This impossibility appears to hold in non-anthropic situations as well.
For example, in the “Tails False Memory” scenario I commented to earlier posts: a coin is flipped and on Tails, Beauty is given a false memory of being asked “what is your credence for heads” and having answered. Then unconditionally Beauty is asked “are you sure”. Beauty knows these rules.
In both possible worlds at every point in time, there is only one agent and therefore at most one agent with any given history. So this is a non-anthropic situation at all times by the given definition.
Let the relevant histories be h1 = “I remember being asked the first question only and not having answered yet”, and h2 = “I remember being asked both questions, having answered the first”. Let the worlds be wT = “the coin comes up Tails” and wH = “the coin comes up heads”.
Beauty knows that (wT, h1) is impossible, and so Q(wT | h1) = 0 and therefore Q(wH | h1) = 1. The subsequent state (wH, h2) is a non-anthropic situation and so Beauty must use the Simple Bayes rule to update, so Q(wH | h2) = 1 also. But Q(wH | h2) + Q(wT | h2) = 1, so Q(wT | h2) = 0 which requires that Q(wT, h2) = 0.
But (wT, h2) is a possible state and must not be assigned zero credence, concluding the impossibility proof.
This impossibility appears to hold in non-anthropic situations as well.
For example, in the “Tails False Memory” scenario I commented to earlier posts: a coin is flipped and on Tails, Beauty is given a false memory of being asked “what is your credence for heads” and having answered. Then unconditionally Beauty is asked “are you sure”. Beauty knows these rules.
In both possible worlds at every point in time, there is only one agent and therefore at most one agent with any given history. So this is a non-anthropic situation at all times by the given definition.
Let the relevant histories be h1 = “I remember being asked the first question only and not having answered yet”, and h2 = “I remember being asked both questions, having answered the first”. Let the worlds be wT = “the coin comes up Tails” and wH = “the coin comes up heads”.
Beauty knows that (wT, h1) is impossible, and so Q(wT | h1) = 0 and therefore Q(wH | h1) = 1. The subsequent state (wH, h2) is a non-anthropic situation and so Beauty must use the Simple Bayes rule to update, so Q(wH | h2) = 1 also. But Q(wH | h2) + Q(wT | h2) = 1, so Q(wT | h2) = 0 which requires that Q(wT, h2) = 0.
But (wT, h2) is a possible state and must not be assigned zero credence, concluding the impossibility proof.
Yep, seems right.
Personally, I consider false memory to be a form of anthropic scenarios—you don’t know exactly who you are.