ksvanhorn

Karma: 516

ksvanhorn Jul 3, 2018, 2:02 AM
1 point
in reply to: AlexMennen’s comment on: Sleeping Beauty Resolved (?) Pt. 2: Identity and Betting
But the development of probability theory and the way that it is applied in practice were guided by implicit assumptions about observers.
I don’t think that’s true, but even if it is an accurate description of the history, that’s irrelevant—we have justifications for probability theory that make no assumptions whatsoever about observers.
You seemed to argue in your first post that selection effects were not routinely handled within standard probability theory.
No, I argued that this isn’t a case of selection effects.
Certainly agreed as to logic (which does not include probability theory).
Why are you ignoring what I wrote about proofs that probability theory is either a or the uniquely determined extension of classical propositional logic to handle degrees of certainty? That places probability theory squarely in the logical camp. It is a logic.
in which we made certain implicit assumptions about observers
No, we made no such implicit assumptions. There are no assumptions, implicit or otherwise, about observers at all. If you think otherwise, show me where they occur in Cox’s Theorem or in my theorem.
I’m going to wait to address that until you clarify what you mean by applying standard probability theory, since you offered a fairly narrow view of what this means in your original post, and seemed to contradict it in your point 3 in the comment.
I have no idea what you’re talking about here.
My position is that “the information available” should not be interpreted as simply the existence of at least one agent making the same observations you are, while declining to make any inferences at all about the number of such agents (beyond that it is at least 1).
Um, there’s only one agent here, but if by “agent” you mean the pair (person, day), then the above is just wrong—it’s very clearly part of the model that if the coin comes up Heads, there is exactly one day on which the remembered observations could be made, and if the coin comes up Tails, there are exactly two days on which the remembered observations could be made. I even worked out the probabilities that the observations occurred on just Monday, just Tuesday, or both Monday and Tuesday.
Listen, if you want to argue against my analysis, you need to
1. Propose a different model of what Beauty knows on Sunday, and/or
2. Propose a different proposition that expresses the additional information Beauty has on Monday/Tuesday and that accounts for her altered probabilities. This proposition should be possible to sensibly state and talk about on Sunday, Monday, Tuesday, or Wednesday, by either Beauty or one of the experimenters, and mean the same thing in all these cases.

ksvanhorn Jul 3, 2018, 1:41 AM
1 point
in reply to: Chris_Leong’s comment on: Sleeping Beauty Resolved?
When Sleeping Beauty wakes up and observes a sequence, they are learning that this sequence occurs on a on a random day out of those days when they are awake.
That would be a valid description if she were awakened only on one day, with that day chosen through some unpredictable process. That is not the case here, though.
What you’re doing here is sneaking in an indexical—“today” is either Monday if Heads, and “today” is either Monday or Tuesday if Tails. See Part 2 for a discussion of this issue. To the extent that indexicals are ambiguous, they cannot be used in classical propositions. The only way to show that they are unambiguous is to show that there is an equivalent way of expressing that same thing that doesn’t use any indexical, and only uses well-defined entities—in which case you might as well use the equivalent expression that has no indexical.

ksvanhorn Jul 3, 2018, 1:33 AM
1 point
in reply to: Lukas Finnveden’s comment on: Sleeping Beauty Resolved?
Yes, that is shown in Part 2.

ksvanhorn Jul 2, 2018, 11:55 PM
1 point
in reply to: ryan_b’s comment on: Book Review: Why Honor Matters
From the OP: “honor requires recognition from others.” That’s not a component of the notion of honor I grew up with. Nor is the requirement of avenging insults.

ksvanhorn Jun 28, 2018, 4:17 PM
7 points
on: Book Review: Why Honor Matters
This is a very, very different concept of honor than the one I grew up with. I was taught that honor means doing what is right (ethical, moral), regardless of personal cost. It meant being unfailingly honest, always keeping your word, doing your duty, etc. How others perceived you was irrelevant. One example of this notion of honor is the case of Sir Thomas More, who was executed by Henry VIII because his conscience would not allow him to cooperate with Henry’s establishment of the Church of England. Another is the Dreyfus Affair and Colonel Georges Picquart, who suffered grave personal consequences for insisting on giving an honest report and refusing to go along with the framing of Alfred Dreyfus for espionage. (There’s a wonderful movie about this, called Prisoner of Honor.)

