Well in that case, it narrows down what we agree about. Mathematical propositions aren’t events that happen. However, someone who doesn’t know a specific digit of Pi would assign likelihood to it’s value with the same rules of probability as they would to an event they don’t know about. I define credence merely as someone’s rational estimate of what’s likely to be true, based on knowledge or ignorance. Credence has no reason to discriminate between the three types of reality I talked about, much less get invalidated.
I would also highlight that almost all external outcomes in the macro world, whether known or unknown, are already determined, as opposed to being truly random. In that sense, an unknown coin flip outcome is just as certain as an unknown mathematical proposition. In the case of Sleeping Beauty being told it’s Monday and that a coin will be flipped tonight, she is arguably already in a Heads world or a Tails world. It’s just that no-one knows which way the coin will land. If so, Lewis’s version of halfing is not as outlandish as it appeared. Beauty’s 2/3 update on Monday that the coin will land Heads is not actually an update on a future random event. It is an update on a reality that already exists but is unknown. From Beauty’s perspective, if she's in a Heads world, she is certain it is Monday. If the world is Tails, she doesn’t have that certainty. Therefore an increased likelihood in Heads, once she learns it’s Monday, is reasonable – assuming that self-locations allow credences. I submit that, since she was previously not certain that the day she found herself awake on was Monday, a non-zero credence that this day was Tuesday legitmately existed before being eliminated.
Below is a version of Sleeping Beauty that mixes the three types of reality I described – contingent, analytic and self-location.
On Sunday night, Beauty is put to sleep and cloned. A coin is flipped. If it lands Heads, the original is woken on Monday and questioned, while the clone stays asleep. If it lands Tails, the clone is woken on Monday and questioned, while the original stays asleep.
The rest of the protocol concerns the Beauty that was not woken, and is determined by the tenth digit of Pi. If it’s between 1-5, the other Beauty is never woken and destroyed. If it’s between 6-0, the other Beauty is woken on Tuesday and questioned.
For any Beauty finding herself awake, do any of the following questions have valid answers?
What is the likelihood that the tenth digit of Pie is 1-5 or 6-0?
What is the likelihood that the coin landed Heads or Tails?
What is the likelihood that today is Monday or Tuesday?
What is the likelihood that the she is the original or clone?
You’ll be unsurprised that I think all these credences are valid. In this example, any credence about her identity or the day happens to be tied in with credence about the other realities.
You’ll also notice how tempting it is for Beauty to apply thirder reasoning to the tenth digit of Pi – i.e it’s 1/3 that it’s 1-5 and 2/3 that it’s 6-0. The plausible thirder argument is that, whatever identity the waking Beauty turns out to have, either original or clone, it was only 50/50 this version would be woken if the Pi digit was 1-5, whereas it was certain this version would be woken if the Pi digit was 6-0. However, I would say that, uniquely from her perspective, her identity is not relevant to her continued conciousness or the likelihood of the tenth digit of Pi. At least one awakening with memories of Sunday was guaranteed. Her status as original or clone doesn’t change her prior certainty of this. All that matters is that one iteration of her consciousness woke up if the digit was 1-5, while two iterations of her consciousness woke up if the digit was 6-0. In either case there is a guaranteed awakening with continuity from Sunday, and no information to indicate whether there are one or two awakenings. This is why I would remain a halfer.
Each question can be asked conditionally, with Beauty being told the answer to one or more of the others. In particular, if she’s told it’s Monday, the likelihood of Pi’s tenth digit being 1-5 must surely increase, whether she's a thirder or halfer. Her reasoning is that if the tenth digit of Pi was 1-5, whatever body she has was certain to wake up on Monday, regardless of the coin. Whereas if the tenth digit of Pi was 6-0, the body she has could have woken on either day, determined by the coin. It would be hard to argue otherwise, since the day she wakes up is a contingent event, not just reflecting a self-location.
My conclusion is that rules of credence and assignments of probability are applicable in all cases where there is uncertainty about what’s true from a first person perspective, regardless of the nature of the reality. This includes self-locations. Self-locations can give rise to different sampling and perspective disagreement between interacting parties, in situations where one party might have more self-locating experiences than the other.
Hi Dadarren. I haven’t forgotten our discussion and wanted to offer further food for thought. It might be helpful to explore definitions. As I see it, there are three kinds of reality about which someone can have knowledge or ignorance.
