Here is yet another variation of the problem that I think perfectly identifies the source of the controversy. The experiment’s methodology is the same as the original, except in these four details:
(1) Two coins, a Nickel and a Quarter, are flipped on Sunday Night.
(2) On either day of the experiment, Beauty is wakened if either of the two coins is showing Tails.
(3) On Monday Night, while Beauty is asleep, the Nickel is flipped over to show its opposite face.
(4) Beauty is asked the same question, but about the Quarter.
The only functional difference is that there is a 50% chance that the “optional” waking occurs on Monday instead of Tuesday. Since Beauty does not know the day in either version, this cannot affect the result; she is still wakened once if the Quarter landed on Heads, and twice if it landed on Tails.
The controversy boils down to whether the current state of the Nickel can be called a random variable. Or, much like Schrodenger’s Cat while its box is unopened, the Nickel has to be considered to be in both states simultaneously for the purposes of the experiment.
Halfers treat it as both. The Nickel shows both Heads and Tails during the experiment, so Beauty cannot use it as a random variable. This is the crux of Radford Neal’s argument, in the original experiment. That “Today” is an indexical becasue it has both the value “Monday” and “Tuesday” during the experiment, so it can’t be used as evidence.
The thirder’s argument is that what the Nickel is currently showing is not an indexical at all. While Beauty is awake, it has only one value. That value is unknown, and can have either value with probability 50%. So there are four states for {Nickel, Quarter} that, at any time during the experiment, are equiprobable in the prior. And that the evidence Beauty has, based on the fact that she is awake, eliminates {Heads, Heads} as a possibility. This makes the probability that the Quarter landed on Heads 1⁄3.
Here is yet another variation of the problem that I think perfectly identifies the source of the controversy. The experiment’s methodology is the same as the original, except in these four details:
(1) Two coins, a Nickel and a Quarter, are flipped on Sunday Night.
(2) On either day of the experiment, Beauty is wakened if either of the two coins is showing Tails.
(3) On Monday Night, while Beauty is asleep, the Nickel is flipped over to show its opposite face.
(4) Beauty is asked the same question, but about the Quarter.
The only functional difference is that there is a 50% chance that the “optional” waking occurs on Monday instead of Tuesday. Since Beauty does not know the day in either version, this cannot affect the result; she is still wakened once if the Quarter landed on Heads, and twice if it landed on Tails.
The controversy boils down to whether the current state of the Nickel can be called a random variable. Or, much like Schrodenger’s Cat while its box is unopened, the Nickel has to be considered to be in both states simultaneously for the purposes of the experiment.
Halfers treat it as both. The Nickel shows both Heads and Tails during the experiment, so Beauty cannot use it as a random variable. This is the crux of Radford Neal’s argument, in the original experiment. That “Today” is an indexical becasue it has both the value “Monday” and “Tuesday” during the experiment, so it can’t be used as evidence.
The thirder’s argument is that what the Nickel is currently showing is not an indexical at all. While Beauty is awake, it has only one value. That value is unknown, and can have either value with probability 50%. So there are four states for {Nickel, Quarter} that, at any time during the experiment, are equiprobable in the prior. And that the evidence Beauty has, based on the fact that she is awake, eliminates {Heads, Heads} as a possibility. This makes the probability that the Quarter landed on Heads 1⁄3.