I think using dice roll to decide Alex’s awakening is an excellent idea.
Back to the topic at hand, what I have been arguing is that the correct halfer argument should not be based on objective chance. Thus my disagreement with Chris so far. I think it should instead be based on the irreducible nature of perspectives, that it roots in the subjective. That’s why I brought up the perspective disagreement in the first place.
What you have argued: that after seeing Alex awake Beauty should eliminate one of the 4 events, reflects the fact that Alex has met Beauty in an awakening. But it does not use the whole information Beauty has: she specifically knows Alex has met Beauty in this awakening. And the specification of this awakening is based on her perspective, as it is the one which she has subjective experience of.
You said you are not thirding, only considering the objective chance. I believe you. But I was saying by considering the objective chance this way one has to endorse Thirderism. For example, if Beauty checks on Alex and see him asleep instead, by the same logic she must discount (Heads, Odd), leaving P(H,E) = P(T,O) = P(T,E) = 1⁄3. So it doesn’t matter whether Beauty finds Alex awake or asleep, she will say P(H)=1/3 regardless. Then Beauty should’ve regard 1⁄3 as the correct answer even before looking at Alex anyway. (This argument is due to Michael Titelbaum ’s Technicolor Beauty)
So do I think the objective chance you presented are correct? Yes, I do. Just as Beauty should think Alex’s answer of 1⁄3 is correct. Do I think Beauty should give the same answer of 1/3? No, I don’t. Because at the end of the day I am a halfer, and I believe perspectives are fundamental, the disagreement between Alex and Beauty is valid.
So do I think the objective chance you presented are correct? Yes, I do.
That’s good, I just wanted to establish that baseline. I think it’s healthy for everyone to confirm their understanding of the objective chances before moving on to debating philosophical issues of credence. Thank you.
“But I was saying by considering the objective chance this way one has to endorse Thirderism”
No. If I asked you to calculate the objective chances then I hope you (and everyone else here) would perform exactly the same calculation. Presented as a maths problem, as I did, it says nothing about my (or your, or anyone else’s) philosophical leanings.
Perhaps you meant that by equating the objective Pr(...) with Credence one has to be a Thirder?
I do like the example though. It confused me enough that I felt I had to post again (replying to myself … bad form). What follows is pure speculation.
I think there’s something here about scopes/closures.
Within the scope of a single wake-up, “Alex is awake at the same time as me” is admissible as an event: specifically the subset { (H,O), (T,O), (T,E) } of the sample space.
Within the scope of the whole experiment, “Alex is awake at the same time as me” is inadmissible as an event: it is not a well-defined subset of the sample space.
So for games whose payout decomposes additively over wake-ups, you may consider each wake-up in isolation and, if you do so, you must use all available events including Alex’s sleeping state, Thirder-style.
For games whose payout does not decompose additively over wake-ups, you must consider the experiment as a whole and, in doing so, find that you have no new information, Halfer-style.
Hey, don’t worry about replying to oneself. I do it quite often :)
I just realized the root cause of our debate after reading this:
That’s good, I just wanted to establish that baseline. I think it’s healthy for everyone to confirm their understanding of the objective chances before moving on to debating philosophical issues of credence. Thank you.
You regard “objective chance” as the baseline, any philosophical leaning is built on top of that. But I think the philosophical aspect of the problem cuts deeper than that.I regard (postulates) perspective reasoning as fundamental and objective reasoning as derived. While in most people recognize (postulate) objective reasoning (i.e. perspective-independent reasoning) as fundamental and indexicals as additional information that need to be incorporated on top of objective reasoning. That’s why I call my approach Perspective Based Reasoning.
So when seeing “objective chance” I was not considering it as the baseline, but treating it as “thinking from a god’s eye view” (which give the same answer as Alex’s perspective). To me it is no more fundamental than thinking from Beauty’s first-person viewpoint. Just two different systems of reasoning. And the two perspective gives different answers, both are valid, thus the disagreement.
It is why when you ask what should Beauty think about the objective chance I keep stressing what should Beauty think from her perspective. You want to leave the philosophical leanings out of it for now, but to me using objective chance as the baseline already has a philosophical leaning.
It is also why I don’t think halfers should change the probability to 1⁄3 after seeing Alex awake. I think the best rebuttal is still the one I just discussed. By the same logic Beauty would change the probability to 1⁄3 after seeing Alex asleep as well. Since it is going to be 1⁄3 regardless of Alex’s status her probability should have been 1⁄3 right from the start, i.e. after waking up.