Contingent – an event in the world that is true or false based on whether it did or did not happen.
Analytic – a mathematical statement or expression that is true or false a priori.
Self-location – an identity or experience in space-time that is true or false at a given moment for an observer.
I’d like to ask two questions that may be relevant.
a) When it comes to mathematical propositions, are credences valid? For example, if I ask whether the tenth digit of Pi is either between 1-5 or 6-0, and you don’t know, is it valid for you to use a principal of indifference and assign a personal credence of 1/2?
b) Suppose you’re told that a clone of you will be created when you’re asleep. A coin will be flipped. If it lands Head, the clone will be destroyed and the original version of you will be woken. If it lands Tails, the original will be destroyed and the clone will be woken. Finding yourself awake in this scenario, is it valid to assign a 1/2 probability that you’re either the original or clone?
I would say that both these are valid and normal Bayesian conditioning applies. The answer to b) reflects both identity and a contingent event, the coin flip. For a), it would be easy to construct a probability puzzle with updatable credences about outcomes determined by mathematical propositions.
However I’m curious what your view is, before I dive further in.
Ok that’s fine. I agree MWI is not proven. My point was only that it is the absolute self-location model. Those endorsing it propose the non-existence of probability, but still assign the mathematics of likelihood based on uncertainty from an observer. Forgive me for stumbling onto the implications of arguments you made elsewhere. I have read much of what you’ve written over time.
I especially agree that perspective disagreement can happen. That's what makes me a Halfer. Self-location is at the heart of this, but I would say it is not because credences are denied. I would say disagreement arises when sampling spaces are different and lead to conflicting first-person information that can’t be shared. I would also say that, whenever you don't know which pre-existing time or identity applies to you, assigning subjective likelihood has much meaning and legitimacy as it does to an unknown random event. I submit that it’s precisely because you do have credences based on uncertainty about self-location that perspective disagreement can happen.
You can also have a situation where a reality might be interpreted subjectively both as random and self-location. Consider a version of Sleeping Beauty where we combine uncertainty of being the original and clone with uncertainty about what day it is.
Beauty is put to sleep and informed of the following. A coin will be (or has already been) flipped. If the coin lands Heads, her original self will be woken on Monday and that is all. If the coin lands Tails, she will be cloned; if that happens, it will be randomly decided whether the original wakes on Monday and the clone on Tuesday, or the other way round.
Is it valid here for Beauty to assign probability to the proposition “Today is Monday” or Today is Tuesday”? I’m guessing you will agree that, in this case, it is. If the coin landed Heads, being woken on Monday was certain and so was being the original. If it landed Tails, being woken on Monday or Tuesday was a separate random event from whichever version of herself she happens to be. Therefore she should assign 3/4 that today is Monday and 1/4 that today is Tuesday. We can also agree as halfers that the coin flip is 1/2 but, once she learns it’s Monday, she would update 2/3 to Heads and 1/3 to Tails. However, if instead of being told it's Monday, she's told that she's the original, then double halfing kicks in for you.
From your PBR position, the day she’s woken does have indpendent probability in this example, but her status as original or clone is a matter of self-location. Whereas I question whether there's any significant difference between the two kinds of determination. Also, in this example, the day she’s woken can be regarded as both externally random and a case of self-location. Whichever order the original and clone wake up if the coin landed Tails, there are still two versions of Beauty with identical memories of Sunday; one of these wakes on Monday, the other on Tuesday. If told that, in the event of Tails, the two awakenings were prearranged to correspond to her original and clone status instead of being randomly assigned, the number of awakenings are the same and nothing is altered for Beauty's knowledge of what day it is. I submit that in both scenarios, validity for credence about the day should the same.
Ok here's some rebuttal. :) I don’t think it’s your communication that’s wrong. I believe it’s the actual concept. You once said that yours is a view that no-one else shares. This does not in itself make it wrong. I genuinely have an open mind to understand a new insight if I’m missing it. However I’ve examined this from many angles. I believe I understand what you’ve put forward.
In anthropic problems, issues of self-location and first person perspective lie at the heart. In anthropic problems, a statement about a person’s self-location, such as ‘'”today is Monday” or “I am the original”, is indeed a first person perspective. Such a statement, if found to be true, is a fact that could not have been otherwise. It was not a random event that had a chance of not happening. From this, you’ve extrapolated – wrongly in my view - that normal rules of credence and Bayesian updating based on information you have or don’t yet have, are invalid when applied to self-location.