ksvanhorn Jun 10, 2018, 9:28 PM
4 points
in reply to: AlexMennen’s comment on: Sleeping Beauty Resolved (?) Pt. 2: Identity and Betting
...the standard formalization of probability… was not designed with anthropic reasoning in mind. It is usually taken for granted that the number of copies of you that will be around in the future to observe the results of experiments is fixed at exactly 1, and that there is thus no need to explicitly include observation selection effects in the formalism.
1. Logic, including probability theory, is not observer-dependent. Just as the conclusions one can obtain with classical propositional logic depend only on the information (propositional axioms) available, and not on any characteristic or circumstance of the reasoner, epistemic probabilities also depend only on the information available. Logic—including probability theory—was designed to be fully general. If you want to argue that probability theory is not, in its standard formulation, suitable for anthropic reasoning, you need to point out the specific points in its rationale that are incompatible with anthropic effects. As I have shown (preprint), all you have to assume to get probability theory from classical propositional logic is that certain properties of propositional logic are retained in the extended logic.
2. No, neither classical logic nor probability theory as the extension of classical propositional logic assumes anything about observers, or their numbers, or experiments, or what may happen in the future.
3. Selection effects are routinely handled within the framework of standard probability theory. You don’t need to go beyond standard probability theory for this.

ksvanhorn Jun 10, 2018, 9:01 PM
7 points
in reply to: agilecaveman’s comment on: Sleeping Beauty Resolved?
No, P(H | X2, M) is $Pr (H ∣ X_{2}, M)$ , and not $Pr (H ∣ X_{2}, Monday)$ . Recall that $M$ is the proposed model. If you thought it meant “today is Monday,” I question how closely you read the post you are criticizing.
I find it ironic that you write “Dismissing betting arguments is very reminiscent of dismissing one-boxing in Newcomb’s”—in an earlier version of this blog post I brought up Newcomb myself as an example of why I am skeptical of standard betting arguments (not sure why or how that got dropped.) The point was that standard betting arguments can get the wrong answer in some problems involving unusual circumstances where a more comprehensive decision theory is required (perhaps FDT).
Re constructing rational agents: this is one use of probability theory; it is not “the point”. We can discuss logic from a purely analytical viewpoint without ever bringing decisions and agents into the discussion. Logic and epistemology are legitimate subjects of their own quite apart from decision theory. And probability theory is the unique extension of classical propositional logic to handle intermediate degrees of plausibility.
You say you have read PTLOS and others. Have you read Cox’s actual paper, or any or detailed discussions of it such as Paris’s discussion in The Uncertain Reasoner’s Companion, or my own “Constructing a Logic of Plausible Inference: A Guide to Cox’s Theorem”? If you think that Cox’s Theorem has too many arguable technical requirements, then I invite you to read my paper, “From Propositional Logic to Plausible Reasoning: A Uniqueness Theorem” (preprint here). That proof assumes only that certain existing properties of classical propositional logic be retained when extending the logic to handle degrees of plausibility. It does not assume any particular functional decomposition of plausibilities, nor does it even assume that plausibilities must be real numbers. As with Cox, we end up with the result that the logic must be isomorphic to probability theory. In addition, the theorem gives the required numeric value for a probability $Pr (A ∣ X)$ when $X$ contains, in propositional form, all of the background information we are using to assess the probability of $A$ . How much more “clear cut” do you demand the relationship between logic and probability be?
Regardless, for my argument about indexicals all that is necessary is that probability theory deals with classical propositions.
I responded to David Chapman’s essay (https://meaningness.com/probability-and-logic) a couple of years ago here.

ksvanhorn Jun 10, 2018, 8:19 PM
6 points
in reply to: AlexMennen’s comment on: Sleeping Beauty Resolved?
The point is that the meaning of a classical proposition must not change throughout the scope of the problem being considered. When we write A1, …, An |= P, i.e. “A1 through An together logically imply P”, we do not apply different structures to each of A1, …, An, and P.
The trouble with using “today” in the Sleeping Beauty problem is that the situation under consideration is not limited to a single day; it spans, at a minimum, both Monday and Tuesday, and arguably Sunday and/or Wednesday also. Any properly constructed proposition used in discussing this problem should make sense and be unambiguous regardless of whether Beauty or the experimenters are uttering the proposition, and whether they are uttering it on Sunday, Monday, Tuesday, or Wednesday.

