c$ if and only if for all $A \subseteq W$ the sequence $({c_n}(A))$ of real numbers converges to $c(A)$ . Another way of putting this is that we are viewing ${\cal C}{ = [0,1]^{{2^W}}}$ as a subset of ${|2^W}|$ -dimensional Euclidean space. A credence function $c$ is probabilistic if and only if it obeys the usual axioms of probability. Let ${\cal P}$ be the set of all probabilistic credence functions. Since both ${\cal P}$ and ${\cal C}$ are closed and bounded, they are compact.

If $w \in W$ , then let

$${v_\omega }(A) = \left\{ {\matrix{ {1,} \ {w \in A} \cr {0,\ } {w \ \notin\ A} \cr } } \right.$$

for all $A \subseteq W$ . The (probabilistic) credence function ${v_w}$ is the omniscient or ideal one at the world $w$ .

An inaccuracy measure ${\cal J}:{\cal C} \times W \to [0,\infty ]$ is a function from pairs of credence functions and worlds into the extended half-line. Intuitively, ${\cal J}(c,w)$ represents how inaccurate the credence function $c$ is in world $w$ .^{Footnote 1} If $p \in {\cal P}$ , let us write

(1) $${\cal J}(c,p) = \sum\limits_{w \,\in\, W} {\cal J}(c,w)p\{ w\}.$$

Then ${\cal J}(c,p)$ is the expected inaccuracy of $c$ according to $p$ . Under this definition, we have ${\cal J}(c,w) = {\cal J}(c,{v_w})$ for all credence functions $c$ and worlds $w$ .

Pettigrew focuses on inaccuracy measures with the following properties.

Strict Propriety: An inaccuracy measure ${\cal J}$ is strictly proper if and only if ${\cal J}(p,p) \lt {\cal J}(c,p)$ for all $p \in {\cal P}$ and all $c \in {\cal C}\backslash \{ p\}$ .

Continuity: An inaccuracy measure ${\cal J}$ is continuous if and only if for all $p \in {\cal P}$ and $c,{c_1},{c_2}, \ldots \in {\cal C}$ the sequence $({\cal J}({c_n},p))$ of extended-real numbers converges to ${\cal J}(c,p)$ whenever ${c_n} \to c$ .^{Footnote 2}

The Brier score is perhaps the most well-known strictly proper and continuous inaccuracy measure, but we will not have need of it here. We will, however, make use of the enhanced log score, defined by

(2) $${\mathcal L}^*(c,w) = \sum_{A \subseteq W}({\hbox{-}}v_w(A) \log c(A) + c(A)),$$

which is also strictly proper and continuous. It is worth pointing out now that if $w \in A$ and $c(A) = 0$ , then $ - {v_w}(A)\log c(A) = \infty $ . So, according to the enhanced log score, any credence function that assigns zero credence to a non-empty set of worlds is infinitely inaccurate at some worlds. The fact that inaccuracy measures can take the value $\infty $ plays an important role in understanding the mistakes in Pettigrew’s proof.

The final definition we will need is the following:

Accuracy-Dominance: Let $c,c' \in {\cal C}$ , and let ${\cal J}$ be an inaccuracy measure. We say that $c$ accuracy-dominates $c'$ (according to ${\cal J}$ ) if and only if ${\cal J}(c,w) \lt {\cal J}(c',w)$ for all $w \in W$ .

As mentioned above, accuracy arguments for probabilism have a normative premise stating that credence functions are irrational if they are accuracy-dominated. We will not question that premise here, as our focus is on the third component of accuracy arguments: the accuracy-dominance theorems.

Here is the result that Pettigrew reports.

In other words, every non-probabilistic credence function is accuracy-dominated by a probabilistic one. If we accept the accuracy argument’s normative premise, then we can use Theorem 1 to infer that rationality forbids violations of the probability axioms.

3. Pettigrew’s proof

Theorem 1 is true, but Pettigrew’s proof of it is incorrect. In this section, I will explain the mistakes and lay the groundwork for a correct proof. We assume throughout this section that ${\cal J}$ is strictly proper and continuous.