I’m reminded of the many worlds quantum interpretation. If we exist in a multiverse, all outcomes take place in different realities and are objectively certain. In a multiverse, credences would be deciding which world your first-person-self is in, not whether events happened. The multiverse is the ultimate self-location model. It denies objective probability. What you have instead are observers with knowledge or uncertainty about their place in the multiverse.
Whether theories of the multiverse prove to be correct or not, there are many who endorse them. In such a model – where probability doesn’t exist – it is still considered both legitimate and necessary for observers to assign likelihood and credence about what is be true for them, and apply rules of updating based on information available.
I have a realistic example that tests your position. Imagine you’re an adopted child. The only information you and your adopted family were given is that your natural parents had three children and that all three were adopted by different families. What is the likelihood that you were the first born natural child? For your adopted parents, it’s straightforward. They assign a 1/3 probability that they adopted the oldest. According to you, as it’s a first-person perspective question about self-location, no likelihood can be assigned.
It won't surprise you to learn that here I find no grounds for you to disagree with your adopted parents, much less to invalidate a credence. Everyone agrees that you are one of three children. Everyone shares the same uncertainty of whether you’re the oldest. Therefore the credence 1/3 that this is the case must also be shared.
I could tweak this situation to allow perspective disagreement with your adopted family, making it closer to Sleeping Beauty - and introducing a coin flip. I may do that later.
I’ll come back with a deeper debate in good time. Meanwhile I’ll point out one immediate anomaly.
I was genuinely unsure which position you’d take when you learnt the two envelopes were the same. I expected you to maintain there was no probability assigned to Envelope A. I didn’t expect you to invalidate probability for the contents of Envelope B. You argued this because any statement about the contents of Envelope B was now linked to your self-location – even though Envelope B’s selection was unmistakably random and your status remained unknown.
It becomes even stranger when you consider your position if told that the two envelopes were different. In that event, any statement about the contents of Envelope B refers just as much to self-location. If the two envelopes are different, a hypothesis that Envelope B contains ‘copy’ is the same hypothesis that you’re the original - and vice versa. Your reasoning would equally compel you to abandon probability for Envelope B.
Therein lies the contradiction. The two envelopes are known in advance to be either the same or different. Whichever turns out to be true will neither reveal or suggest the contents of Envelope B. Before you discover whether the envelopes are the same or different, Envelope B definitely had a random 1/2 chance of containing ‘original’ or ‘copy’. Once you find out whether they’re the same or different, regardless of which turn out to be true, you're saying that Envelope B can no longer be assigned probability.
Something is wrong... :)
I’ve allowed some time to digest on this occasion. Let's go with this example.
A clone of you is created when you’re asleep. Both of you are woken with identical memories. Under your pillow are two envelopes, call them A and B. You are told that inside Envelope A is the title ‘original’ or ‘copy’, reflecting your body’s status. Inside Envelope B is also one of those titles, but the selection was random and regardless of status. You are asked the likelihood that each envelope contains ‘original’ as the title.
I’m guessing you'd say you can't assign any valid probability to the contents of Envelope A. However, you’d say it’s legitimate to assign a 1/2 probability that Envelope B contains ‘original’.
Is there a fundamental difference here, from your point of view? Admittedly if Envelope A contains 'original', this reflects a pre-existing self-location that was previously known but became unknown while you were asleep. Whereas if Envelope B contains 'original', this reflects an independent random selection that occured while you were asleep. However, your available evidence is identical for what could be inside each envelope. You therfore have identical grounds to assign likelihood about what is true in both.
Suppose it's revealed that both envelopes contain the same word. You are asked again the likelihood that the envelopes contain ‘original’. What rules do you follow? Would you apply the non-existence of Envelope A’s probability to Envelope B? Or would you extend the legitimacy of Envelope B’s probability to Envelope A?