ksvanhorn Jun 9, 2018, 5:51 PM
6 points
in reply to: Jeff Jo’s comment on: Sleeping Beauty Resolved (?) Pt. 2: Identity and Betting
...the details of the experiment do provide context for “today.” But as a random variable, not an explicit value.
You seem to think that “random” variables are special in some way that avoids the problems of indexicals. They are not. When dealing with epistemic probabilities, a “random” variable is any variable whose precise value is not known with complete certainty.
Still, there are ways to avoid using an indexical in a solution. I suggested one in a comment to part 1: use four Beauties, where each is left asleep under a different combination of {Coin,Day}. Three are wakened each day of the experiment. One of those three will be awakened only once during the experiment. They are asked, essentially, “what is the probability that you will be awake only once?” It was agreed that this question is equivalent the original problem.
This is not what I understood you to be proposing. As described here, I would say that this is not the same as the original question, and does not avoid using an indexical. You have simply camouflaged the indexical by omission when you write that “One of those three [who are awake today] will be awakened only once during the experiment.”

ksvanhorn Jun 9, 2018, 5:43 PM
4 points
in reply to: Jeff Jo’s comment on: Sleeping Beauty Resolved (?) Pt. 2: Identity and Betting
The situation with indexicals is similar to the situation with “irrelevant” information. If there is any dispute over whether some information is irrelevant, you condition on it and see if it changes the answer. If it does, the judgment that the information was irrelevant was wrong.
Same thing with indexicals. You may claim that use of an indexical in a proposition is unambiguous. The only way to prove this is to actually remove the ambiguity—replace it with a more explicit statement that lacks indexicals—and see that this doesn’t change anything. So for your burned paper analogy, “today” and “here” are replaced by “the day on which Carl wrote this note” and “the city of origin for the call for which Carl took this note”. For the dice-throwing example, “What is the probability that today is Wednesday?” can be replaced by “What is the probability that the day on Beauty experiences $y$ is Wednesday” because there can only be one such day, in which her last memories before her last sleep were from Sunday.
When we try this for the SB problem, however, a nonzero probability of ambiguity remains. Neal gives one way of removing the ambiguity in terms of information to which Beauty actually has access, that is, her memories and experiences. Doing that leads to an answer that is close to, but not quite the same as, treating “today” as unambiguous. If Beauty has exactly the same experiences $y$ on both Monday and Tuesday, she cannot disambiguate “today”.
That leaves you with a choice: either you must agree that “today” is ambiguous in this problem, or you need to propose a different way of rephrasing the statement “today is Monday” into a form that removes the indexicals and then condition on information Beauty actually has.

ksvanhorn Jun 9, 2018, 4:52 PM
4 points
in reply to: jessicata’s comment on: Sleeping Beauty Resolved (?) Pt. 2: Identity and Betting
You’re right, my argument wasn’t quite right. Thanks for looking into this and fixing it.

ksvanhorn Jun 9, 2018, 4:46 PM
4 points
in reply to: Radford Neal’s comment on: Sleeping Beauty Resolved (?) Pt. 2: Identity and Betting
I think a variation of my approach to resolving the betting argument for SB can also help deal with the very large universe problem. I’ve taken a look at the following setup:
- There are $N$ Experimenters scattered throughout the universe, where $N$ is very, very large. Each Experimenter tries to determine which of two hypotheses $A$ and $B$ about the universe are correct by running some experiment and collection some data. Let $d$ be the data collected, and let $y$ be the remaining information (experiences, memories) that could distinguish this Experimenter from others.
- It is possible to choose $N$ so large that the prior probability approaches one that there will be some Experimenter with that particular $d$ and $y$ , regardless of whether $A$ or $B$ is true. This means that the Experimenter’s posterior probability for $A$ versus $B$ will update only slightly from its prior probability.
- And yet if the Experimenter has to make a choice based on whether $A$ or $B$ is true, and we weight the payoffs according to how many Experimenters there are with the same $y$ and $d$ (as done in my analysis for SB), then the maximum-expected-utility answer does not depend on $N$ : from the standpoint of decision-making, we can ignore the possibility of all those other Experimenters and just assume $N = 1$ .

ksvanhorn Jun 2, 2018, 1:36 AM
2 points
on: Editor Mini-Guide
How do I do things like tables using the WYSIWYG interface? There doesn’t seem to be any way to insert markdown in that interface. And once you’ve already been using WYSIWYG on an article, you can’t really switch to markdown—I tried, and it was a complete mess.

