The standard textbook definition of a proposition is this:
A proposition is a sentence that is either true or false. If a proposition is true, we say that its truth value is “true,” and if a proposition is false, we say that its truth value is “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}.
As for the rest of your argument, my request is this: show me the math. That is, define the joint probability distribution describing what Beauty knows on Sunday night, and tell me what additional information she has after awakening on Monday/Tuesday. As I argued in the OP, purely verbal arguments are suspect when it comes to probability problems; it’s too easy to miss something subtle.
BTW, in one place you say “if the two coins are not both showing Heads,” and in another you say “if the two coins show the same face”; which is the one you intended?
(Sorry about the typo—I waffled between several isomorphic versions. The one I ultimately chose should have “both showed Heads.”)
In the OP, you said:
Another serious error in many discussions of this problem is the use of supposedly mutually exclusive “propositions” that are neither mutually exclusive nor actually legitimate propositions. HM, TM, and TT can be written as
HM=H and (it is Monday)TM=(not H) and (it is Monday)TT=(not H) and (it is Tuesday).
These are not truly mutually exclusive because, if not H, then Beauty will awaken on both Monday and Tuesday.
Now you say:
A proposition is a sentence that is either true or false.
Are you really claiming that the statement “today is Monday” is not a sentence that is either true or false? That it is not “mutually exclusive” with “today is Tuesday”? 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? That she is asked to evaluate it at the current moment, and not over the entire time frame of the experiment?
Let me insert an example here, to illustrate the problem with your assertion about functions. One half of a hidden, spinning disk is white; the other, black. It spins at a constant rate V, but you don’t know its position at any previous time. There is a sensor aligned along its rim that can detect the color at the point in time when you press a button. 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. It is valid because it doesn’t ask you to evaluate the proposition at every time, but at a fixed point in time.
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}.
It does have a simple true/false truth value if you are asked to evaluate it at fixed point in time. Your assertion applies to functions where every value of the dependent variable are considered to be “true” simultaneously.
I did give you the math, but I’ll repeat it in a slightly different form. Consider the point in time just before (A) in my version, when Beauty is awake and could be interviewed, or (B) in yours, when Beauty could be awakened. At this point in time, there are two valid-by-your-definition propositions: H, the proposition that “the coin lands Heads” and M, the proposition that “today is Monday.” Each is asking about a specific moment in time, so your unsupported assertion that we need to consider all possible values of the time parameter is wrong. The two propositions are independent, because at the moment in time where I asked you to evaluate it, H does not influence M.
The sample space (the set set of possible outcomes described by {H,M}) is {{t,t},{t,f},{f,t},{f,f}}. The probability distribution for this sample space is {1/4,1/4,1/4,1/4}. If Beauty is (A) interviewed or (B) awakened, she knows that the outcome that applies to the current moment in time is not {t,f}. So the probability distribution can be updated to {1/3,0,1/3/,1/3}.
+++++
The error that halfer’s make, is considering all values of the time parameter to applicable when Beauty is asked to make an assessment at a single, unknown time.
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”.
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.
It most certainly does. 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.
A better random variable might be D, which takes values in {0,1,2,3} for these four days. What you refuse to deal with, is that its uninformed distribution depends on the stage of the experiment: {1,0,0,0} when she knows it is Sunday, {0,1/2,1/2,0} when she is awakened but not told the experiment is over, and {0,0,0,1} when she is told it is over.
Or you could just recognize that the probability space when she awakes is not derived by removing outcomes from Sunday’s. Which is how conventional problems in conditional probability work. That a new element of randomness is introduced by the procedures you use in steps 2 and 3.
To illustrate this without obfuscation, ignore the amnesia part. Wake Beauty just once. It can happen any day during the rest of the week, as determined by a roll of a six-sided die. When she is awake, “Die lands 3” is just as valid a proposition—in fact, the same proposition—as “today is Wednesday.” It has probability 1⁄6.
If you add in the amnesia drug, and roll two dice (re-rolling if you get doubles so that you wake her on two random days), the probability for “a die lands 3” is 1⁄3, but for “today is Wednesday” it is 1⁄6.
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.
The proposition “coin lands heads” is sometimes true, and sometimes false, as well. In fact, you have difficulty expressing the tense of the statement for that very reason.