I'm guessing you would continue to distinguish the two, stating that 1/2 was a still valid probability for Envelope B containing 'original' but no such likelihood existed for Envelope A - even knowing that whatever is true for Envelope B is true for Envelope A. If so, then it appears to be a semantic difference. Indeed, from a first person perspective, it seems like a difference that makes no difference. :)
I doubt we’ll persuade each other :) As I understand it, in my example you’re saying that the moment a self-location is ruled out, any present and future updating is impossible – but the last known probability of the coin stands. So if Beauty rules out Heads/Peter and nothing else, she must update Heads from 1/2 to 1/3. Then if she subsequently rules out Tails/Peter, you say she can’t update, so she will stay with the last known valid probability of 1/3. On the other hand, if she rules out Tails/Peter first, you say she can’t update so it’s 1/2 for Heads. However, you also say no further updating is possible even if if she then rules out Heads/Peter, so her credence will remain 1/2, even though she ends up with identical information. That is strange, to say the least.
I’ll make the following argument. When it comes to probability or credence from a first person perspective, what matters is knowledge or lack of it. Poeple can use that knowledge to judge what is likely to be true for them at that moment. Their ignorance doesn’t discriminate between unknown external events and unknown current realities, including self-locations. Likewise, their knowledge is not invalidated just because a first person perspective might happen to conflict with a third person perspective or because the same credence may not be objectively verfiable in the frequentist senes. In either case, it’s their personal evidence for what’s true that gives them their credence, not what kind of truth it is. That credence, based on ignorance and knowledge, might or might not correspond to an actual random event. It might reflect a pre-existing reality - such as there is a 1/10 chance that the 9th digit of Pi is 6. Or it might reflect an unknown self-location – such as “today is Monday or Tuesday”, or “I’m the original or clone”. Whatever they’re unsure about doesn’t change the validity of what they consider likely.
You could have exactly the same original Sleeping Beauty problem, but translated entirely to self-location without the need for a coin flip. Consider this version. Beauty enters the experiment on a Saturday and is put to sleep. She will be woken in both Week 1 and in Week 2. A memory wipe will prevent her from knowing which week it is, but whenever she wakes, she will definitely be in one or the other. She also knows the precise protocol for each week. In Week 1, she will be woken on Sunday, questioned, put back to sleep without a memory wipe, woken on Monday and questioned. This completes Week 1. She will return the following Saturday and be put to sleep as before. She is now given a memory wipe of both her awakenings from Week 1. She is then woken on Sunday and questioned but her last memory is of the previous Saturday when she entered the experiment. She doesn’t know whether this is the Sunday from Week 1 or Week 2. Next she is put to sleep without a memory wipe, woken on Monday and questioned. Her last memory is the Sunday just gone, but she still doesn’t know if it’s Week 1 or Week 2. Next she is put back to sleep and given a memory wipe, but only of todays’s awakening. Finally she woken on Tuesday and questioned. Her last memory is the most recent Sunday. She still won’t know which week she’s in.
The questions asked of her are as follows. When she awakens for what seems to be the first time – always a Sunday – what is her credence for being in Week 1 or Week 2? When she awakens for what seems to be the second time – which might be a Monday or Tuesday – what is her credence for being in Week 1 or Week 2?
Essentially this is the same Sleeping Beauty problem. The fact that she has uncertainty of which week she’s in rather than a coin flip, doesn’t prevent her from assigning a credence/probability based on the evidence she has. On her Sunday awakening, she has equal evidence to favour Week 1 and Week 2, so it is valid for her to assign 1/2 to both. On her weekday awakenings, Halfers and Thirders will disagree whether it’s 1/2 or 1/3 that she’s in Week 1. If she's told that today is Monday, they will disagree whether it's 2/3 or 1/2 that she's in Week 1.
We could add Bob to the experiment. Like Beauty, Bob enters the experiment on Saturday. His protocol is the same, except that he is kept asleep on the Sundays in both week. He is only woken Monday of Week 1, then Monday and Tuesday of Week 2. Each time he’s woken, his last memory is the Saturday he entered the experiment. He therefore disagrees with Beauty. From his point of view, it’s 1/3 that they’re in Week 1, whereas she says it’s 1/2. If told it's Monday, for him it's 1/2 that they're in Week 1, and for her it's 2/3.
This recreates perspective disagreement – but exclusively using self-location. You might be tempted to argue that neither Beauty or Bob can ever assign any probability or likelihood as to which week they’re in. I say it’s legitimate for them to do so, and to disagree.
Ok let’s see if we can pin this down! Either Beauty learns something relevant to the probability of the coin flip, or she doesn't. We can agree on this, even if you think updating can't happen with self-location.
