2. Predd et al.’s conditions

In this section, I describe Predd et al.’s set of conditions—the ones we numbered (IV) in our list above. This will furnish us with statements of strict propriety and continuity, the assumptions we’ll use in our new arguments for Probabilism, the Principal Principle, and Conditionalization; and it will also introduce us to additivity, the assumption that we’re dropping from the existing best arguments for these conclusions. We will explain the problems with additivity in section 3 below.

First, let’s lay out the framework in which we’re working:

• We write ${\cal{W}}$ for the set of possible worlds. We assume ${\cal{W}}$ is finite. ^{Footnote 2} So ${\cal{W}} = \{w_1, \ldots, w_n\}$ .

• We write ${\cal{F}}$ for the full algebra of propositions built over ${\cal{W}}$ . That is, ${\cal{F}}$ is the set of all subsets of ${\cal{W}}$ .

• We write ${\cal{C}}$ for the set of credence functions defined on ${\cal{F}}$ . That is, ${\cal{C}}$ is the set of functions $c: {\cal{F}} \rightarrow [0, 1]$ .

• We write ${\cal{P}}$ for the set of probabilistic credence functions defined on ${\cal{F}}$ . That is, p is in ${\cal{P}}$ iff p is in ${\cal{C}}$ and (i) $p(\top) = 1$ , and (ii) $p(A \vee B) = p(A) + p(B)$ when A and B are mutually exclusive, that is, when there is no possible world at which A and B are both true.

• Given a credence function c, we write $c_i$ for the credence that c assigns to world $w_i$ .

• We write $w^i$ for the ideal credence function on ${\cal{F}}$ at world $w_i$ . That is, for X in ${\cal{F}}$ ,

  $${w^i}(X) = \left\{ {\begin{array}{*{20}{c}} {1\,{\rm{if}}\,X\,{\rm{is}}\,{\rm{true}}\,{\rm{at}}\,{w_i}} \\ {0\,{\rm{if}}\,X\,{\rm{is}}\,{\rm{false}}\,{\rm{at}}\,{w_i}} \\ \end{array}} \right.$$
  So, in particular, $w^i(w_j) = w^i_j = 1$ if $i = j$ and 0 if $i \neq j$ .

• An inaccuracy measure is a function ${\mathfrak{I}}: {\cal{C}} \times {\cal{W}} \rightarrow [0, \infty]$ . For c in ${\cal{C}}$ and $w_i$ in ${\cal{W}}$ , ${\mathfrak{I}}(c, i)$ is the inaccuracy of c at world $w_i$ .

Here are the three properties that Predd et al. demand of inaccuracy measures.

Continuity For each world $w_i$ , ${\mathfrak{I}}(c, i)$ is a continuous function of c.

Additivity For each X in ${\cal{F}}$ , there is a scoring rule ${\mathfrak{s}}_X: \{0, 1\} \times [0, 1] \rightarrow [0, \infty]$ such that, for all c in ${\cal{C}}$ and $w_i$ in ${\cal{W}}$ ,

${\mathfrak{I}}(c, i) = \sum\limits_{X \in {\cal{F}}} {\mathfrak{s}}_X(w^i(X), c(X))$

We say that the scoring rules ${\mathfrak{s}}_X$ for each X in ${\cal{F}}$ generate ${\mathfrak{I}}$ .

Additivity says that the inaccuracy of a credence function is the sum of the inaccuracies of the individual credences it assigns.

Strict Propriety For all p in ${\cal{P}}$ , $\sum^n_{i=1} p_i {\mathfrak{I}}(c, i)$ is minimized uniquely, as a function of c, at $c = p$ .