ksvanhorn May 31, 2018, 2:52 AM
1 point
in reply to: Jeff Jo’s comment on: Sleeping Beauty Resolved?
At any point in the history that Beauty remembers in step 2 of step 3, the proposition has a simple, single truth value.
No, it doesn’t. This boils down to a question of identity. Absent any means of uniquely identifying the day—such as, “the day in which a black marble is on the dresser”—there is a fundamental ambiguity. If Beauty’s remembered experiences and mental state are identical at a point in time on Monday and another point in time on Tuesday, then “today” becomes ill-defined for her.
In some instances of the experiment
What instances are you talking about? We’re talking about a single experiment. We’re talking about epistemic probabilities, not frequencies. You need to relinquish your frequentist mindset for this problem, as it’s not a problem about frequentist probabilities.
to an awake Beauty, the “experiment” she sees consists of Sunday and a single day after it.
No, it doesn’t. She knows quite well that if the coin lands Tails, she will awaken on two separate days. It doesn’t matter that she can only remember one of them.
The problem that I am analyzing is the problem that Beauty was asked to analyze. Not what an outside observer sees.
Epistemic probabilities are a function, not of the person, but of the available information. Any other person given the same information must produce the same epistemic probabilities. That’s fundamental.
No, “time” is an indexical.
Go read the quotes again. Are you a greater authority on this subject than the authors of the Stanford Encyclopedia of Philosphy?
you didn’t answer my questions, about the variable Sleeping Beauty Problem.
They’re irrelevant. You added an extra layer of randomness on top of the problem. Each of the four card outcomes leads to a problem equivalent to the first. But randomly choosing one of four problems equivalent to the first problem doesn’t tell you what the solution to the first problem is.
I do not understand why you are so insistent on using “propositions” that include indexicals, especially when there is no need to do so—we can express the information Beauty has in a way that does not involve indexicals. When we do so, we get an answer that is not quite the same as the answer you get when you play fast and loose with indexicals. Since you’ve never been able to point out a flaw in the argument—all you’ve done is presented a different argument you like better—you should consider this evidence that indexicals are, in fact, a problem, just like Epstein and others have said.

ksvanhorn May 31, 2018, 2:26 AM
1 point
in reply to: Jeff Jo’s comment on: Sleeping Beauty Resolved?
Note the clause “in general.”
Now you’re really stretching.
And over the duration of when Beauty considers the meaning of “today,” it does not change.
That duration potentially includes both Monday and Tuesday.
“Today” means the same thing every time Beauty uses it.
This is getting ridiculous. “Today” means a different thing on every different day. That’s why the article lists it as an indexical. Going back to the quote, the “discussion” is not limited to a single day. There are at least two days involved.
I notice you carefully ignored the quote from Epstein’s book, which was very clear that a classical proposition must not contain indexicals.