The main technique for proving accuracy-dominance theorems involves analyzing the divergences associated with inaccuracy measures. Given ${\cal J}$ , the associated divergence ${{\cal D}_{\cal J}}:{\cal C} \times {\cal P} \to [0,\infty ]$ is defined by

(3) $${{\cal D}_{\cal J}}(c,p) = {\cal J}(c,p) - {\cal J}(p,p).$$

The equation in (3) is well defined—never of the form $\infty - \infty $ —because ${\cal J}$ is strictly proper, and this implies that ${\cal J}(p,p) \lt \infty $ for all $p \in {\cal P}$ . We can think of ${{\cal D}_{\cal J}}$ as measuring the distance between the $p$ -expected inaccuracy of $c$ and the minimal $p$ -expected inaccuracy, which is ${\cal J}(p,p)$ by strict propriety.

Now, let $c \in {\cal C}\backslash {\cal P}$ be a non-probabilistic credence function. Then ${{\cal D}_{\cal J}}(c, \cdot )$ is a function from the set of probability functions ${\cal P}$ into the extended half-line. The proof of Theorem 1 has two steps. Step 1: one argues that ${{\cal D}_{\cal J}}(c, \cdot )$ can be minimized. That is, one argues that there is some ${p^*} \in {\cal P}$ such that ${{\cal D}_{\cal J}}(c,{p^*}) \le {{\cal D}_{\cal J}}(c,p)$ for all $p \in {\cal P}$ . Step 2: one shows that the minimizer ${p^*}$ of ${{\cal D}_{\cal J}}(c, \cdot )$ can be used to construct a probability function that accuracy-dominates $c$ .

Pettigrew’s proof has mistakes in both steps. It will be easier to describe the mistake he makes in the second step in the next section (Remark 1). In this section, we will focus on the mistakes in the first step.

Why is it possible to minimize ${{\cal D}_{\cal J}}(c, \cdot )$ ? Pettigrew gives two arguments, both contained in the proof of his Lemma 5. Here is the first argument: “Suppose $c$ is not in ${\cal P}$ . Then since ${\cal P}$ is a closed convex set and since Lemma 4(iii) shows that ${{\cal D}_{\cal J}}(c, \cdot )$ is strictly convex, there is a unique ${p^*} \in {\cal P}$ that minimizes ${{\cal D}_{\cal J}}(c, \cdot )$ .”^{Footnote 3}

There are two problems with this argument. (1) The assumptions in Theorem 1, namely that ${\cal J}$ is strictly proper and continuous, do not imply that ${{\cal D}_{\cal J}}(c, \cdot )$ is strictly convex (that is, Pettigrew’s Lemma 4(iii) is false). (2) Even if ${{\cal D}_{\cal J}}(c, \cdot )$ were strictly convex, this would not imply the existence of a minimizer. In particular, it is not the case that every strictly convex function on a closed convex set attains a minimum on that set.

I will demonstrate (1) and (2) in a moment. But first, let us take a look at the beginning of Pettigrew’s second argument. He writes: “We now briefly sketch an alternative proof … that is available if ${{\cal D}_{\cal J}}(c, \cdot )$ is not only continuous … but also differentiable.”

The problem with this is: (3) The assumptions in Theorem 1 do not imply that ${{\cal D}_{\cal J}}(c, \cdot )$ is continuous and therefore don’t imply that ${{\cal D}_{\cal J}}(c, \cdot )$ is differentiable.^{Footnote 4} This is important to realize, because if ${{\cal D}_{\cal J}}(c, \cdot )$ were continuous, then it would attain a minimum on ${\cal P}$ due to the basic topological fact that continuous functions attain minima on compact sets. In other words, if ${{\cal D}_{\cal J}}(c, \cdot )$ were continuous, then Pettigrew’s argument would be easy to repair.^{Footnote 5} As we will soon see, proving Pettigrew’s theorem requires a bit more argumentation.

Now let us establish the three claims made in response to Pettigrew’s arguments. Claims (1) and (3) are established by the following counterexample.

Let $W = \{ 1,2\} $ , and let the inaccuracy measure be the enhanced log score ${{\mathcal L}^*}$ , defined in (2). Let $c$ be the probabilistic credence function defined by $c\{ 1\} = 0$ . Then ${{\mathcal L}^*}(c,1) = \infty $ . Let $({p_n})$ be the sequence of probability functions defined by ${p_n}\{ 1\} = {n^{ - 1}}$ . Then ${{\mathcal L}^*}(c,{p_n}) = \infty $ for all $n$ , which implies ${{\cal D}_{{{\mathcal L}^*}}}(c,{p_n}) = \infty $ for all $n$ . But ${p_n} \to c$ . So,

$${{\cal D}_{{{\mathcal L}^*}}}(c,{p_n}) \to \infty \ne 0 = {{\cal D}_{{{\mathcal L}^*}}}(c,c)$$

shows that ${{\cal D}_{{{\mathcal L}^*}}}(c, \cdot )$ is not continuous.