That is, for all p in ${\cal{P}}$ and $c \neq p$ in ${\cal{C}}$ ,

$\sum\limits^n_{i=1} p_i {\mathfrak{I}}(p, i) \lt \sum\limits^n_{i=1} p_i {\mathfrak{I}}(c, i)$

Strict propriety says that each probabilistic credence function should expect itself to be most accurate. ^{Footnote 3}

A few examples of inaccuracy measures:

• Brier score ${\mathfrak{B}}(c, i) = \sum_{X \in {\cal{F}}} {(w^i(X) - c(X))}^2$

• Log score ${\mathfrak{L}}(c, i) = -\log c_i$

• Enhanced log score ${\mathfrak{L}}^\star(c, i) = \sum_{X \in {\cal{F}}} \left(-w^i(X)\log c(X) + c(X) \right)$

• Absolute value score ${\mathfrak{A}}(c, i) = \sum_{X \in {\cal{F}}} |w^i(X) - c(X)|$

• Logsumexp score

  ${\mathfrak{LSE}}(c, i) = -\log (1+\sum\limits_{X \in {\cal{F}}} e^{c(X)}) + {{\sum_{X \in {\cal{F}}}(w^i(X) - c(X))e^{c(X)}}\over{1+ \sum_{X \in {\cal{F}}} e^{c(X)}}$

Then:

	Continuity	Additivity	Strict Propriety
${\mathfrak{B}}$	$\checkmark$	$\checkmark$	$\checkmark$
${\mathfrak{L}}$	$\checkmark$	$\times$	$\times$
${\mathfrak{L}}^\star$	$\checkmark$	$\checkmark$	$\checkmark$
${\mathfrak{A}}$	$\checkmark$	$\checkmark$	$\times$
${\mathfrak{LSE}}$	$\checkmark$	$\times$	$\checkmark$

Some notes:

• The Brier score is additive. It is generated by using the quadratic scoring rule ${\mathfrak{q}}$ for every proposition, where

  1. - ${\mathfrak{q}}(1,x) = {(1-x)}^2$ ;

  2. - ${\mathfrak{q}}(0, x) = x^2$ .

     Since ${\mathfrak{q}}$ is continuous and strictly proper, so is ${\mathfrak{B}}$ .

• The log score is not additive, and it is not strictly proper. A credence function that assigns credence 1 to each world dominates any credence function that assigns less than credence 1 to each world. The log score is, however, strictly ${\cal{P}}$ -proper: That is, for all p in ${\cal{P}}$ , $\sum_i p_i{\mathfrak{L}}(q, i)$ is minimized uniquely, among credence functions q in ${\cal{P}}$ , at $q = p$ .

• The enhanced log score is additive. It is generated using the enhanced logarithmic scoring rule ${\mathfrak{l}}^\star$ for every proposition, where

  1. - ${\mathfrak{l}}^\star(1, x) = -\log x + x$ ;

  2. - ${\mathfrak{l}}^\star(0, x) = x$ .

     Since ${\mathfrak{l}}$ is continuous and strictly proper, so is ${\mathfrak{L}}^\star$ .

• The absolute value score is additive. It is generated by using the absolute scoring rule ${\mathfrak{a}}$ for every proposition, where

  1. - ${\mathfrak{a}}(1, x) = 1-x$ ;

  2. - ${\mathfrak{a}}(0, x) = x$ .

     But ${\mathfrak{a}}$ is not strictly proper. If $$p \lt {1 \over 2}$$ , then $p{\mathfrak{a}}(1, x) + (1-p){\mathfrak{a}}(0, x) = p(1-x) + (1-p)x$ is minimized at $x = 0$ ; if $$p \gt {1 \over 2}$$ , it is minimized at $x = 1$ ; if $$p = {1 \over 2}$$ , it is minimized at any $0 \leq x \leq 1$ . So ${\cal{A}}$ is not strictly proper either.

• The logsumexp score is strictly proper and continuous, but it is not additive. This shows that assuming strict propriety and continuity, as we do below, is strictly weaker than assuming strict propriety, additivity, and continuity. Lewis and Fallis (Reference Lewis and Fallis2019) give another example of an inaccuracy measure that satisfies strict propriety and continuity, but not additivity: They call it the asymmetric spherical rule.