Let’s go back to a straightforward version of the original problem. It's similar to one you came up with. If Heads there is one awakening, if Tails there are two. If Heads, it will be decided randomly whether she wakes on Monday or Tuesday. If Tails, she will be woken on both days with amnesia in between. She is told in advance that, regardless of the coin flip, Bob and Peter will be in the room. Bob will be wake on Monday, while Peter is asleep. Peter will be awake on Tuesday, while Bob asleep. Whichever of the men wakes up, it will be two minutes before Beauty (if she’s woken). All this is known in advance. Neither man knows the coin result, nor will they undergo amnesia. Beauty has not met either of them, so although she knows the protocol, she won’t know the name of who's awake with her or the day unless he reveals it.
Bob and Peter’s perspective when they wake up is not controversial. Each is is guaranteed to find the other guy asleep. In the first two minutes, each will find Beauty asleep. During that time, both men’s probability is 1/2 for Heads and Tails. If Beauty is still asleep after two minutes, its definitely Heads. If she wakes up, it’s 1/3 Heads and 2/3 Tails.
A Thirder believes that Beauty shares the same credence as Bob or Peter when she wakes. As a Halfer, I endorse perspective disagreement. Unlike the guys, Beauty was guaranteed to encounter someone awake. Her credence therefore remains 1/2 for Heads and Tails, regardless of who she enounters.
What happens when the man awake reveals his name - say Bob? This reveals to Beauty that today is Monday. I would say that the probability of the coin remains unchanged from whatever it was before she got this information. For a Thirder this is 1/3 Heads, 2/3 Tails. For a Halfer it is still 1/2. I submit that the reason the probability remains unchanged is because something in both Heads and Tails was eliminated with parity. But suppose she got the information in stages.
She first asks the experimenter: is it true that the coin landed Tails and I'm talking to Peter? She’s told that this is not true. I regard this as a legitimate update that halves the probability of Tails, whereas you don't. Such an update would make her credence 2/3 Heads and 1/3 Tails. You would claim that no update is possible because what’s been ruled out is the self-location Tuesday/Tails. For you, ruling out Tuesday/Tails says nothing new about the coin. You argue that, whether the coin landed Heads or Tails, there is only one 'me' for Beauty and a guaranteed awakening applied to her. So the probability of Heads or Tails must be 1/2 before and after Tuesday/Tails is ruled out.
We might disagree whether ruling out the self-location Tuesday/Tails permitted an update or whether her credence must remain 1/2. But we can agree that, if she gets further information about a random event that had prior probability, she must update in the normal way. Even if ruling out a self-location told her nothing about the coin probability, it can't prevent her from updating if she does get this information.
So now she asks the experimenter: is it true that the coin landed Heads and I’m talking to Peter? She’s told that this is not true. This tells her that Bob is the one she's interacting with, plus she knows it's Monday. What's more, this is not a self-location that's been ruled out like before. The prior possibility of the coin landing Heads plus the prior possibility of her encountering Peter was a random sequence that has just been eliminated. It definitely requires an update. If her credence immediately before was 1/2 for Heads, it must be 1/3 now. If her credence was 2/3 for Heads – which I think was correct - then it is 1/2 now. Which is it?
That brings us back to Beauty’s position before the man says his name. Her credence for the coin is 1/2, before she learns who she’s with. Learning his identity rules out a possibility in both coin outcomes, as described above. The order in which she got the information makes no difference to what she now beleives. The fact remains that a random event with Heads, and a self-location with Tails, were both ruled out. It's the parity in updating that makes her credence still 1/2, whoever turns out to be awake with her.
Interesting Dadarren.
I sense that you’re close to being converted to become a ‘pure halfer’, willing to assign probability to self-locations. Let me address what you said.
“Your latter example: after seeing Red/Blue, I will not say Heads' probability is halved while Tails' remain the same. I will say there is no way to update. “
I assume you mean that the probability of 1/2 for Heads or Tails – before and after she sees either colour – remains correct. We can agree on that. What matters is why it is true. You’re arguing that no updating is possible after she sees a colour. I invite you to examine this again.
The moment she sees – for example – blue, we can agree that Tails/Red and Heads/Red are definitely ruled out. The difference is that Tails/Red reflects a self-location whereas Heads/Red does not. You’re claiming that because Tails/Red reflects a self-location - i.e the room colour isn’t a random event - no updating is allowed. But you can’t make the same claim about Heads/Red. With Heads, the room being painted red is a random event. For Beauty, the Heads/Red outcome had an unequivocal 1/4 chance of being encountered and it has just been eliminated by her seeing the colour blue. So how can Heads not be halved?