To see that ${{\cal D}_{{{\mathcal L}^*}}}(c, \cdot )$ is not strictly convex, note that ${p_n} = {n^{ - 1}}{v_1} + (1 - {n^{ - 1}})c$ . But then,

$${{\cal D}_{{{\mathcal L}^*}}}(c,{p_n}) = \infty \ge {n^{ - 1}}{{\cal D}_{{{\mathcal L}^*}}}(c,{v_1}) + (1 - {n^{ - 1}}){{\cal D}_{{{\mathcal L}^*}}}(c,c),$$

so ${{\cal D}_{{{\mathcal L}^*}}}(c, \cdot )$ is not strictly convex.

And Claim (2) is established by the following counterexample.

Consider the function $f:[0,1] \to {\mathbb R} $ defined by

$$f(x) = \left\{ {\matrix{ {{{(x - 1)}^2},} \ {x \in [0,1]} \cr \hskip -35pt{1,\ } \ {x = 1.} \cr } } \right.$$

The set $[0,1]$ is closed (indeed, compact) and convex, and $f$ is strictly convex. But $f$ does not attain a minimum on $[0,1]$ .

The key points to take away from the examples are: (1) In general, a divergence is not strictly convex in its second argument; (2) Strict convexity does not guarantee the existence of minimizers; (3) In general, a divergence is not continuous in its second argument.^{Footnote 6} We now move on to providing an alternative proof of Theorem 1.

4. An accuracy-dominance theorem

In fact, we will generalize Theorem 1 a bit by weakening both strict propriety and continuity. I do not know whether the properties that I am about to introduce are more philosophically compelling than strict propriety and continuity. My aim here is simply to lay bare as much as possible the mathematical arguments needed to support accuracy-based approaches to justifying probabilism.

Quasi-Strict Propriety: An inaccuracy measure ${\cal J}$ is quasi-strictly proper if and only if ${\cal J}(p,p) \le {\cal J}(c,p)$ for all $p \in {\cal P}$ and all $c \in {\cal C}$ , with strict inequality if $c \in {\cal C}\backslash {\cal P}$ .

Continuity on ${\cal P}\!:$ An inaccuracy measure ${\cal J}$ is continuous on ${\cal P}$ iff for all $p,q,{q_1},{q_2}, \ldots \in {\cal P}$ the sequence $({\cal J}({q_n},p))$ of extended-real numbers converges to ${\cal J}(q,p)$ whenever ${q_n} \to q$ .

Unlike strict propriety, quasi-strict propriety allows for ${\cal J}(p,p) = {\cal J}(q,p)$ , provided $q$ is probabilistic. A strict inequality is required only for non-probabilistic credence functions. Continuity on ${\cal P}$ differs from continuity by allowing the inaccuracy measure to be discontinuous on non-probabilistic credence functions.

Here is the result that we will prove.^{Footnote 7}

Throughout the rest of this section, let ${\cal J}$ be a quasi-strictly proper inaccuracy measure that is continuous on ${\cal P}$ , and let $c \in {\cal C}\backslash {\cal P}$ . As outlined in the previous section, the proof of Theorem 2 has two steps that involve analyzing the divergence ${{\cal D}_{\cal J}}(c, \cdot )$ . The first thing we should note is that ${{\cal D}_{\cal J}}$ continues to be well defined under quasi-strict propriety, because we still have ${\cal J}(p,p) \lt \infty $ for all $p \in {\cal P}$ .

Another consequence of quasi-strict propriety worth noting right away is that we can assume without loss of generality that ${\cal J}(c,w) \lt \infty $ for some $w \in W$ . Otherwise, any regular probability—any probability such that $p\{ w\} \gt 0$ for all $w \in W$ —would accuracy-dominate $c$ , because if $p$ is regular, then ${\cal J}(p,p) \lt \infty $ implies ${\cal J}(p,w) \lt \infty $ for all $w \in W$ .