My concern in this paper is to strengthen the accuracy-first arguments for Probabilism, the Principal Principle, and Conditionalization based on continuity + additivity + strict propriety by showing that they go through if we assume only continuity + strict propriety. Thus, for instance, they go through for the logsumexp score defined above and Lewis and Fallis’ asymmetic spherical rule as well as the Brier and enhanced log scores. Of course, weakening the premises of an argument always strengthens it. So there seems good reason to note this fact regardless of your view of additivity. Nonetheless, in the next section, I explain why you might be suspicious of additivity. Then, in section 4, I give the arguments for Probabilism, the Principal Principle, and Conditionalization without appealing to it. In section 5, I conclude. The appendix contains the proofs of all the theorems on which these arguments are based.

3. Why additivity?

I should begin by pointing out that, while I appealed to Predd et al.’s mathematical results in my presentation of the accuracy dominance argument for probabilism in Accuracy and the Laws of Credence, those authors weren’t themselves working in the accuracy-first framework (Pettigrew Reference Pettigrew2016a). What we are calling inaccuracy measures are for them loss functions; and in the context of loss functions, the additivity assumption is perfectly natural—providing the loss is stated in units of some commodity in which your utility is linear, the total loss to which your credence function is subject is the sum of the individual losses to which your credences are subject. But what about the accuracy-first framework? Is additivity still so natural there? ^{Footnote 4} Some claim it is not: Kotzen (Reference Kotzen2018, 778) claims that, by assuming additivity, we rule out certain plausible ways of measuring inaccuracy “more or less by fiat”; and, as I mentioned above, Lewis and Fallis (Reference Lewis and Fallis2019) describe a continuous and strictly proper inaccuracy measure that seems reasonable, but note that it is not additive.

On the other hand, I wrote:

My reason? Your credence function is not a single, unified doxastic state, but rather merely the motley agglomeration of all of your individual credal doxastic states. We might mistakenly think of a credence function as unified because we represent it by a single mathematical object, but mathematical functions are anyway just collections of assignments of values to arguments. Or so I argued.

However, while this may be so, it does not entail that there are not features of a credal state that partly determine its accuracy but which cannot be captured by looking only at the credences individually—we might call these global features of the credence function. For instance, we might think that, at a world at which two propositions A and B have the same truth value, it is more accurate to have equal credence in A and in B than to have different credences in them. After all, the ideal credence function will have the same credence in them, namely, 1 in both or 0 in both. And perhaps resembling the ideal credence function in this respect gets you closer to it and therefore more accurate. But having the same credence in A and in B is a global feature of a credence function. To determine whether or not a credence function has it, you can’t just look at the credences it assigns individually.

However, interestingly, while this is indeed a global feature of a credence function, some additive inaccuracy measures will in fact reward it. Take the Brier score, for instance. If A and B are both true, then the inaccuracy of assigning credences a and b to them, respectively, is ${(1-a)}^2 + {(1-b)}^2$ . But it’s easy to see that, if $a \neq b$ , then assigning their average, $${1 \over 2}a + {1 \over 2}b$$ , to both is more accurate. Since ${(1-x)}^2$ is a strictly convex function of x,

$${(1 - a)^2} + {(1 - b)^2} \gt {\left( {1 - \left( {{1 \over 2}a + {1 \over 2}b} \right)} \right)^2} + {\left( {1 - \left( {{1 \over 2}a + {1 \over 2}b} \right)} \right)^2}$$

Similarly, if A and B are both false,

$${a^2} + {b^2} \gt {\left( {{1 \over 2}a + {1 \over 2}b} \right)^2} + {\left( {{1 \over 2}a + {1 \over 2}b} \right)^2}$$

So, if we measure inaccuracy using the Brier score, which is additive, having the global property in question—that is, assigning equal credences to A and B when A and B are either both true or both false—improves accuracy.

In Accuracy and the Laws of Credence, I argued that we should simply assume that, for any global feature of a credence function that we consider relevant to accuracy, it must be possible to capture it using additive measures along the lines just sketched.

