Centering the Principal Principle

Abstract

I show that centered propositions—also called de se propositions, and usually modeled as sets of centered worlds—pose a serious problem for various versions of Lewis’s Principal Principle. The problem, put roughly, is that in scenarios like Elga’s ‘Sleeping Beauty’ case, those principles imply that rational agents ought to have obviously irrational credences. To solve the problem, I propose a centered version of the Principal Principle. My version allows centered propositions to be objectively chancy.

Appendix

In this Appendix, I derive the equation

$$\begin{aligned} Cr_{1}(H) = \frac{1}{2} \end{aligned}$$

(3)

from the Principal Principle, the New Principle, and the General Recipe. Then I show how this equation, together with equations

$$\begin{aligned} Cr_{1}(H_{t})=0 \end{aligned}$$

(1)

and

$$\begin{aligned} Cr_{1}(H \mid Mo)=\frac{1}{2} \end{aligned}$$

(2)

implies that, when Susie wakes up on Monday but before she is told that it is Monday, she ought to be completely confident that it is Monday. Finally, I argue that equation

$$\begin{aligned} Cr_{2}(H) = \frac{1}{2} \end{aligned}$$

(5)

also follows from the Principal Principle; as discussed in Sect. 2.2, one of my arguments for (2) assumes this equation.

Here is the derivation of (3) from the Principal Principle. To start, recall that the Principal Principle concerns agents’ initial credence functions: their credence functions at some earlier time. So let Cr be Susie’s initial credence function; for simplicity, suppose that Cr is Susie’s credence function right after being told all the details of Susie’s experiment. Following Elga (2000) and Lewis (2001), we may stipulate that $Cr_{1}$ is equal to Susie’s initial credence function Cr conditionalized on the proposition $\mathcal {H}_{1}$, where $\mathcal {H}_{1}$ expresses the complete history of the world up to time $t=1$. That is, for all propositions A,

$$\begin{aligned} Cr_{1}(A) = Cr(A\mid \mathcal {H}_{1}) \end{aligned}$$

It follows, of course, that

$$\begin{aligned} Cr_{1}(H) = Cr(H\mid \mathcal {H}_{1}) \end{aligned}$$

(6)

Moreover, according to the Principal Principle,

$$\begin{aligned} Cr\left( H\;\Big \vert \; \mathcal {H}_{1}\; \& \;Ch_{1}(H)=\frac{1}{2}\right) = \frac{1}{2} \end{aligned}$$

(7)

This assumes, of course, that $\mathcal {H}_{1}$ is admissible. But that is quite reasonable. Propositions like $\mathcal {H}_{1}$, about the history of the world, are paradigmatic examples of admissible propositions: Lewis, for example, says as much (1980, p. 276). So this is a reasonable assumption to adopt.

Equations (6) and (7) jointly imply equation (3). To see why, note that $Cr\left( Ch_{1}(H)=\frac{1}{2}\right) =1$, since Susie is told that the coin is fair. Because of this, (7) reduces to $Cr(H\mid \mathcal {H}_{1})=\frac{1}{2}$. And this, in conjunction with (6), implies (3).

One might object to this derivation by objecting to (6). Perhaps the credence function which Susie ought to have, at time $t=1$, is not obtained by standard Bayesian conditionalization on a history proposition like $\mathcal {H}_{1}$. Perhaps the credence function which Susie ought to have, at time $t=1$, is obtained by some other update procedure. Or perhaps the credence function which Susie ought to have, at time $t=1$, is obtained by conditionalizing her initial credence function on some proposition other than $\mathcal {H}_{1}$.

Regardless, (6) still holds. For starters, other update procedures can be used to derive that equation. Given certain reasonable assumptions, for instance, (6) follows from Meacham’s compartmentalized conditionalization rule.

In addition, $\mathcal {H}_{1}$ is the right proposition for Susie to use when updating her credences. For $\mathcal {H}_{1}$ need not be an uncentered proposition: it need not describe the uncentered history of the world. $\mathcal {H}_{1}$ may be a centered proposition: it can describe various centered and uncentered facts that obtain throughout the world’s history. So $\mathcal {H}_{1}$ can be a history proposition which implies that at time $t=1$, it is either Monday or Tuesday. $\mathcal {H}_{1}$ could be Susie’s total evidence at $t=1$, for instance.

I now derive equation (3) from a different version of the Principal Principle, one proposed by Hall (1994, p. 511) and Lewis (1994, p. 487). Let A be a proposition. Let t be a time. Let Ch be a chance function defined, at time t, over an algebra of which A is a member. Let $\mathcal {H}_{t}$ be the proposition that completely characterizes the history of the world up to time t. Let Th be the proposition expressing the complete theory of chance at the actual world.²⁰ Then a rational agent’s initial credence function Cr ought to satisfy the following equation.

$$\begin{aligned} Cr\big (A\;\big \vert \; \mathcal {H}_{t}\; \& \; Th\big ) = Ch(A\mid Th) \end{aligned}$$

Call this the ‘New Principle’.

To streamline the derivation of (3) from the New Principle, suppose that the complete chance theory Th only specifies the chances of the coin landing heads; no other chancy events happen in the world. So Th is the proposition that the chance of the coin landing heads is $\frac{1}{2}$ and the chance of the coin landing tails is $\frac{1}{2}$. Therefore, $Ch_{1}(H\mid Th)=\frac{1}{2}$. Since Susie was told all the details of the experiment, Susie knows the chance theory Th. In conjunction with (6), it follows that $\begin{aligned} Cr(H\mid \mathcal {H}_{1}\; \& \; Th)=Cr_{1}(H) \end{aligned}$. By the New Principle, $\begin{aligned} Cr(H\mid \mathcal {H}_{1}\; \& \; Th)=Ch_{1}(H\mid Th) \end{aligned}$. Therefore, $Cr_{1}(H)=\frac{1}{2}$; that is, (3) holds.