I don’t think we’re a million miles apart on this. We can both tolerate Halfing and Thirding, in your case because they’re both reasonable perspectives and in my case because it doesn’t matter anyway. Your characterisation of our disagreement seems about right: let’s come back to that in the comments of some future unsuspecting author’s post on Sleeping Beauty :)
I think using dice roll to decide Alex’s awakening is an excellent idea.
Back to the topic at hand, what I have been arguing is that the correct halfer argument should not be based on objective chance. Thus my disagreement with Chris so far. I think it should instead be based on the irreducible nature of perspectives, that it roots in the subjective. That’s why I brought up the perspective disagreement in the first place.
What you have argued: that after seeing Alex awake Beauty should eliminate one of the 4 events, reflects the fact that Alex has met Beauty in an awakening. But it does not use the whole information Beauty has: she specifically knows Alex has met Beauty in this awakening. And the specification of this awakening is based on her perspective, as it is the one which she has subjective experience of.
You said you are not thirding, only considering the objective chance. I believe you. But I was saying by considering the objective chance this way one has to endorse Thirderism. For example, if Beauty checks on Alex and see him asleep instead, by the same logic she must discount (Heads, Odd), leaving P(H,E) = P(T,O) = P(T,E) = 1⁄3. So it doesn’t matter whether Beauty finds Alex awake or asleep, she will say P(H)=1/3 regardless. Then Beauty should’ve regard 1⁄3 as the correct answer even before looking at Alex anyway. (This argument is due to Michael Titelbaum ’s Technicolor Beauty)
So do I think the objective chance you presented are correct? Yes, I do. Just as Beauty should think Alex’s answer of 1⁄3 is correct. Do I think Beauty should give the same answer of 1/3? No, I don’t. Because at the end of the day I am a halfer, and I believe perspectives are fundamental, the disagreement between Alex and Beauty is valid.
That’s good, I just wanted to establish that baseline. I think it’s healthy for everyone to confirm their understanding of the objective chances before moving on to debating philosophical issues of credence. Thank you.
No. If I asked you to calculate the objective chances then I hope you (and everyone else here) would perform exactly the same calculation. Presented as a maths problem, as I did, it says nothing about my (or your, or anyone else’s) philosophical leanings.
Perhaps you meant that by equating the objective Pr(...) with Credence one has to be a Thirder?
I do like the example though. It confused me enough that I felt I had to post again (replying to myself … bad form). What follows is pure speculation.
I think there’s something here about scopes/closures.
Within the scope of a single wake-up, “Alex is awake at the same time as me” is admissible as an event: specifically the subset { (H,O), (T,O), (T,E) } of the sample space.
Within the scope of the whole experiment, “Alex is awake at the same time as me” is inadmissible as an event: it is not a well-defined subset of the sample space.
So for games whose payout decomposes additively over wake-ups, you may consider each wake-up in isolation and, if you do so, you must use all available events including Alex’s sleeping state, Thirder-style.
For games whose payout does not decompose additively over wake-ups, you must consider the experiment as a whole and, in doing so, find that you have no new information, Halfer-style.
Hey, don’t worry about replying to oneself. I do it quite often :)
I just realized the root cause of our debate after reading this:
You regard “objective chance” as the baseline, any philosophical leaning is built on top of that. But I think the philosophical aspect of the problem cuts deeper than that.I regard (postulates) perspective reasoning as fundamental and objective reasoning as derived. While in most people recognize (postulate) objective reasoning (i.e. perspective-independent reasoning) as fundamental and indexicals as additional information that need to be incorporated on top of objective reasoning. That’s why I call my approach Perspective Based Reasoning.
So when seeing “objective chance” I was not considering it as the baseline, but treating it as “thinking from a god’s eye view” (which give the same answer as Alex’s perspective). To me it is no more fundamental than thinking from Beauty’s first-person viewpoint. Just two different systems of reasoning. And the two perspective gives different answers, both are valid, thus the disagreement.
It is why when you ask what should Beauty think about the objective chance I keep stressing what should Beauty think from her perspective. You want to leave the philosophical leanings out of it for now, but to me using objective chance as the baseline already has a philosophical leaning.
It is also why I don’t think halfers should change the probability to 1⁄3 after seeing Alex awake. I think the best rebuttal is still the one I just discussed. By the same logic Beauty would change the probability to 1⁄3 after seeing Alex asleep as well. Since it is going to be 1⁄3 regardless of Alex’s status her probability should have been 1⁄3 right from the start, i.e. after waking up.
I don’t think we’re a million miles apart on this. We can both tolerate Halfing and Thirding, in your case because they’re both reasonable perspectives and in my case because it doesn’t matter anyway. Your characterisation of our disagreement seems about right: let’s come back to that in the comments of some future unsuspecting author’s post on Sleeping Beauty :)