Recall the two steps of the proof. Step 1: show that ${{\cal D}_{\cal J}}(c, \cdot )$ is minimized at some ${p^*} \in {\cal P}$ . Step 2: use ${p^*}$ to construct a probabilistic credence function that accuracy-dominates $c$ . I will dispense with Step 2 first, as this will allow us to see the final mistake in Pettigrew’s proof. The proof of the following lemma is similar to arguments given by Pettigrew, but special care has to be taken to handle cases where ${\cal J}(c,w) = \infty $ .

$${{\cal D}_{\cal J}}(c,{p^*}) \le {{\cal D}_{\cal J}}(c,{v_w}) = {\cal J}(c,w) - {\cal J}({v_w},w) \lt \infty.$$

This, in turn, implies that ${\cal J}(c,{p^*}) \lt \infty $ .

Now let $w \in W$ be arbitrary. If $n \in {\mathbb N}$ , let ${p_n} = {n^{ - 1}}{v_w} + (1 - {n^{ - 1}}){p^*}$ . Note that because ${\cal J}({p_n},{p_n}) \lt \infty $ and

$${\cal J}({p_n},{p_n}) = {n^{ - 1}}{\cal J}({p_n},w) + (1 - {n^{ - 1}}){\cal J}({p_n},{p^*}),$$

we must have both ${\cal J}({p_n},w) \lt \infty $ and ${\cal J}({p_n},{p^*}) \lt \infty $ for all $n$ . Using the fact that ${p^*}$ is a minimizer of ${{\cal D}_{\cal J}}(c, \cdot )$ , we have, for all $n$ ,

(4) $$\eqalign{0 & \le n({{\cal D}_{\cal J}}(c,{p_n}) - {{\cal D}_{\cal J}}(c,{p^*})) \cr & = n({\cal J}(c,{p_n}) - {\cal J}({p_n},{p_n}) - {\cal J}(c,{p^*}) + {\cal J}({p^*},{p^*}))\cr & = n({n^{ - 1}}{\cal J}(c,w) + (1 - {n^{ - 1}}){\cal J}(c,{p^*})- {n^{ - 1}}{\cal J}({p_n},w) \cr & - (1 - {n^{ - 1}}){\cal J}({p_n},{p^*}) - {\cal J}(c,{p^*}) + {\cal J}({p^*},{p^*}))\cr & = ({\cal J}(c,w) - {\cal J}(c,{p^*}) - {\cal J}({p_n},w) + {\cal J}({p_n},{p^*})) + n({\cal J}({p^*},{p^*}) - {\cal J}({p_n},{p^*}))\cr & \le {\cal J}(c,w) - {\cal J}(c,{p^*}) - {\cal J}({p_n},w) + {\cal J}({p_n},{p^*}),}$$

where the final inequality uses quasi-strict propriety. Next, (4) rearranges to

(5) $${\cal J}({p_n},w) \le {\cal J}(c,w) - {\cal J}(c,{p^*}) + {\cal J}({p_n},{p^*}),$$

and ${p_n} \to {p^*}$ implies

(6) $${\cal J}({p^*},w) \le {\cal J}(c,w) - {{\cal D}_{\cal J}}(c,{p^*}).$$

Quasi-strict propriety implies ${{\cal D}_{\cal J}}(c,{p^*}) \gt 0$ . So from (6), we have ${\cal J}({p^*},w) \lt {\cal J}(c,w)$ for all $w \in W$ such that ${\cal J}(c,w) \lt \infty $ . But if ${\cal J}(c,w) = \infty $ , then (6) implies only the weak inequality ${\cal J}({p^*},w) \le {\cal J}(c,w)$ . To deal with this, we can choose any regular probability $q$ and let ${q_n} = {n^{ - 1}}q + (1 - {n^{ - 1}}){p^*}$ . Then ${q_n} \to {p^*}$ . So by continuity, for large enough $n$ , we have ${\cal J}({q_n},w) \lt {\cal J}(c,w)$ whenever ${\cal J}(c,w) \lt \infty $ . And we also have ${\cal J}({q_n},w) \lt {\cal J}(c,w)$ whenever ${\cal J}(c,w) = \infty $ , simply in virtue of the fact that the regularity of ${q_n}$ implies ${\cal J}({q_n},w) \lt \infty $ for all $w \in W$ . Thus, for large enough $n$ , ${q_n}$ accuracy-dominates $c$ . This concludes the proof of the lemma.^{Footnote 8}

So much for Step 2. In order to conclude the proof of Theorem 2, we must now show that ${{\cal D}_{\cal J}}(c, \cdot )$ actually attains a minimum.