Suppose Beauty had her eyes closed with the same set up. Suppose, before she opens her eyes, she must be told straight away whether she’s in a Heads/Red awakening. It’s confirmed that she’s not. You’d agree that the probability of Heads is now halved while for Tails it isn’t. No problem updating. It’s 1/3 heads 2/3 Tails. Next, suppose she must be told whether she’s in a Tails/Tuesday awakening. It’s confirmed that she’s not. By your reasoning, this would not reduce the probability of Tails. Moreover, Heads has already been halved and there is no reason to change it back. Therefore, if self-locations are un-updatable, her probability must still be 1/3 for Heads and 2/3 for Tails. What’s more, the information she just received is the same as she would have got by opening her eyes and seeing blue.
The only logical reason the coin outcome is still 1/2, after she sees either colour, is because both Heads and Tails must both get halved. This means ignorance or information about self-location status, such as Monday/Tuesday or Original/Clone, are subject to the normal rules of probability and conditionalisation.
There’s a general consensus that, although quantum theory has changed our understanding of reality, Newtonian physics remains a reliable short term guide to the macro world. In principle, the vast majority of macro events that are just about to happen are thought to be 99.9999% inevitable, as opposed to 100% like Newton thought. From that I deduce that if a coin is shortly to be flipped, the outcome is unknown but, is as good as determined as makes no odds. Whereas if a coin is flipped farther into the future from a point of prediction, the outcome is proportionately more likely to be undetermined.
I’m willing to concede debate about this. What I do recognise is that Beauty’s answer of 2/3 Heads, after she learns it’s Monday, depends on it being an already certain but unknown outcome. Whereas if the equivalent of a quantum coin were to be flipped on Monday night, this makes a difference. In that case, awaking on Monday morning, Beauty would not yet be in a Heads world or a Tails world. Her answer would certainly be 1/2 , after she learns it’s Monday. What it would be before she learns it’s Monday would depend on what quantum theory model is used. I can consider this another time.
Perspective disagreement between interacting parties, as a result of someone having more than one possible self-locating identity, is something I can certainly see a reason for. Invalidating someone’s likelihood of what that identity might be, I can't find a reason for. I’ve looked hard.
I’d like to explore your simplified experiment. First it’s important to distinguish precisely what happens with Heads to the version of me that is not woken during the experiment. If the other me is woken after the experiment and told this fact, then there’s no controversy. On finding myself awake in the experiment, my answer is definitely 1/3 for Heads and 2/3 for Tails. Furthermore, it should makes no difference which version might have woken inside the experiment and which outside, assuming the coin landed Heads. Nor does it matter if that potential selection was made before the flip and I’m subsequently told what the choice was. I’d argue that this information about my possible identity is irrelevant to my credence for the coin.
This takes us to a controversy at the heart of anthropic debate. In the event of Heads, if the version of me that is not woken in the experiment never wakes up at all, it becomes like standard Sleeping Beauty and the answer is 1/2 for Heads or Tails. This is because all awakenings will now be inside the experiment and at least one awakening is guaranteed. Regardless of identity, my mind was certain to continue, so long as at least one version woke up. Whether it’s the original or clone, either share the same memories and there is no qualitative difference for the guaranteed continuity of my consciousness. All that matters is that there is no possible experience outside the experiment.
Even if it is an uncertain event as to which body woke up, that uncertainty doesn’t apply to my mind. This was guaranteed to carry on in whichever body it found itself. For the unconscious body that never wakes up, no mind is present. If that body was the original, it’s former mind now continues in the clone body, complete with memories. There is no qualitative difference if I continue in my original body or if I continue as the clone. In terms of actual consciousness, my primitive self has no greater or lesser claim to identical memories of my past, because of the body I have. For some, this will be controversial.
It’s also irrelevant whether the potential sole awakening of original or clone was decided before the flip or whether I’m told what the choice was. Would you actually claim it’s 1/3 for Heads providing that, in the event of that outcome, you know don’t whether you woke as the original or clone? However, if you learn what the potential Heads selection was – regardless of whether this turns out to be original or clone – Heads goes up to 1/2? We've touched on this before. It wouldn’t be a perspective disagreement with a third party. It would be a perspective disagreement with yourself.