But why? Why think that this is true? I imagined that this follows from the fact that credence functions are not unified entities. I assumed that this warrants a defeasible assumption in favour of additivity. That assumption could be defeated if we were to find a global feature we wished to reward but which could not be rewarded by additive measures. But no such feature has presented itself.

Perhaps. But even if we grant the move from the disunified nature of credence functions to this defeasible assumption in favour of additivity, such a defeasible assumption is a flimsy basis for an argument, particularly since we have not systematically investigated the sorts of global features we might consider relevant to accuracy and so have rather sparse evidence that there is no defeater to be found. As we showed above, we can explain why one particular global feature is conducive to accuracy, namely, having the same credence in two propositions that have the same truth value. And indeed you can view the accuracy dominance argument for probabilism as furnishing us with another example. After all, probabilism makes two demands of a credence function. The first is local: The credence function should assign maximal credence to a tautology and minimal credence to a contradiction. The second is global: The credence it assigns to a disjunction of mutually exclusive propositions should be the sum of the credences it assigns to the disjuncts. To tell whether a credence function satisfies this latter condition, you must look at the relationships between the credences it assigns. However, the fact that you can run the accuracy dominance argument for probabilism using additive inaccuracy measures like the Brier score shows that you can show that this global feature of a credence function is conducive to accuracy without building into your inaccuracy measure that it should be rewarded explicitly. Indeed, that is one of the remarkable features of de Finetti’s original proof, which is the ultimate source of Predd et al.’s theorem. But that’s about it. Those are the only two global features of credence functions we’ve succeeded in capturing using additive inaccuracy measures.

So, in the end, to the extent that there still exists doubt that we have considered all global features of credence functions that are relevant to their accuracy and showed how their relevance can be captured by additive inaccuracy measures, there still exists doubt over additivity. And while there still exists doubt over additivity, removing that assumption from the arguments for Probabilism, the Principal Principle, and Conditionalization strengthens them.

4. Arguments without additivity

Let me begin this section by spelling out the arguments we wish to give. Then I’ll move on to explaining and proving the theorems to which they appeal. As I noted above, each argument has the same form:

(NC) Necessary conditions on being a legitimate inaccuracy measure.

(BP) Accuracy–rationality bridge principle.

(MT) Mathematical theorem.

Therefore,

(PR) Principle of rationality.

The first component will be the same in each argument that we give. We will assume only that inaccuracy measures satisfy continuity and strict propriety. But the accuracy–rationality principles will differ from case to case. In the remainder of this section, I’ll state each principle for which we’re arguing and then the bridge principle to which we’ll appeal in that argument.

To state some of the principles of credal rationality, we need some moderately technical notions. I’ll lay them out here. Suppose ${\cal{X}}$ is a set of credence functions. Then:

• ${\cal{X}}$ is convex if it is closed under taking mixtures.

  That is, for any c, c’ in ${\cal{X}}$ and any $0 \leq \lambda \leq 1$ , $\lambda c + (1-\lambda) c'$ is also in ${\cal{X}}$ .

• ${\cal{X}}$ is closed if it is closed under taking limits.

  That is, for any infinite sequence $c_1, c_2, \ldots$ of credence functions in ${\cal{X}}$ that tends to c in the limit, c is in ${\cal{X}}$ . ^{Footnote 5}

• The convex hull of ${\cal{X}}$ is the smallest convex set that contains ${\cal{X}}$ . We write it ${\cal{X}}^+$ .

  That is, if ${\cal{Z}}$ is convex and ${\cal{X}} \subseteq {\cal{Z}}$ , then ${\cal{X}}^+ \subseteq {\cal{Z}}$ .

• The closed convex hull of ${\cal{X}}$ is the smallest closed set that contains the convex hull of ${\cal{X}}$ . We write it ${\textrm{cl}}({\cal{X}}^+)$ .

  Thus, if ${\cal{Z}}$ is closed and ${\cal{X}}^+ \subseteq {\cal{Z}}$ , then ${\textrm{cl}}({\cal{X}}^+) \subseteq {\cal{Z}}$ .

