Why follow the royal rule?

Abstract

This note is a sequel to Huber (Synthese 191:2167–2193, 2014). It is shown that obeying a normative principle relating counterfactual conditionals and conditional beliefs, viz. the royal rule, is a necessary and sufficient means to attaining a cognitive end that relates true beliefs in purely factual, non-modal propositions and true beliefs in purely modal propositions. Along the way I will sketch my idealism about alethic or metaphysical modality.

Appendix

Someone interested in studying safety, and not bothered by Huber’s (2014) thought that it is important to distinguish between the factual components $f_w$ and the modal components $r_w$ of a possible world w, will find the following models more elegant: $\left( W,\left( R_w\right) _{w\in W},\left( S_w\right) _{w\in W}\right)$, where W is a non-empty set of possible worlds, $\left( R_w\right) _{w\in W}$ is a family of arbitrary functions from $\wp \left( W\right)$ into ${\mathbb {N}}\cup \left\{ \infty \right\}$, and $\left( S_w\right) _{w\in W}$ is a family of regular subjective ranking functions from $\wp \left( W\right)$ into ${\mathbb {N}}\cup \left\{ \infty \right\}$.

In this set-up the royal rule says: for all $v\in W$ and all $A\subseteq W$, and all “v, A-admissible” propositions $E\subseteq W$,

$$\begin{aligned} S_v\left( A\mid \left\{ w\in W:R_w\left( A\right) =n\right\} \cap E\right) =n. \end{aligned}$$

With the right assumptions about admissibility it follows from these assumptions that $R_w$ is a ranking function for each $w\in W$. In particular, the condition from sections 3 and 4 that $r_w$ be a ranking function is not an assumption of the present paper, but rather a consequence of the royal rule that I have taken for granted to simplify the presentation of this paper (for a derivation see Huber 2014).

In this set-up, instead of considering cases $\omega$ one considers possible worlds $w=\left( \llbracket \rrbracket _w,R_w,S_w\right)$. Sensitivity says: for all $v\in W$ and all $A\subseteq W$, all A-worlds w that are $R_v$-minimal are such that $S_w\left( A\right) =0$. Adherence says: for all $v\in W$ and all $A\subseteq W$, all A-worlds w that are $R_v$-minimal are such that $S_w\left( W{\setminus } A\right) >0$. Safety says: for all $v\in W$ and all $A\subseteq W$, A is true in all $R_v$-minimal worlds w in which $S_w\left( W{\setminus } A\right) >0$.

One conditional version of sensitivity says: for all $v\in W$ and all $A\subseteq W$, all A-worlds w that are $R_v$-minimal are such that

$$\begin{aligned} S_w\left( A\mid \left\{ x\in W:R_x\left( A\right) =R_w\left( A\right) \right\} \cap E\right) =0 \end{aligned}$$

for all “w, A-admissible” $E\subseteq W$. One conditional version of adherence says: for all $v\in W$ and all $A\subseteq W$, all A-worlds w that are $R_v$-minimal are such that

$$\begin{aligned} S_w\left( W{\setminus } A\mid \left\{ x\in W:R_x\left( W{\setminus } A\right) =R_w\left( W{\setminus } A\right) \right\} \cap E\right) >0 \end{aligned}$$

for all “$w,\left( W{\setminus } A\right)$-admissible” $E\subseteq W$. One conditional version of safety says: for all $v\in W$ and all $A\subseteq W$, A is true in all $R_v$-minimal worlds w in which

$$\begin{aligned} S_w\left( W{\setminus } A\mid \left\{ x\in W:R_x\left( W{\setminus } A\right) =R_w\left( W{\setminus } A\right) \right\} \cap E\right) >0 \end{aligned}$$

for some “$w,\left( W{\setminus } A\right)$-admissible” $E\subseteq W$.

These conditional versions consider the truth about the modal status of A in the word w the agent is in. Quite different conditional versions consider the truth about the modal status of A in the world v where the counterfactual conditional is evaluated. The conditions studied in this paper considered if the agent believes A in w conditional on what is true about the modal status of A in the world w she is in. The different conditions consider if she believes A in w conditional on what is true about the modal status of A elsewhere, in a different possible world v.

One way to think of the former conditions is in terms of what is true from the first person perspective of the agent, and to think of the latter conditions in terms of what is true from the third person perspective of some tracking-ascriber. The latter conditional versions of sensitivity and adherence and safety say: for all $v\in W$ and all $A\subseteq W$, all A-worlds w that are $R_v$-minimal are such that $S_w\left( A\mid \left\{ x\in W:R_x\left( A\right) =R_v\left( A\right) \right\} \cap E\right) =0$ for all “v, A-admissible” $E\subseteq W$. And: for all $v\in W$ and all $A\subseteq W$, all A-worlds w that are $R_v$-minimal are such that $S_w\left( W{\setminus } A\mid \left\{ x\in W:R_x\left( W{\setminus } A\right) =R_v\left( W{\setminus } A\right) \right\} \cap E\right) >0$ for all “$v,\left( W{\setminus } A\right)$-admissible” $E\subseteq W$. As well as: for all $v\in W$ and all $A\subseteq W$, A is true in all $R_v$-minimal worlds w in which $S_w\left( W{\setminus } A\mid \left\{ x\in W:R_x\left( W{\setminus } A\right) =R_v\left( W{\setminus } A\right) \right\} \cap E\right) >0$ for some “$v,\left( W{\setminus } A\right)$-admissible” $E\subseteq W$.

Bayesians may want to interpret $R_w$ as the objective chance distribution of world w, $ch_w$, and $S_w$ as the ideal doxastic agent’s subjective credence function in w, $cr_w$ (as well as change the domain and range to some suitable algebra as well as the unit interval, respectively). On this probabilistic interpretation the royal rule becomes Lewis’ (1980) principal principle, and the probabilistically re-interpreted set-up allows one to study which ends the principal principle furthers (Pettigrew 2013 is an excellent study of this kind). In addition one can study which norms further the following cognitive ends that also consider what is true about the chance of A from the third person perspective mentioned above:

• The objective chance is high that one’s subjective credence in A conditional on the truth about the objective chance of A is high given that A is true. For all $A\subseteq W$ and all $v\in W$ and all “v, A-admissible” $E\subseteq W$, $ch_v\left( cr\left( A\mid \left\{ x\in W:ch_x\left( A\right) =ch_v\left( A\right) \right\} \cap E\right) =high\mid A\right) =high$.

• The objective chance equals c that A is true given that one’s subjective credence in A conditional on the truth about the objective chance of A equals c. For all $A\subseteq W$ and all $v\in W$ and all “ v, A-admissible” $E\subseteq W$, $ch_v\left( A\mid cr\left( A\mid \left\{ x\in W:ch_x\left( A\right) =ch_v\left( A\right) \right\} \cap E\right) =c\right) =c$.

Here $cr\left( A\mid C\right) =high$ is the proposition $\left\{ x\in W:cr_x\left( A\mid C\right) =high\right\}$.

Of course, in this more elegant and general set-up, it is not possible anymore to say that admissible information is purely modal information, as we have done in this paper. The reason is that, in this more elegant and general set-up, the very distinction between factual information and modal information cannot be drawn without further assumptions. Admissibility now is relative to a proposition and a world (and, of course, a credence function), and this is only the tip of the iceberg that is known as the “big bad bug” (Rachael 2009) and that is the reason for Huber’s (2014) complicated set-up in the first place.¹¹