ksvanhorn May 30, 2018, 2:49 PM
3 points
in reply to: Jeff Jo’s comment on: Sleeping Beauty Resolved?
It is true on Monday when Beauty is awake, and false on Sunday Night, on Tuesday whether or not Beauty is awake, and on Wednesday.
That’s not a simple, single truth value; that’s a structure built out of truth values.
The proposition “coin lands heads” is sometimes true, and sometimes false, as well.
No, it is not. It has the same truth value throughout the entire scenario, Sunday through Wednesday. On Sunday and Monday it is impossible to know what that truth value is, but it is either true that the coin will land heads, or false that it will land heads—and by definition, that is the same truth value you’ll assign after seeing the coin toss. In contrast, the truth of “it is Monday” keeps on changing throughout the scenario. Likewise, the truth of “the sensor detects white” changes throughout the scenario you are considering in your button-and-sensor example.
Day is an independent parameter that defines the randomness of the situation
I don’t know what it means to “define the randomness of the situation.” In any event, the point you are missing is that Day changes throughout the problem you are analyzing—not just that there are different possible values for it, and you don’t know which is the correct one, but at different points in the same problem it has different values.
Things like “today” and “now” are known as indexicals, and there is an entire philosophical literature on them because they are problematic for classical logic. Various special logics have been devised specifically to handle them. It would not have been necessary to devise such alternative logics if they posed no problem for classical logic. You can read about them in the article Demonstratives and Indicatives in The Internet Encyclopedia of Philosophy. Some excerpts:
In the philosophy of language, an indexical is any expression whose content varies from one context of use to another. The standard list of indexicals includes… adverbs such as “now”, “then”, “today”, “yesterday”, “here”, and “actually”...
Contemporary philosophical and linguistic interest in indexicals and demonstratives arises from at least four sources. …(iii) Indexicals and demonstratives raise interesting technical challenges for logicians seeking to provide formal models of correct reasoning in natural language...
The problem with indexicals is that they have meanings that may change over the course of the problem being discussed. This is simply not allowed in classical logic. In classical logic, a proposition must have a stable, unvarying truth value over the entire argument. I’m going to appeal to authority here, and give you some quotes.
Section 3.2, “Meanings of Sentences”, in Propositions, Stanford Encyclopedia of Philosophy:
The problem is this: it seems propositions, being the objects of belief, cannot in general be spatially and temporally unqualified. Suppose that Smith, in London, looks out his window and forms the belief that it is raining. Suppose that Ramirez, in Madrid, relying on yesterday’s weather report, awakens and forms the belief that it is raining, before looking out the window to see sunshine. What Smith believes is true, while what Ramirez believes is false. So they must not believe the same proposition. But if propositions were generally spatially unqualified, they would believe the same proposition. An analogous argument can be given to show that what is believed must not in general be temporally unqualified.
(Emphasis added.) The above is telling us that a “proposition” involving an indexical is not a single proposition, but a set of propositions that you get by specifying a particular time/location.
Classical Mathematical Logic: The Semantic Foundations of Logic, by Richard L. Epstein, is clear that indexicals are not allowed in classical logic. On p. 4, “Exercises for Sections A and B,” one of the exercises is this:
Explain why we cannnot take sentence types as propositions if we allow the use of indexicals in our reasoning.
The explanation is given on the previous page (p. 3):
When we reason together, we assume that words will continue to be used in the same way. That assumption is so embedded in our use of language that it’s hard to think of a word except as a type, that is, as a representative of inscriptions that look the same and utterances that sound the same. I don’t know how to make precise what we mean by ‘look the same’ or ‘sound the same.’ But we know well enough in writing and conversation what it means for two inscriptions or utterances to be equiform.
Words are types. We will assume that throughout any particular discussion equiform words will have the same properties of interest to logic. We therefore identify them and treat them as the same word. Briefly, a word is a type.
This assumption, while useful, rules out many sentences we can and do reason with quite well. Consider ‘Rose rose and picked a rose.’ If words are types, we have to distinguish the three equiform inscriptions in this sentence, perhaps as ‘Rose_{name} rose_{verb} and picked a rose_{noun}’.
Further, if we accept this agreement, we must avoid words such as ‘I’, ‘my’, ‘now’, or ‘this’, whose meaning or reference depends on the circumstances of their use. Such words, called indexicals, play an important role in reasoning, yet our demand that words be types requires that they be replaced by words that we can treat as uniform in meaning or reference throughout a discussion.
...
Propositions are types. In any discussion in which we use logic we’ll consider a sentence to be a proposition only if any other sentence or phrase that is composed of the same words in the same order can be assumed to have the same properties of concern to logic during that discussion. We therefore identify equiform sentences or phrases and treat them as the same sentence. Briefly, a proposition is a type.
Notice the following statements made above:
- “words will continue to be used in the same way” They do not change meaning within the discussion.
- “equiform words will have the same properties of interest to logic” In particular, the same word used at different points in the argument must have the same meaning.
- “we must avoid words such as… ‘now’, … whose meaning or reference depends on the circumstances of their use.”
- “our demand that words be types requires that they be replaced by words that we can treat as uniform in meaning or reference throughout a discussion.”

ksvanhorn May 29, 2018, 4:29 AM
1 point
in reply to: Jeff Jo’s comment on: Sleeping Beauty Resolved?
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?

ksvanhorn May 29, 2018, 3:07 AM
3 points
in reply to: Jeff Jo’s comment on: Sleeping Beauty Resolved?
Your whole analysis rests on the idea that “it is Monday” is a legitimate proposition. I’ve responded to this many other places in the comments, so I’ll just say here that a legitimate proposition needs to maintain the same truth value throughout the entire analysis (Sunday, Monday, Tuesday, and Wednesday). Otherwise it’s a predicate. The point of introducing R(y,d) is that it’s as close as we can get to what you want “it is Monday” to mean.