As well as clarifying the technical notions just presented, I also want to flag that these arguments appeal to a notion of possibility at various points: In setting up the framework, we have already talked of possible worlds and algebras built on those worlds; in accuracy dominance arguments for Probabilism, we will quantify over possible worlds; in the accuracy dominance argument for Conditionalization we quantify over possible worlds, but also over possible future credence functions that you currently endorse conditional on you receiving certain evidence; in expected inaccuracy arguments like Greaves and Wallace’s argument for Conditionalization, we will sum over inaccuracies at possible worlds weighted by credences in those worlds to give expectations; and in chance dominance arguments for the Principal Principle, we’ll quantify over possible objective chance functions. What notion of possibility is at play here? Joyce (Reference Joyce1998) takes it to be logical possibility; in a recent paper (Reference Pettigrew2020), I take it to be something like epistemic possibility. It is beyond the scope of this essay to argue for one or the other in detail, so I will leave the notion as a placeholder. But my own preference is for epistemic possibility. What makes it irrational to violate Probabilism, for instance, is that there is another credence function that is more accurate than yours at all epistemically possible worlds—that is, you can tell from the inside, so to speak, that this alternative is better than yours because the only worlds you need to consider are those you consider possible.

4.2 Argument for the Principal Principle

The principal principle is usually stated as follows, where, for any probability function ch, $C_{ch}$ is the proposition that says that ch is the objective chance function: ^{Footnote 6}

Principal principle. If $C_{ch}$ is in ${\cal{F}}$ and $c(C_{ch}) \gt 0$ , then, for all A in ${\cal{F}}$ , we should have

$c(A | C_{ch}) = ch(A)$

The version we’ll consider here is more general than this version. And indeed, the more general version entails the usual version.

Generalized Principal Principle. Suppose ${\cal{A}}$ is the set of possible objective chance functions. Then your credence function should be in the closed convex hull of ${\cal{A}}$ .

Thus, for instance, if all the possible objective chance functions consider A and B equally likely, your credence function should consider them equally likely; and if all the possible objective chance functions consider A more likely than B, then you should consider A at least as likely as B; and if all the possible objective chance functions consider A to be between 30% and 70% likely, then you should not assign credence 0.8 to A; and so on.

Suppose ch is a possible objective chance function and $C_{ch}$ is in ${\cal{F}}$ . And suppose further that ch is not a self-undermining chance function. That is, it is certain that it gives the chances. That is, $ch(C_{ch}) = 1$ . Then, if you satisfy the Generalized Principal Principle, then $c(A | C_{ch}) = ch(A)$ . ^{Footnote 7}

So that’s the version of the Principal Principle that we’ll consider. And here’s the bridge principle: A credence function is irrational if there is another that is guaranteed to have greater expected accuracy from the point of view of the objective chance function. That is:

Chance Dominance. If there is $c^\star$ such that, for all possible chance functions ch, $\sum_i ch_i {\mathfrak{I}}(c^\star, i) \lt \sum_i ch_i {\mathfrak{I}}(c, i)$ , then c is irrational.

Thus, to move from the claim that an inaccuracy measure must satisfy continuity and strict propriety, together with Chance Dominance, to the Generalized Principal Principle, we need the following theorem:

Theorem 2 Suppose ${\mathfrak{I}}$ is continuous and strictly proper. If c is not in ${\textrm{cl}}({\cal{A}}^+)$ , there is $c^\star$ in ${\textrm{cl}}({\cal{A}}^+)$ , such that $\sum_i ch_i{\mathfrak{I}}(c^\star, i) \lt \sum_i ch_i{\mathfrak{I}}(c, i)$ , for all ch in ${\cal{A}}$ .

Again, we prove this in the appendix.