Now consider a version of the Principal Principle due to Ismael (2008, p. 298). Let A and t be as before. For each complete theory of chance Th, let $Ch_{Th}(A)$ be the chance of A at t according to Th, and let $a_{Th}$ be an agent’s subjective assessment of the probability of Th at t. Then in order for this agent to be rational, her credence function Cr at t ought to satisfy the following equation.

$$\begin{aligned} Cr(A) = \sum _{Th}a_{Th}Ch_{Th}(A) \end{aligned}$$

Call this the ‘General Recipe’.

To streamline the derivation of (3) from the General Recipe, suppose once again that the actual world’s complete chance theory Th only specifies the chances of the coin landing heads. Let $Ch_{1,Th}$ be the chances, according to Th, at time $t=1$. Then $Ch_{1,Th}(H)=\frac{1}{2}$. Because Susie was told all the details of the experiment, it follows that at $t=1$, her subjective assessment of the probability of Th is 1. So her subjective assessments of the probabilities of all other chance theories are equal to 0. Substituting these values into the General Recipe yields $Cr_{1}(H)=\frac{1}{2}$; that is, (3) holds.

Now let us see why equations (1), (2), and (3) imply that $Cr_{1}(Mo) = 1$. To start, note that since the conditional credence in (2) is well-defined, it follows that $Cr_{1}(Mo)>0$. Therefore,

$$\begin{aligned} \frac{1}{2}&= Cr_{1}(H\mid Mo)\\&= \frac{Cr_{1}(H\; \& \; Mo)}{Cr_{1}(Mo)}\\&= \frac{Cr_{1}\big ((H_{m}\vee H_{t})\; \& \; (H_{m}\vee T_{m})\big )}{Cr_{1}(Mo)}\\&= \frac{Cr_{1}(H_{m})}{Cr_{1}(Mo)}\\&= \frac{Cr_{1}(H_{m})+Cr_{1}(H_{t})}{Cr_{1}(Mo)}\\&= \frac{Cr_{1}(H_{m}\vee H_{t})}{Cr_{1}(Mo)}\\&= \frac{Cr_{1}(H)}{Cr_{1}(Mo)}\\&= \frac{\frac{1}{2}}{Cr_{1}(Mo)} \end{aligned}$$

where (2) yields the first line, (1) yields the fifth line, and (3) yields the eighth line. Multiplying through by $2Cr_{1}(Mo)$ yields $Cr_{1}(Mo)=1$. In other words, at time $t=1$—that is, before Susie is told that it is Monday—Susie must be completely confident that it is Monday.

Now for the derivation of (5) from the Principal Principle.²¹ To start, let $Ch_{2}$ be the objective chance function at time $t=2$. Because the coin is fair, the chance of heads at $t=2$ is $\frac{1}{2}$; that is, $Ch_{2}(H)=\frac{1}{2}$. By definition, $Cr_{2}$ is equal to Cr conditionalized on the proposition $\mathcal {H}_{2}$, where $\mathcal {H}_{2}$ expresses the complete history of the world up to time $t=2$. That is, for all propositions A,

$$\begin{aligned} Cr_{2}(A) = Cr(A\mid \mathcal {H}_{2}) \end{aligned}$$

It follows, of course, that

$$\begin{aligned} Cr_{2}(H) = Cr(H\mid \mathcal {H}_{2}) \end{aligned}$$

(8)

Moreover, according to the Principal Principle,

$$\begin{aligned} Cr\left( H\;\Big \vert \; \mathcal {H}_{2}\; \& \;Ch_{2}(H)=\frac{1}{2}\right) = \frac{1}{2} \end{aligned}$$

(9)

These two equations—along with the fact that Susie knows the chance, at time $t=2$, of the coin landing heads—jointly imply (5).²²

Equation (9) assumes, of course, that $\mathcal {H}_{2}$ is admissible. And one might deny that. In fact, Lewis (2001) does. For according to Lewis, Mo is inadmissible. Since $\mathcal {H}_{2}$ implies Mo, it follows that $\mathcal {H}_{2}$ is inadmissible as well.

I disagree with Lewis: Mo is admissible, and $\mathcal {H}_{2}$ is as well. For that is the most intuitively plausible view, and I see no good reasons to think otherwise. And in particular, I think that Lewis’s argument for the inadmissibility of Mo does not succeed. According to Lewis, Mo is inadmissible because (1) it is “about the future” (2001, p. 175), and (2) Mo changes Susie's credence in H, the proposition that the coin lands heads. But it is not clear that Mo is ‘about’ the future in any sense relevant to Susie’s experiment. Mo seems to be ‘about’ the present. And even if Mo is ‘about’ the future in some way, it does not follow that Mo must be inadmissible simply because it changes Susie’s credence in H. Plenty of propositions are about the future, and change agents’ credences in other propositions, but are perfectly admissible. For example, suppose Billy is told that two fair coins will be tossed in two days. Because of that, his credence in the proposition that at least one coin comes up heads—call this proposition C—is $\frac{3}{4}$, as that is the current chance of C. The next day, Billy learns the proposition L: one of the coins has been lost, and so only one coin will be flipped. This proposition is ‘about’ the future, in the sense that it tells Billy something about the future coin-flipping event. And it changes Billy’s credence in C to $\frac{1}{2}$, since it changes the chance of C to $\frac{1}{2}$ as well. But L is not inadmissible. Billy does everything correctly when he uses the Principal Principle to adjust his credence in C to the new chance of C.