But, it is a function of the parameters that define how you flip a coin: start position, force applied, etc. What you refuse to deal with, is that in this odd experiment, the time parameter Day is also one of the independent parameters that defines the randomness of Beauty’s situation, and not one that makes Monday’s state predicated on Sunday’s.
It is not a fixed moment in time; if it were, the SB problem would be trivial and nobody would write papers about it.
By being asked about the proposition H, Beauty knows that she is in either step 2 or step 3 of your experiment. This establishes a fixed value of the time parameter Day. And the problem is trivial—people write papers about it because they don’t understand how Day is an independent parameter that defines the randomness of the situation, and not one that predicates one state on another.
The sentence “the sensor detects white” is not a valid proposition.
Then “Coin lands heads” is similarly a predicate, and so not a valid proposition.
But your argument about being a predicate, and not a valid proposition, does apply to the statement “It is the 9 o’clock hour.” Because “hour” it is not a parameter you use to define the randomness of the situation.
+++++
Here’s some simple questions for you, to illustrate how randomness is being defined. Write the four labels {”Heads,Monday”, “Tails,Monday”,”Heads,Tuesday”, “Tails,Tuesday”} on four cards. Deal one at random to Beauty. Change step 3 of your experiment so that the day and coin result it mentions, are those on the dealt card. Change step 2 so that it mentions the other day. Change the proposition Beauty is asked to evaluate a probability for to “Coin lands on the face written on the dealt card.”
If, on Sunday, she is shown that she was dealt, “Heads,Tuesday,” this is identically your problem.
If, on Sunday, she is shown a different label, does this represent an equivalent problem with the same answer?
If she is not shown the label, does have the same answer? And is it still an equivalent problem?
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.”
The problem is this: it seems propositions, being the objects of belief, cannot in general be spatially and temporally unqualified.
Note the clause “in general.” Any assertion that applies “in general” can have exceptions in specific contexts.
We similarly cannot deduce, in general, that a coin toss which influences the path(s) of an experiment, is a 50:50 proposition when evaluated in the context of only one path.
“In the philosophy of language, an indexical is any expression whose content varies from one context of use to another.”
An awake Beauty is asked about her current assessment of the proposition “The coin will/has landed Heads.” Presumably, she is supposed to answer on the same day. So, while the content of the expression “today” may change with the changing context of the overarching experiment, that context does not change between asking and answering. So this passage is irrelevant.
The problem with indexicals is that they have meanings that may change over the course of the problem being discussed.
And the problem with using this argument on the proposition “Today is Monday,” is that neither the context, nor the meaning, changes within the problem Beauty addresses.
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.
No, it analyzed two specific usages of an indexical, and showed that they represented different propositions. And concluded that, in general, indexicals can represent different propositions. It never said that multiple usages of a time/location word cannot represent the same proposition, or that we can’t define a situation where we know they represent the same proposition.
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.
So my corner bar can post a sign saying “Free Beer Tomorrow,” without ever having to pour free suds. But if it says “Free Beer Today,” they will, because the context of the sign is the same as the context when somebody asks for it. Both are indexicals, but the conditions that would make it ambiguous are removed.
“words will continue to be used in the same way” They do not change meaning within the discussion.
And over the duration of when Beauty considers the meaning of “today,” it does not change.
the same word used at different points in the argument must have the same meaning.
“Today” means the same thing every time Beauty uses it. This is different than saying the truth value of the statement is the same at different points in Beauty’s argument; but it is. She is making a different (but identical) argument on the two days.
“we must avoid words such as… ‘now’, … whose meaning or reference depends on the circumstances of their use.”
Only if those circumstances might change within the scope of their use.
requires that [words] be replaced by words that we can treat as uniform in meaning or reference throughout a discussion.
And throughout Beauty’s discussion of the probability she was asked for, the meaning of “Today” does not change.
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.
[The proposition “today is Monday” is] not a simple, single truth value; that’s a structure built out of truth values.
At any point in the history that Beauty remembers in step 2 of step 3, the proposition has a simple, single truth value. But she cannot determine what it that value is. This is basis for being able to describe its truth value with probabilities.
“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.
In some instances of the experiment, it is true. In others, it is false.