What’s more, Theorem 2 allows us to improve not only my argument for the Principal Principle, but also my argument for linear pooling as the correct method of probabilistic judgment aggregation (Pettigrew Reference Pettigrew2019a). In that argument, I appealed to the following bridge principle: If we have a group of individuals with probabilistic credence functions $c_1, \ldots, c_m$ , and we wish to find a credence function that aggregates them, then we should not pick an aggregate c if there is an alternative $c^\star$ such that each member of the group expects $c^\star$ to be more accurate than they expect c to be. I assumed that inaccuracy measures satisfy additivity, continuity, and strict propriety. And I showed that, if you use an aggregation method other than linear pooling, then there will be an alternative aggregate that everyone expects to be better; but if you use linear pooling, there won’t be. Armed with Theorem 2, we obtain the same result assuming only continuity and strict propriety and thereby strengthen my original argument.

4.3 Argument for Conditionalization and the Weak Reflection Principle

The version of conditionalization for which Greaves and Wallace as well as Briggs and I argue is this:

Plan Conditionalization Suppose your prior is $c_0$ and you know that your evidence will come from a partition $E_1, \ldots, E_m$ . And suppose you will learn a particular cell of this partition iff it is true. And suppose you plan to adopt credence function $c_k$ in response to evidence $E_k$ . Then, for all $E_k$ with $c_0(E_k) \gt 0$ and for all X in ${\cal{F}}$ , we should have

$${c_k}(X) = {c_0}(X|{E_k}) = {{{c_0}(X{E_k})} \over {{c_0}({E_k})}}$$

That is, you should plan to update by applying Bayes’ rule to your prior.

As we mentioned above, we won’t argue for Plan Conditionalization directly. Rather, we’ll argue for a more general principle of rationality called the Weak Reflection Principle; and we’ll show that it entails Plan Conditionalization (Pettigrew Reference Pettigrew2020).

Weak Reflection Principle. Suppose ${\cal{R}} = \{c_1, \ldots, c_m\}$ is the set of possible future credence functions you endorse. Then your current credence function should be in the convex hull of ${\cal{R}}$ .

Here’s the idea behind this principle. A lot might happen between now and tomorrow. You might see new sights, think new thoughts; you might forget things you know today, take mind-altering drugs that enhance or impair your thinking; and so on. So perhaps there is a set of credence functions, any one of which you think you might have tomorrow. Some of those you’ll endorse—perhaps those you’d adopt if you saw certain new things, or enhanced your cognition in various ways. And some of them you’ll disavow—perhaps those that you’d adopt if you were to forget certain things, or were to impair your cognition in some way. The Weak Reflection Principle asks you to separate out the wheat from the chaff, and once you’ve identified the ones you endorse, it tells you that your current credence function should lie within the convex hull of those future ones.

Now, suppose that you are in the situation that Plan Conditionalization covers. That is, (i) you know that you will receive evidence from the partition $E_1, \ldots, E_m$ , (ii) you will learn $E_k$ iff $E_k$ is true, and (iii) you form a plan for how to respond to these different possible pieces of evidence—you will adopt $c_1$ if you learn $E_1$ , $c_2$ if you learn $E_2$ , and so on. Thus, the possible future credence functions that you endorse are $c_1, \ldots, c_m$ , for they are the possible outcomes of a plan that you have made and which you know covers all the bases. Then, by the Weak Reflection Principle, $c_0$ should be in the convex hull of $c_1, \ldots, c_m$ . Then, if $c_k(E_k) = 1$ and $c_0(E_k) \gt 0$ , then $c_k(X) = c_0(X |E_k)$ , as Plan Conditionalization requires. After all, by the Weak Reflection Principle, there are $0 \leq \lambda_1, \ldots, \lambda_m \leq 1$ with $\sum^m_{i=1} \lambda_i = 1$ such that $c_0(X) = \sum^m_{i=1} \lambda_i c_i(X)$ for all X in ${\cal{F}}$ . What’s more, by assumption, $c_i(E_k) = 1$ if $i = k$ and $c_i(E_k) =0$ if $i \neq k$ . So, $c_i(XE_k) = c_i(X)$ if $i = k$ and $c_i(XE_k) =0$ if $i \neq k$ . So,

$${c_0}(X|{E_k}) = {{\sum_{i = 1}^m {{\lambda _i}} {c_i}(X{E_k})} \over {\sum_{i = 1}^m {{\lambda _i}} {c_i}({E_k})}} = {{{\lambda _k}{c_k}(X{E_k})} \over {{\lambda _k}{c_k}({E_k})}} = {c_k}(X)$$

as required.

