Here is yet another variation of the problem that I think perfectly identifies the source of the controversy. The experiment's methodology is the same as the original, except in these four details:
(1) Two coins, a Nickel and a Quarter, are flipped on Sunday Night.
(2) On either day of the experiment, Beauty is wakened if either of the two coins is showing Tails.
(3) On Monday Night, while Beauty is asleep, the Nickel is flipped over to show its opposite face.
(4) Beauty is asked the same question, but about the Quarter.
The only functional difference is t...
Well, I never checked back to see replies, and just tripped back across this.
The error made by halfers is in thinking "the entire analysis" spans four days. Beauty is asked for her assessment, based on her current state of knowledge, that the coin landed Heads. In this state of knowledge, the truth value of the proposition "it is Monday" does not change.
But there is another easy way to find the answer, that satisfies your criterion. Use four Beauties to create an isomorphic problem. Each will be told all of the details on Sunday; that e...
You said: "The standard textbook definition of a proposition is a sentence that has a truth value of either true or false.
This is correct. And when a well-defined truth value is not known to an observer, the standard textbook definition of a probability (or confidence) for the proposition, is that there is a probability P that it is "true" and a probability 1-P that it is "false."
For example, if I flip a coin but keep it hidden from you, the statement "The coin shows Heads on the face-up side" fits your definition of a pr...
And the purpose of a thought experiment, is to define how ideal concepts work when you can't run them in principle. And strawman arguments do not change that.
She is allowed any reasoning she wants to use. The condition explicitly stated in the thought problem (see https://en.wikipedia.org/wiki/Thought_experiment, for why we shouldn't care about realism) is that experiences during the day will not help her to deduce what day it is, not that she can't use it to determine her initial belief about the day or the coin.
What this means, is that if Xi represents her ordered experiences, with X0 representing only the experience of waking up as defined by the experiment, that Pr(Today=Monday|Xi+1) = Pr(Today=Mo...
"You're failing to distinguish between though experiments that are only mildly-fantastic, like ones assuming perfectly fair coins, when real ones have (say) a 50.01% chance of landing heads, versus highly-fantastic thought experiments, such as ones assuming that on Sunday you know exactly, in complete detail, what all your experiences will be on Monday."
I'm not failing to distinguish anything. I'm intentionally not bothering to distinguish what the problem statement says we should treat as indistinguishable. "While awake...
"[Elga and Lewis] don't realize that that is relevant information. They're mistaken." They are not. The very premise of the problem is that it cannot be relevant. The same reasoning suggests we don't need to accept that the coin is fair, or that Beauty might wake on Tuesday after Heads.
"Surely you would agree that a thought experiment is uninteresting if the conditions for it are actually impossible?" I absolutely would not. There is no coin, or a methodology for flipping one, that produces exactly 50%. In fact, if we cou...
It is consensus on how one uses experiences as evidence, not the usage itself, that is only possible if the method is uncontroversial. Controversy just means that two people see its applicability differently. Not that it is impossible to use, or that either is correct or incorrect.
But neither Lewis, nor Elga, say anything about using Beauty's experiences during the day of an awakening as evidence. Elga's footnote is defining what he means by "new information," which we are calling "evidence." He never relates it to experiences...
Lewis says that the evidence that it is Monday or Tuesday is identical, not the totality of her thoughts and experiences is identical. A window and rain on only one of the days constitutes different experiences, but requires knowledge of the weather forecast to extrapolate that difference into evidence.
The context you omitted from the Elga quote was comparing Sunday's knowledge to Monday's, with no mention of Tuesday. He even added a footnote: "To say that an agent receives new information (as I shall use that expression) is to say that the ...
You said "Discussion of Sleeping Beauty often slides [into a] problem in which Beauty's thoughts and experiences on Tuesday (if she is woken then) are ABSOLUTELY IDENTICAL to her thoughts and experiences on Monday." No, they don't. They assume that nothing in the set of experiences can change the assessment of what day it currently is. But that, of course, requires one to recognize that "today" is a valid random variable.
Carl works Monday through Friday in the European Rain Recording Society in Berlin. He records daily data from two field agents: Colin in London, and Carlos in Madrid.
On Sunday Night, the temporary janitor in his office is careless with his cigarette, and accidentally sets fire to some papers on Carl's desk. Most are totally destroyed; just the bottom half of one piece of paper remains. It says "It rained here today." Knowing nothing of the work that is performed in the office, he thinks this is a very odd message. "Today" and "...
"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 ste...
(Not in order)
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 u...
[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, S...
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,...
(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 Mo...
You point out that Elga's analysis is based on an unproven assertion; that "it is Monday” and “it is Tuesday” are legitimate propositions. As far as I know, there is no definition of what can, or cannot, be used as a proposition. In other words, your analysis is based on the equally unproven assertion that they are not valid. Can remove the need to decide?
You mis-characterize what Elga does. He never directly formulates the state M1, where Beauty is awake. Instead, he formulates two states that are derived from information being added to M1. I'll call them M2A (Beauty learns the outcome is Tails) and M2B (Beauty learns that it is Monday). While he may not do it as formally as you want, he works backwards to show that three of the four components of a proper description of state M1 must have the same probability. What he skips over, is identifying the fourth component (whose probability is now zero).
Wha...
Sleeping Beauty (SB) volunteers for this experiment, and is told all these details by a Lab Assistant (LA):
- I will put you to sleep tonight (Sunday) with a drug that lasts 12 hours. After you are asleep, I will flip two coins - a Dime and a Nickel. I will lock them in an opaque box that has a sensor which can tell if at least one coin inside is showing Tails.
- I will then administer a drug to myself, that erases my memory of the last 12 hours, and go to sleep in the next room.
- Until I am stopped (which will happen on Wednesday morning), when I wake up in the m
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