On the first read I didn’t understand what you were proposing, because of the confusion over “If the two coins show the same face” versus “If the two coins are not both heads.” Now that it’s clear it should be “if the two coins are not both heads” throughout, and after rereading, I now see your argument.
The problem with your argument is that you still have “today” smuggled in: one of your state components is which way the nickel is lying “today.” That changes over the course of the time period we are analyzing, so it does not give a legitimate proposition. To get a legitimate proposition we’ll have to split it up into two propositions: “The nickel lies Heads up on Monday” and “The nickel lies Heads up on Tuesday”.
So in truth, the actual four possible outcomes are HHT, HTH, THT, and TTH. None of these is ruled out by the mere fact of waking up. Not until Beauty receives sufficient sensory input to provide a label for “today” that is nearly certain to be unique do we arrive at a situation in which your analysis is approximately correct.
BTW, is this argument your own? Although I don’t think it’s right, it is an interesting argument. Is there a citation I should use if I want to reference it in future writing?
On the first read I didn’t understand what you were proposing, because of the confusion over “If the two coins show the same face” versus “If the two coins are not both heads.” Now that it’s clear it should be “if the two coins are not both heads” throughout, and after rereading, I now see your argument.
The problem with your argument is that you still have “today” smuggled in: one of your state components is which way the nickel is lying “today.” That changes over the course of the time period we are analyzing, so it does not give a legitimate proposition. To get a legitimate proposition we’ll have to split it up into two propositions: “The nickel lies Heads up on Monday” and “The nickel lies Heads up on Tuesday”.
So in truth, the actual four possible outcomes are HHT, HTH, THT, and TTH. None of these is ruled out by the mere fact of waking up. Not until Beauty receives sufficient sensory input to provide a label for “today” that is nearly certain to be unique do we arrive at a situation in which your analysis is approximately correct.
BTW, is this argument your own? Although I don’t think it’s right, it is an interesting argument. Is there a citation I should use if I want to reference it in future writing?