Just like “today is Monday” has the same truth value at any point in the history that Beauty remembers in step 2 of step 3. Your error is in falling to understand that, to an awake Beauty, the “experiment” she sees consists of Sunday and a single day after it. She just doesn’t know which. In her experiment, the proposition “today is Monday” has a simple, single truth value. The truth of “it is Monday” never changes in any point of the scenario she sees after being wakened.
The point you are missing is that Day changes throughout the problem you are analyzing.
And the point I am trying to get across to you is that it cannot change at any point of the problem Beauty is asked to analyze.
The problem that I am analyzing is the problem that Beauty was asked to analyze. Not what an outside observer sees. She was told some details on Sunday, put to sleep, and is now awake on an indeterminate day.
She is asked about a coin that may have been flipped, or has already been flipped, but to her that difference is irrelevant. “Today is Monday” is either true, or false (which means “Today is Tuesday”). She doesn’t know which, but she does know that this truth value cannot change within the scope of the problem as she sees it now.
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.
No, “time” is an indexical. That means that the value of time can change the context of the problem when you consider different values to be part of the same problem. Not that a problem that deals with only one specific value, and so an unchanging context, has that property.
While Beauty is awake, the day does not change. While Beauty is awake, the context of the problem does not change. While Beauty is awake, the other day of the experiment does not exist in her context. So for our problem, this resolves the issue that classical logic has with the word “today.”
The problem with indexicals is that they have meanings that may change over the course of the problem being discussed.
But the meaning of “Today” does not change of the course of the problem Beauty is asked to address. This is different than her not know what that value is.
+++++
And you didn’t answer my questions, about the variable Sleeping Beauty Problem. They really are simple.
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.
“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.
At any point in the history that Beauty remembers when she is in one of those steps, the proposition M, “Today is Monday,” has a simple, single truth value. All day. Either day. If she is in step 2, it is “true.” If she is in step 3, it is “false.”
The properties of “indexicals” that you are misusing apply when, within her current memory state, the value of “today” could change. Not within the context of the overarching experiment.
This has nothing to do with whether she knows what that truth value is. In fact, probability is how we represent the “fundamental ambiguity” that the simple, single truth value belonging to a proposition is unknown to us. If you want to argue this point, I suggest that you try looking for the forest through the trees.
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.
I tell you that I will flip a coin, ask a question, and then repeat the process.
If the question is “What is the probability that the coin is showing Heads?”, and I require an answer before I repeat the flip, then coin’s state has a simple, single truth value that you can represent with a probability.
If the question is “What is the probability that the coin is showing Heads?”, and I require an answer only at after the second flip, the question only applies to the second since it asks about a current state.But it has a simple, single truth value that you can represent with a probability.
If the question is “What is the probability of showing Heads?” then the we have the logical conundrum you describe.
“Showing” is an indexical. It can change over time. But it is only an issue if we refer to it in the context of a range of time where it does change. That’s why indexicals are a problem in general, but maybe not in a specific case.
“Today” is never ill-defined for Beauty.
“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 entirety of the experiment includes Sunday, Wednesday, and two other days. She knows that. The portion that exists in her memory state at the time she is asked to provide an answer consists of Sunday (when she learned it all), which cannot be “Today,” and Today, which has a simple, single value.
I do not understand why you are so insistent on using “propositions” that include indexicals
Because the property that defines an indexical is that it can change over the domain where it is evaluated. Beauty is asked for her answer within a domain where “Today” does not change.
You didn’t answer my questions, about the variable Sleeping Beauty Problem.
They’re irrelevant.
I’ve learned from experience that I need halfers to answer them while they seem irrelevant. Otherwise, they argue that there is a difference, but can’t say what that difference is. Yes, this has happened more than once.
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.
Not yet, but it does tell you that the same answer applies to the original problem, and to the random-card problem.
So use four Beauties. Deal one card to each, but don’t show them. And flip the coin on Sunday (necessary since we need the result on Monday).
In your step 2, bring the three awake volunteers together to discuss their answers. Tell them, truthfully, what they already know: “One of you was dealt card where the coin value matches the flip we performed on Sunday. Two were dealt a card with the opposite coin result. What probability should you assign the propositions that each of you is the one whose card matches?”
