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 proposition. But since you do not know whether it is true or false, you can assign a 50% probability to the result where “It shows Heads” is true, and a 50% probability the event where “it shows Heads” is false. This entire debate can be reduced to you confusing a truth value, with the probability of that truth value.
On Monday Beauty is awakened. While awake she obtains no information that would help her infer the day of the week. Later in the day she is put to sleep again.
During this part of the experiment, the statement “today is Monday” has the truth value “true”, and does not have the truth value “false.” So by your definition, it is a valid proposition. But Beauty does not know that it is “true.”
On Tuesday the experimenters flip a fair coin. If it lands Tails, Beauty is administered a drug that erases her memory of the Monday awakening, and step 2 is repeated.
During this part of the experiment, the statement “today is Monday” has the truth value “false”, and does not have the truth value “true.” So by your definition, it is a valid proposition. But Beauty dos not know that it is “false.”
In either case, the statement “today is Monday” is a valid proposition by the standard definition you use. What you refuse to acknowledge, is that it is also a proposition that Beauty can treat as “true” or “false” with probabilities P and 1-P.
[Moderator Note:] I am reasonably confident that this current format of the discussion is not going to cause any participant to change their mind, and seems quite stressful to the people participating in it, at least from the outside. While I haven’t been able to read the whole debate in detail, it seems like you are repeating similar points over and over, in mostly the same language. I think it’s fine for you to continue and comment, but I just really want to make sure that people don’t feel an obligation to respond and get dragged into a debate that they don’t expect to get any value from.
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 proposition. But since you do not know whether it is true or false, you can assign a 50% probability to the result where “It shows Heads” is true, and a 50% probability the event where “it shows Heads” is false. This entire debate can be reduced to you confusing a truth value, with the probability of that truth value.
On Monday Beauty is awakened. While awake she obtains no information that would help her infer the day of the week. Later in the day she is put to sleep again.
During this part of the experiment, the statement “today is Monday” has the truth value “true”, and does not have the truth value “false.” So by your definition, it is a valid proposition. But Beauty does not know that it is “true.”
On Tuesday the experimenters flip a fair coin. If it lands Tails, Beauty is administered a drug that erases her memory of the Monday awakening, and step 2 is repeated.
During this part of the experiment, the statement “today is Monday” has the truth value “false”, and does not have the truth value “true.” So by your definition, it is a valid proposition. But Beauty dos not know that it is “false.”
In either case, the statement “today is Monday” is a valid proposition by the standard definition you use. What you refuse to acknowledge, is that it is also a proposition that Beauty can treat as “true” or “false” with probabilities P and 1-P.
[Moderator Note:] I am reasonably confident that this current format of the discussion is not going to cause any participant to change their mind, and seems quite stressful to the people participating in it, at least from the outside. While I haven’t been able to read the whole debate in detail, it seems like you are repeating similar points over and over, in mostly the same language. I think it’s fine for you to continue and comment, but I just really want to make sure that people don’t feel an obligation to respond and get dragged into a debate that they don’t expect to get any value from.