ksvanhorn May 29, 2018, 2:25 AM
3 points
in reply to: Jeff Jo’s comment on: Sleeping Beauty Resolved?
Are you really claiming that the statement “today is Monday” is not a sentence that is either true or false?
Yes. It does not have a simple true/false truth value. Since it is sometimes true and sometimes false, its truth value is a function from time to {true, false}. That makes it a predicate, not a proposition.
Or are you simply ignoring the fact that the frame of reference, within which Beauty is asked to assess the proposition “The coin lands Heads,” is a fixed moment in time?
It is not a fixed moment in time; if it were, the SB problem would be trivial and nobody would write papers about it. The questions about day of week and outcome of coin toss are potentially asked both on Monday and on Tuesday. This makes the rest of your analysis invalid. You keep on asserting that “today is Monday” is evaluated at a fixed moment in time, when in reality it is evaluated at at least two separate moments in time with different answers.
You are asked to assess the probability of the proposition W, that the sensor will detect “white” when you first press the button. This is a valid proposition, even though it varies with time.
The sentence “the sensor detects white” is not a valid proposition; it is a predicate, because it is a function of time. Let’s write $P (t)$ for this predicate. But yes, the sentence “the sensor detects white when you first press the button” is a legitimate proposition, precisely because specifies a particular time $t$ for which $P (t)$ is true, and so the truth value of the statement itself does not vary with time.
This gets us to the whole point of defining $R (y, d)$ : saying “Beauty has a stream of experiences $y$ on day $d$ ” is as close as we can get to identifying a specific moment in time corresponding to the “this” in “this is day $d$ ”. The more nearly that $y$ uniquely identifies the day, the more nearly that $R (y, d)$ can be interpreted to mean “this is day $d$ ”.

ksvanhorn May 29, 2018, 1:42 AM
3 points
in reply to: agilecaveman’s comment on: Sleeping Beauty Resolved?
By bayes rule, Pr (H | M) * Pr(X2 |H, M) / Pr(X2 |M) = Pr(H∣X2, M), which is not the same quantity you claimed to compute Pr(H∣X2).
That’s a typo. I meant to write $Pr (H ∣ X_{2}, M)$ , not $Pr (H ∣ X_{2})$ .
Second, the dismissal of betting arguments is strange.
I’ll have more to say soon about what I think is the correct betting argument. Until then, see my comment in reply to Radford Neal about disagreement on how to apply betting arguments to this problem.
“probability theory is logically prior to decision theory.” Yes, this is the common view because probability theory was developed first and is easier but it’s not actually obvious this *has* to be the case.
I said logically prior, not chronologically prior. You cannot have decision theory without probability theory—the former is necessarily based on the latter. In contrast, probability theory requires no reference to decision theory for its justification and development. Have you read any of the literature on how probability theory is either an or the uniquely determined extension of propositional logic to handle degrees of certainty? If not, see my references. Neither Cox’s Theorem nor my theorem rely on any form of decision theory.
Third, dismissal of “not H and it’s Tuesday” as not propositions doesn’t make sense. Classical logic encodes arbitrary statements within AND and OR -type constructions. There isn’t a whole lot of restrictions on them.
I’ll repeat my response to Jeff Jo: The standard textbook definition of a proposition is a sentence that has a truth value of either true or false. The problem with a statement whose truth varies with time is that it does not have a simple true/false truth value; instead, its truth value is a function from time to the set {true,false}. In logical terms, such a statement is a predicate, not a proposition. For example, “Today is Monday” corresponds to the predicate $P (t) ≜ (d a y o f (t) = M o n d a y)$ . It doesn’t become a proposition until you substitute in a specific value for $t$ , e.g. “Unix timestamp 1527556491 is a Monday.”
The paradox still stands for the moment when you wake up
You have not considered the possibility that the usual decision analysis applied to this problem is wrong. There is, in fact, disagreement as to what the correct decision analysis is. I will be writing more on this in a future post.
You seem to simply declare [Beauty’s probability at the moment of awakening] to be ½, by saying:
The prior for H is even odds: Pr(H∣M)=Pr(¬H∣M)=1/2.
This is generally indistinguishable from the ½ position you dismiss that argues for that prior on the basis of “no new information.”
In fact, I explicitly said that at the instant of awakening, Beauty’s probability is the same as the prior, because at that point she does not yet have any new information. As she receives sensory input, her probability for Heads decreases asymptotically to ¹⁄₂. All of this is just standard probability theory, conditioning on the new information as it arrives. I dismissed the naive halfer position because it incorrectly assumes that Beauty’s sensory input is irrelevant to the determination of her probability for Heads.
You still don’t know how to handle the situation of being told that it’s Monday and needing to update your probability accordingly,
Uh, yes I do—it’s just standard probability theory again. Just do the math. If Beauty finds out that it is Monday, her new information since Sunday changes from $(R (y, M) or R (y, T))$ to just $R (y, M)$ , and since the problem definition assumes that
$Pr (R (y, M) ∣ H, M) = Pr (R (y, M) ∣ not H, M)$
we get equal posterior probabilities for $H$ and $not H$ , which is generally accepted to be the right answer.