There are three possibilities. Each must have the same probability, since they have no information that distinguishes any one from the other. The probabilities must add up to 1.
[Kinda speaking from my experience as a moderator here, but not actually really doing anything super mod-related]: I haven’t been able to follow the details from this conversation, and I apologize for that, but from the outside it does really look like you two are talking past each other. I don’t know what the best way to fix that is, or even whether I am right, but my guess is that it’s better to retire this thread for now and continue some other time. I am also happy to offer some more moderation if either of you requests that.
Also feel free to ignore this and just continue with your discussion, but it seemed better to give you two an out, if either of you feels like you are wasting time but are forced to continue talking for some reason or another.
The standard textbook definition of a proposition is this:
(Adapted from https://www.cs.utexas.edu/~schrum2/cs301k/lec/topic01-propLogic.pdf.)
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}.
As for the rest of your argument, my request is this: show me the math. That is, define the joint probability distribution describing what Beauty knows on Sunday night, and tell me what additional information she has after awakening on Monday/Tuesday. As I argued in the OP, purely verbal arguments are suspect when it comes to probability problems; it’s too easy to miss something subtle.
BTW, in one place you say “if the two coins are not both showing Heads,” and in another you say “if the two coins show the same face”; which is the one you intended?
(Sorry about the typo—I waffled between several isomorphic versions. The one I ultimately chose should have “both showed Heads.”)
In the OP, you said:
Now you say:
Are you really claiming that the statement “today is Monday” is not a sentence that is either true or false? That it is not “mutually exclusive” with “today is Tuesday”? 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? That she is asked to evaluate it at the current moment, and not over the entire time frame of the experiment?
Let me insert an example here, to illustrate the problem with your assertion about functions. One half of a hidden, spinning disk is white; the other, black. It spins at a constant rate V, but you don’t know its position at any previous time. There is a sensor aligned along its rim that can detect the color at the point in time when you press a button. 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. It is valid because it doesn’t ask you to evaluate the proposition at every time, but at a fixed point in time.
It does have a simple true/false truth value if you are asked to evaluate it at fixed point in time. Your assertion applies to functions where every value of the dependent variable are considered to be “true” simultaneously.
I did give you the math, but I’ll repeat it in a slightly different form. Consider the point in time just before (A) in my version, when Beauty is awake and could be interviewed, or (B) in yours, when Beauty could be awakened. At this point in time, there are two valid-by-your-definition propositions: H, the proposition that “the coin lands Heads” and M, the proposition that “today is Monday.” Each is asking about a specific moment in time, so your unsupported assertion that we need to consider all possible values of the time parameter is wrong. The two propositions are independent, because at the moment in time where I asked you to evaluate it, H does not influence M.
The sample space (the set set of possible outcomes described by {H,M}) is {{t,t},{t,f},{f,t},{f,f}}. The probability distribution for this sample space is {1/4,1/4,1/4,1/4}. If Beauty is (A) interviewed or (B) awakened, she knows that the outcome that applies to the current moment in time is not {t,f}. So the probability distribution can be updated to {1/3,0,1/3/,1/3}.
+++++
The error that halfer’s make, is considering all values of the time parameter to applicable when Beauty is asked to make an assessment at a single, unknown time.
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.
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.
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”.
It most certainly does. 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.
A better random variable might be D, which takes values in {0,1,2,3} for these four days. What you refuse to deal with, is that its uninformed distribution depends on the stage of the experiment: {1,0,0,0} when she knows it is Sunday, {0,1/2,1/2,0} when she is awakened but not told the experiment is over, and {0,0,0,1} when she is told it is over.
Or you could just recognize that the probability space when she awakes is not derived by removing outcomes from Sunday’s. Which is how conventional problems in conditional probability work. That a new element of randomness is introduced by the procedures you use in steps 2 and 3.
To illustrate this without obfuscation, ignore the amnesia part. Wake Beauty just once. It can happen any day during the rest of the week, as determined by a roll of a six-sided die. When she is awake, “Die lands 3” is just as valid a proposition—in fact, the same proposition—as “today is Wednesday.” It has probability 1⁄6.
If you add in the amnesia drug, and roll two dice (re-rolling if you get doubles so that you wake her on two random days), the probability for “a die lands 3” is 1⁄3, but for “today is Wednesday” it is 1⁄6.