To do this, we will argue that ${{\cal D}_{\cal J}}(c, \cdot )$ has a property that is weaker than continuity but still strong enough to imply the existence of a minimizer on compact sets. Here is that property.

Lower Semicontinuity: Let $X$ be a topological space, and let $f:X \to [ - \infty ,\infty ]$ . We say that $f$ is lower semicontinuous (lsc) if and only if for every $r \in {\mathbb R}$ the set $\{ x \in X:f(x) \le r\} $ is closed.

Our aim is to argue that ${{\cal D}_{\cal J}}(c, \cdot )$ is lsc, and use the following lemma to conclude.

We will need the following basic facts about lsc functions.

Finally, it is worth isolating the following key fact as a separate lemma.

$$r \ge {\cal J}({p_n},{p_n}) \ge \sum\limits_{w \,\in\, A} {\cal J}({p_n},w){p_n}\{ w\} \to \sum\limits_{w \,\in \,A} {\cal J}(p,w)p\{ w\} = {\cal J}(p,p).$$

This proves that $p \in {F_r}$ . So $f$ is lsc.

Next, begin by observing that quasi-strict propriety implies that $f(p) = \mathop {\inf }\nolimits_{c \,\in \,{\cal C}} {\cal J}(c,p)$ for all $p \in {\cal P}$ . We claim that the infimum can be taken over only those $c \in {\cal C}$ for which ${\cal J}(c, \cdot )$ is real-valued. Let the set of such $c$ be denoted by ${{\cal C}_{\mathbb R}}$ , and recall that if $p$ is a regular probability, then $p \in {{\cal C}_{\mathbb R}}$ . Now, any $p \in {\cal P}$ is the limit of a sequence $({p_n})$ of regular probabilities. Continuity on ${\cal P}$ implies that ${\cal J}({p_n},p) \to {\cal J}(p,p)$ . Thus, for all $\varepsilon \gt 0$ , there is some ${p_n} \in {{\cal C}_{\mathbb R}}$ such that

$$\mathop {\inf }\limits_{\ c \in \ {{\cal C}_{\mathbb R}}} {\cal J}(c,p) \le {\cal J}({p_n},p) \le {\cal J}(p,p) + \varepsilon = \mathop {\inf }\limits_{c \ \in \ {\cal C}} {\cal J}(c,p) + \varepsilon.$$

This proves the claim, that is, $f(p) = \mathop {\inf }\nolimits_{c \in {{\cal C}_{\mathbb R}}} {\cal J}(c,p)$ for all $p \in {\cal P}$ .

Now, if $c \in {{\cal C}_{\mathbb R}}$ , then the function ${\cal J}(c, \cdot )$ is continuous on ${\cal P}$ . Indeed, if ${p_n} \to p$ , then

$${\cal J}(c,{p_n}) = \sum\limits_{w \,\in\, W} {\cal J}(c,w){p_n}\{ w\} \to \sum\limits_{w\, \in \,W} {\cal J}(c,w)p\{ w\} = {\cal J}(p,p).$$

Then by the duality

$$ - f(p) = \mathop {\sup }\limits_{c \in {{\cal C}_{\mathbb R}}} - {\cal J}(c,p),$$

we see that $ - f$ is the pointwise supremum of a family of continuous (and therefore lsc) functions. It follows from Lemma 3(ii) that $ - f$ is lsc, and the proof is complete.

We can finally show that divergences are lsc in their second argument.

$$\eqalign{r & \ge {{\cal D}_{\cal J}}(c,{p_n}) \cr & = {\cal J}(c,{p_n}) - {\cal J}({p_n},{p_n})\cr & = \sum\limits_{w \,\in \,A} {\cal J}(c,w){p_n}\{ w\} - {\cal J}({p_n},{p_n})\cr & \to \sum\limits_{w \,\in \,A} {\cal J}(c,w)p\{ w\} - {\cal J}(p,p)\cr & = {\cal J}(c,p) - {\cal J}(p,p) \cr & = {{\cal D}_{\cal J}}(c,p)}.$$

Thus, $p \in {F_r}$ , and the lemma is proved.

Let us conclude by summarizing how all of the pieces fit together.