Hello Marc,

Appreciate the reply. Allow me to explain why the perceived ambiguity is in my opinion caused by mixing of perspectives. When we try to analysis an anthropic related problem, such as sleeping beauty or an equivalent cloning experiment, there are actually two school of thoughts. One is to reason from the participant's first person perspective. The other is to reason purely objectively as an impartial observer (third-person perspective).

Reasoning from the participant's first-person perspective makes concepts such as I, now or here primitively understood. These indexical concepts inherently standout to the first-person as they are defined by their subjective immediacy to the perspective center. (an everyday example would be an identical twin can inherently distinguish himself from his brother. He can do this without knowing any difference between them which is impossible for an impartial outsider.) At the same time, reasoning from the first-person perspective would affirm self-existence/consciousness. It is necessarily true that from first-person perspective "I am, here, now." (cogito ergo sum)

If reasoned objectively then no agent, time or place would be inherently unique. It is uncentered. So technically from the third-person perspective there is no I now or here. e.g. to objectively specify an agent instead of using the indexical I some feature such like the proper name dadadarren have to be used. (In everyday language we tend to use indexical terms even when trying to reason objectively. In such instances words such as I should not be regarded as indexicals pointing to a perspective center but a conventional shorthand representing various objective features defining an agent.) From this objective third-person perspective no agent's existence/consciousness is guaranteed. It is logical to ask about any agent's probability of existence. Furthermore, from this perspective all agents/times are logical equals. So a principle of indifference can be applied and they can belong to the same reference class.

Now if we are to ask the probability density function of today being Monday or Tuesday, or the pdf of I being the original or clone, problems arise. At one end, to use the indexicals I and now to specify an agent or date requires we take the participant's first-person perspective. They cannot be used to specify a particular agent or time if we are reasoning objectively as an impartial observer since these indexicals have no objective significance. This is why using these terms in anthropic problems seems ambiguous. On the other hand if we reason from the first-person perspective of a participant, yes the meaning of I or now are clear. Yet a sample space containing all possible agents (the original and the clone) or all times (Monday and Tuesday) cannot be constructed. Because from a participant's first-person perspective all perspective centres are not logical equals. A principle of indifference among the two days was thrown out of the window the minute a particular day is regarded as inherently special such that it can be specified not by its objective difference from other days but by a simple utterance of of the word "today". Therefore even though it is correct to say that I am either the original or the clone it is impossible to put a probability on either alternative. This is also backed by frequentist analysis. Which btw, if a consistent perspective is used the long run frequency of Heads should be 1/2.

When I ask my coworker "what day is it today?" The today is still primitively defined from the first-person perspective by both me and my coworker. I don't think it is a shared concept. It is just that the time taken of our communication is minuscule compare to the duration of interest (day) such that one's perspective center can be used to approximate the other's perspective center without causing problems in communication. If instead you find a message in a bottle on the beach which says "what day is it today?" Then this question becomes impossible to answer. Since you would have no idea what this today refers to.

Since my entire position is that first-person (centered) and third-person (uncentered) reasoning should not mix I cannot agree with your mirror argument. Although I would say due to its simplicity (this is a compliment) thirders would have a hard time countering it. Yet the current reality of the Sleeping Beauty discussion is less of finding mistakes in other’s argument but more of whose position have less undesirable consequences. So I won’t be surprised if the mirror argument fail to convince many thirders. In your paper you seems to agree with David Lewis that if the coin toss has already happened and beauty is told it is Monday then the probability of Heads should be rightfully 2/3. Only when the coin toss is yet to happen then the probability shall be 1/2. That the time of coin toss has a material influence on the probability. If this is the correct understanding then I feel the argument is quite problematic. I don’t think there is any logical significance to the physical coin toss. You argued the toss is a chancy event whose timing could affect probability. Yet it can also be deemed as a deterministic event: as long as one has the detailed information of the various variables, such as magnitude and direction of force and air resistance and impact surface shape etc, the result can be readily predicted. The randomness could be interpreted as entirely due to the lack of information. Whether it is a truly random event or a pseudo one shouldn’t affect the probability calculation. Yet this version of double-halving argument depends on that.