The proposition “coin lands heads” is sometimes true, and sometimes false, as well. In fact, you have difficulty expressing the tense of the statement for that very reason.
But, it is a function of the parameters that define how you flip a coin: start position, force applied, etc. What you refuse to deal with, is that in this odd experiment, the time parameter Day is also one of the independent parameters that defines the randomness of Beauty’s situation, and not one that makes Monday’s state predicated on Sunday’s.
By being asked about the proposition H, Beauty knows that she is in either step 2 or step 3 of your experiment. This establishes a fixed value of the time parameter Day. And the problem is trivial—people write papers about it because they don’t understand how Day is an independent parameter that defines the randomness of the situation, and not one that predicates one state on another.
Then “Coin lands heads” is similarly a predicate, and so not a valid proposition.
But your argument about being a predicate, and not a valid proposition, does apply to the statement “It is the 9 o’clock hour.” Because “hour” it is not a parameter you use to define the randomness of the situation.
+++++
Here’s some simple questions for you, to illustrate how randomness is being defined. Write the four labels {”Heads,Monday”, “Tails,Monday”,”Heads,Tuesday”, “Tails,Tuesday”} on four cards. Deal one at random to Beauty. Change step 3 of your experiment so that the day and coin result it mentions, are those on the dealt card. Change step 2 so that it mentions the other day. Change the proposition Beauty is asked to evaluate a probability for to “Coin lands on the face written on the dealt card.”
If, on Sunday, she is shown that she was dealt, “Heads,Tuesday,” this is identically your problem.
If, on Sunday, she is shown a different label, does this represent an equivalent problem with the same answer?
If she is not shown the label, does have the same answer? And is it still an equivalent problem?
That’s not a simple, single truth value; that’s a structure built out of truth values.
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.
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:
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:
(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:
The explanation is given on the previous page (p. 3):
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.”
(Not in order)
Note the clause “in general.” Any assertion that applies “in general” can have exceptions in specific contexts.
We similarly cannot deduce, in general, that a coin toss which influences the path(s) of an experiment, is a 50:50 proposition when evaluated in the context of only one path.
An awake Beauty is asked about her current assessment of the proposition “The coin will/has landed Heads.” Presumably, she is supposed to answer on the same day. So, while the content of the expression “today” may change with the changing context of the overarching experiment, that context does not change between asking and answering. So this passage is irrelevant.
And the problem with using this argument on the proposition “Today is Monday,” is that neither the context, nor the meaning, changes within the problem Beauty addresses.
No, it analyzed two specific usages of an indexical, and showed that they represented different propositions. And concluded that, in general, indexicals can represent different propositions. It never said that multiple usages of a time/location word cannot represent the same proposition, or that we can’t define a situation where we know they represent the same proposition.
So my corner bar can post a sign saying “Free Beer Tomorrow,” without ever having to pour free suds. But if it says “Free Beer Today,” they will, because the context of the sign is the same as the context when somebody asks for it. Both are indexicals, but the conditions that would make it ambiguous are removed.
And over the duration of when Beauty considers the meaning of “today,” it does not change.
“Today” means the same thing every time Beauty uses it. This is different than saying the truth value of the statement is the same at different points in Beauty’s argument; but it is. She is making a different (but identical) argument on the two days.
Only if those circumstances might change within the scope of their use.
And throughout Beauty’s discussion of the probability she was asked for, the meaning of “Today” does not change.
Now you’re really stretching.
That duration potentially includes both Monday and Tuesday.
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.
At any point in the history that Beauty remembers in step 2 of step 3, the proposition has a simple, single truth value. But she cannot determine what it that value is. This is basis for being able to describe its truth value with probabilities.
In some instances of the experiment, it is true. In others, it is false.
Just like “today is Monday” has the same truth value at any point in the history that Beauty remembers in step 2 of step 3. Your error is in falling to understand that, to an awake Beauty, the “experiment” she sees consists of Sunday and a single day after it. She just doesn’t know which. In her experiment, the proposition “today is Monday” has a simple, single truth value. The truth of “it is Monday” never changes in any point of the scenario she sees after being wakened.
And the point I am trying to get across to you is that it cannot change at any point of the problem Beauty is asked to analyze.