However, the Weak Reflection Principle applies in many other situations beyond those imagined by Plan Conditionalization. For instance, suppose you know you’ll receive evidence from $E_1, \ldots, E_m$ , but, while these propositions are mutually exclusive, they do not exhaust the logical space. Then again, if you plan to adopt $c_k$ upon learning $E_k$ , and $c_k(E_k) = 1$ and $c_0(E_k) \gt 0$ , then the Weak Reflection Principle says that you should plan so that $c_k(X) = c_0(X|E_k)$ . That follows from the working at the end of the previous paragraph. Or suppose your evidence is not perfectly reliable. So it’s not certain that you’ll learn $E_k$ iff $E_k$ is true. Then again, if you plan to adopt $c_k$ upon learning $E_k$ , and $c_k(E_k) = 1$ and $c_0(E_k) \gt 0$ , then the Weak Reflection Principle says that you should plan so that $c_k(X) = c_0(X|E_k)$ . Again, this follows from the working above. What’s more, suppose you know you’ll receive evidence from $E_1, \ldots, E_m$ , but you don’t make a deterministic plan. Perhaps you plan as follows: If I learn $E_1$ , then I’ll adopt $c_1$ or I’ll adopt $c'_1$ ; if I learn $E_2$ , then I’ll adopt $c_2$ or I’ll adopt $c'_2$ ; and so on. Then Plan Conditionalization no longer applies. But the Weak Reflection Principle does. It constrains your plans. It says that your current credence function should be in the convex hull of $c_1, c'_1, c_2, c'_2, \ldots, c_m, c'_m$ (Pettigrew Reference Pettigrew2019b).

So that’s the Weak Reflection Principle. Now for the bridge principle we use to establish it: It is irrational for you to have a prior credence function and a set of possible future credence functions you endorse if there is some alternative prior and, for each possible future credence function you endorse, an alternative to that, such that the sum of the inaccuracy of your prior and the inaccuracy of one of the possible future credence functions you endorse is always greater than the inaccuracy of the alternative prior and the corresponding alternative possible future credence function. That is:

Diachronic Worldly Dominance. Suppose $c_0$ is your current credence function and $c_1, \ldots, c_m$ are the possible future credence functions you endorse. Then if there are $c^\star_0, c^\star_1, \ldots, c^\star_m$ such that, for each $1 \leq k \leq m$ and all $w_i$ in ${\cal{W}}$ ,

${\mathfrak{I}}(c^\star_0, i) + {\mathfrak{I}}(c^\star_k, i) \lt {\mathfrak{I}}(c_0, i) + {\mathfrak{I}}(c_k, i)$

then you are irrational.

Thus, to move from the claim that an inaccuracy measure must satisfy continuity and strict propriety, together with diachronic worldly dominance, to the Weak Reflection Principle, we need the following theorem:

Theorem 3 Suppose ${\mathfrak{I}}$ is continuous and strictly proper. If $c_0$ is not in ${\{c_1, \ldots, c_n\}}^+$ , then there are $c^\star_0, c^\star_1, \ldots, c^\star_n$ such that, for all $w_i$ in ${\cal{W}}$ ,

${\mathfrak{I}}(c^\star_0, i) + {\mathfrak{I}}(c^\star_k, i) \lt {\mathfrak{I}}(c_0, i) + {\mathfrak{I}}(c_k, i)$

We prove this in the appendix. Note, however, as we mentioned above: While our new proof removes one way in which an additivity enters into the argument that Briggs and I gave, it doesn’t remove another. We still take the inaccuracy of your prior and your posterior, taken together, to be the sum of the inaccuracy of your prior and the inaccuracy of your posterior. It must await future work to explore whether that assumption might also be removed.