Sleeping beauty and the dynamics of de se beliefs

Abstract

This paper examines three accounts of the sleeping beauty case: an account proposed by Adam Elga, an account proposed by David Lewis, and a third account defended in this paper. It provides two reasons for preferring the third account. First, this account does a good job of capturing the temporal continuity of our beliefs, while the accounts favored by Elga and Lewis do not. Second, Elga’s and Lewis’ treatments of the sleeping beauty case lead to highly counterintuitive consequences. The proposed account also leads to counterintuitive consequences, but they’re not as bad as those of Elga’s account, and no worse than those of Lewis’ account.

Appendices

Appendix

The many brains argument

For simplicity, assume that there are only two worlds under consideration, one normal world and one brain-duplicating world; it’s easy to see how the result generalizes to multiple worlds. Let S be the stable world, and D be the duplicating world.

Consider the alternatives focused on the original (non-brain) individual at the S and D worlds. As time changes you will replace these alternatives with new alternatives at those worlds, centered on the same individual and a later time. (At the D world, of course, you will also be replacing old brain-centered alternatives with their temporal successors, as well as adding entirely new brain alternatives.) The new non-brain alternatives and the old non-brain alternatives they replaced satisfy the three conditions of Elga’s Continuity Principle. We saw in the “Continuity” section that given centered conditionalization, the Continuity Principle requires that the ratios of priors between new and old continuous alternatives be the same. So the ratio of your priors in the non-brain alternatives at the D and S worlds at a time will be constant. That is, if we let ${\hbox {pr}}_{S_t}$ and ${\hbox {pr}}_{D_t}$ be your priors in the non-brain alternatives at the D and S worlds at t, the Continuity Principle entails that $\forall t \left(\frac{{\hbox {pr}}_{D_t}}{{\hbox {pr}}_{S_t}}=k\right)$, for some constant k.

Elga’s Indifference Principle entails that one’s credences in alternatives at a world be the same, and thus (given centered conditionalization) that one’s priors in alternatives at a world be the same. So one’s prior in the brain alternatives centered on world D and time t will be the same as your prior in the non-brain alternative centered on world D and time t, ${\hbox {pr}}_{D_t}$.

Now, let $N_{W_t}$ be the number of alternatives you have at time t that are centered on a world W, and let cr _t (W) be your credence at t in W. Assume the brains are created one at a time, and choose temporal units and a temporal origin such that (a) $N_{D_0}=N_{S_0} = 1,$ and (b) $N_{D_t}=t+1.$ Since you only ever have one alternative centered on S, ∀t ($(N_{S_t}=1)$.

Centered conditionalization and the above then entail that:

$$\begin{aligned} {\hbox {cr}}_t (D) &=& \frac{N_{D_{t}} \cdot {\hbox {pr}}_{D_t}}{N_{D_{t}} \cdot {\hbox {pr}}_{D_t} + N_{S_{t}} \cdot {\hbox {pr}}_{S_t} }\\ &=&\frac{N_{D_{t}} \cdot {\hbox {pr}}_{D_{t}}}{N_{D_{t}} \cdot {\hbox {pr}}_{D_{t}} + N_{S_{t}} \cdot \frac {{\hbox {pr}}_{D_{t}}} {k} }\\ &=&\frac{N_{D_{t}}}{N_{D_{t}} + \frac {N_{S_{t}}} {k}}\\ &=&\frac{t+1}{t+1 + \frac {1}{ k} }. \end{aligned}$$

Thus:

$$\begin{aligned} \lim_{t\to\infty} \left(\hbox {cr}_t (D) \right)=\lim_{t\to\infty} \left(\frac{t+1}{t + 1 + \frac{1}{k}} \right)=1. \end{aligned}$$

The sadistic scientists argument

Again, for simplicity assume that there are only two worlds under consideration, one normal world and one brain-duplicating-and-destroying world. Let S be the stable world, and D be the duplicating-and-destroying world.

As before, let $N_{W_t}$ be the number of alternatives you have at time t that are centered on a world W, and let cr _t (W) be your credence at t in W. Choose temporal units and a temporal origin such that if t < 0 or t > n, then $N_{D_t}$ = 1, and if 0 ≤ t ≤ n, then $N_{D_t}$ = (n + 1) − t. (So n is the number of brains that will be created in D at time t = 0, and one of these brains will be destroyed every unit of time thereafter.)

As before, let ${\hbox {pr}}_{S_t}$ and ${\hbox {pr}}_{D_t}$ be your priors in the non-brain alternatives at the D and S worlds at t. Now consider the alternatives focused on the original (non-brain) individual at the S and D worlds. As time changes you will replace these alternatives with new alternatives at those worlds, centered on the same individual and a later time. (At the D world, of course, you will also be replacing old brain-centered alternatives with their temporal continuants, as well as adding entirely new brain alternatives.) The new non-brain alternatives and the old non-brain alternatives they replaced satisfy the conditions of Lewis’ Continuity Principle until time t = 0, when the brains are created. So the continuity principle entails that for t < 0, $\left(\frac{{\hbox {pr}}_{D_t}}{{\hbox {pr}}_{S_t}}=k\right)$, for some constant k. The conditions also hold after the brains are created, so the continuity principle entails that for t ≥ 0, $\left(\frac{{\hbox {pr}}_{D_t}}{{\hbox {pr}}_{S_t}}=l\right)$, for some constant l.

Elga’s Indifference Principle entails that one’s credences in alternatives at a world be the same, and thus (given centered conditionalization) that one’s priors in alternatives at a world be the same. So one’s prior in the brain alternatives at D at t will be the same as your prior in the non-brain alternative at D, ${\hbox {pr}}_{D_t}$. The No-Increase Principle entails that your credence in D shouldn’t change when the new brains are created at t = 0. This, centered conditionalization and the above entail that l = k / (n + 1).

Centered conditionalization and the above then entail that:

$$\begin{aligned} {\hbox {cr}}_{t=n} (D) &=& \frac{N_{D_{n}} \cdot {\hbox {pr}}_{D_{n}}}{N_{D_{n}} \cdot {\hbox {pr}}_{D_{n}} + N_{S_{n}} \cdot {\hbox {pr}}_{S_n }}\\ &=&\frac{N_{D_{n}} \cdot {\hbox {pr}}_{D_{n}}}{N_{D_{n}} \cdot {\hbox {pr}}_{D_{n}} + N_{S_{n}} \cdot {{\hbox {pr}}_{D_{n}} \cdot (n+1) / k} }\\ &=&\frac{{\hbox {pr}}_{D_{n}}}{{\hbox {pr}}_{D_{n}} + {{\hbox {pr}}_{D_{n}} \cdot (n+1) / k} }\\ &=&\frac{1}{1 + {(n+1) / k}}. \end{aligned}$$

Thus:

$$\begin{aligned} \lim_{n\to\infty} \left({\hbox {cr}}_{t=n} (D) \right)= \lim_{n\to\infty} \left(\frac{1}{1 + \frac{n+1}{k}} \right)=0. \end{aligned}$$