The problem that I am analyzing is the problem that Beauty was asked to analyze. Not what an outside observer sees. She was told some details on Sunday, put to sleep, and is now awake on an indeterminate day.
She is asked about a coin that may have been flipped, or has already been flipped, but to her that difference is irrelevant. “Today is Monday” is either true, or false (which means “Today is Tuesday”). She doesn’t know which, but she does know that this truth value cannot change within the scope of the problem as she sees it now.
No, “time” is an indexical. That means that the value of time can change the context of the problem when you consider different values to be part of the same problem. Not that a problem that deals with only one specific value, and so an unchanging context, has that property.
While Beauty is awake, the day does not change. While Beauty is awake, the context of the problem does not change. While Beauty is awake, the other day of the experiment does not exist in her context. So for our problem, this resolves the issue that classical logic has with the word “today.”
But the meaning of “Today” does not change of the course of the problem Beauty is asked to address. This is different than her not know what that value is.
+++++
And you didn’t answer my questions, about the variable Sleeping Beauty Problem. They really are simple.
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.
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.
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.
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.
Go read the quotes again. Are you a greater authority on this subject than the authors of the Stanford Encyclopedia of Philosphy?
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.
At any point in the history that Beauty remembers when she is in one of those steps, the proposition M, “Today is Monday,” has a simple, single truth value. All day. Either day. If she is in step 2, it is “true.” If she is in step 3, it is “false.”
The properties of “indexicals” that you are misusing apply when, within her current memory state, the value of “today” could change. Not within the context of the overarching experiment.
This has nothing to do with whether she knows what that truth value is. In fact, probability is how we represent the “fundamental ambiguity” that the simple, single truth value belonging to a proposition is unknown to us. If you want to argue this point, I suggest that you try looking for the forest through the trees.
I tell you that I will flip a coin, ask a question, and then repeat the process.
If the question is “What is the probability that the coin is showing Heads?”, and I require an answer before I repeat the flip, then coin’s state has a simple, single truth value that you can represent with a probability.
If the question is “What is the probability that the coin is showing Heads?”, and I require an answer only at after the second flip, the question only applies to the second since it asks about a current state.But it has a simple, single truth value that you can represent with a probability.
If the question is “What is the probability of showing Heads?” then the we have the logical conundrum you describe.
“Showing” is an indexical. It can change over time. But it is only an issue if we refer to it in the context of a range of time where it does change. That’s why indexicals are a problem in general, but maybe not in a specific case.
“Today” is never ill-defined for Beauty.
The entirety of the experiment includes Sunday, Wednesday, and two other days. She knows that. The portion that exists in her memory state at the time she is asked to provide an answer consists of Sunday (when she learned it all), which cannot be “Today,” and Today, which has a simple, single value.
Because the property that defines an indexical is that it can change over the domain where it is evaluated. Beauty is asked for her answer within a domain where “Today” does not change.
I’ve learned from experience that I need halfers to answer them while they seem irrelevant. Otherwise, they argue that there is a difference, but can’t say what that difference is. Yes, this has happened more than once.
Not yet, but it does tell you that the same answer applies to the original problem, and to the random-card problem.
So use four Beauties. Deal one card to each, but don’t show them. And flip the coin on Sunday (necessary since we need the result on Monday).
In your step 2, bring the three awake volunteers together to discuss their answers. Tell them, truthfully, what they already know: “One of you was dealt card where the coin value matches the flip we performed on Sunday. Two were dealt a card with the opposite coin result. What probability should you assign the propositions that each of you is the one whose card matches?”
There are three possibilities. Each must have the same probability, since they have no information that distinguishes any one from the other. The probabilities must add up to 1.
They are all 1⁄3.
[Kinda speaking from my experience as a moderator here, but not actually really doing anything super mod-related]: I haven’t been able to follow the details from this conversation, and I apologize for that, but from the outside it does really look like you two are talking past each other. I don’t know what the best way to fix that is, or even whether I am right, but my guess is that it’s better to retire this thread for now and continue some other time. I am also happy to offer some more moderation if either of you requests that.
Also feel free to ignore this and just continue with your discussion, but it seemed better to give you two an out, if either of you feels like you are wasting time but are forced to continue talking for some reason or another.