Sadly, Mathew has accidentally misinterpreted my views quite a bit, so I'll be explaining them here in the comments.
> Now, the reason AITC doesn’t buy this is that he says that you can’t have credences in de se facts.
This is not a correct description of my view. I agree that there are some valid usages of probability theory that fall under the categorization of "de se reasoning". But there are also a lot of invalid ones. What I'm saying is that this category is very unhelpful and so we better off simply stop using it all together. Instead, we need to think in terms of probability experiments: https://en.wikipedia.org/wiki/Experiment_(probability_theory)
When we can formalize a particular setting as a probability experiment with mutually exclusive outcomes, we, therefore, can lawfully use probability theory to describe our knowledge state of this setting. When we can't - we do not have any reason to expect that probability theory is applicable. The whole confusion started with David Lewis simply assuming without any justification that we should be able to use probability theory in a more broad class of situations than we have reasons to assume.
> Imagine, to give a case from Lewis, that there are two identical gods located on opposite sides of the world.
Here we can indeed construct a probability experiment with two mutually exclusive outcomes: being in place 1 or place 2, while not having any reason to think that being in place 1 is more likely than place 2 or vice versa. In every iteration of the experiment only one place will be occupied by the person in question. No problem with applying probability theory here.
Now compare it with a Sleeping Beauty case where both Tails awakening happen in the same iteration of the experiment, and yet, proponents of "de se reasoning" attempt to treat them as different outcomes. Such application is invalid.
To correctly reason about Sleeping Beauty we can talk about mutually exclusive outcomes: Heads&Monday; Tails&Monday&Tuesday, where Monday/Tuesday means "Monday/Tuesday awakening happens in this iteration of the experiment"
P(Monday) = 1; P(Tuesday) = 1/2;
P(Heads&Monday) = P(Tails&Monday&Tuesday) = 1/2
> I think you can assign probabilities—perhaps imprecisely—to any proposition, any fact about the world.
Probability is a mathematical function and it's domain is event space. If your statement can not be interpreted as well-defined event of probability experiment, then you can't assign probability value to it. Easy as that. This is not a matter of philosophical disagreement. It's the way math works as a truth preserving mechanism. Now, you are free to define a different function so that it had a coherent value for you statement. But that wouldn't be "probability" anymore.
> Imagine that you’ll be woken up each day of a week. Each day, you have no memories of previous days. On AITC’s view, you can’t coherently say “it’s more likely to be Monday through Saturday today than to be Sunday.”
The problem is in the word "today", that is ill-defined in any experiment where you experience amnesia and repetition of the same experience.
Suppose you were awakened on a random day of a week, now you could formalize "today" as "Monday xor Tuesday xor Wednesday xor Thursday xor Friday xor Saturday xor Sunday". In any iteration of a probability experiment there is exactly one awakening and therefore "today" refers a one particular day from 7 possibilities.
Suppose you were awakened on every day but remembered all your awakenings. Now you can treat this sequence of awakenings as a chain seven different experiments. In first experiment you are always awakened on Monday. In the second you are always awakened on Tuesday. In the third you are always awakened on Thursday and so on. And you have a way to formally define "today" in any of these experiments. In the first experiment "today" means Monday. In the second experiment "today" means Tuesday and so on. Once again today" refers a one particular day from 7 possibilities.
But when you are awakened on every day and experience amnesia and is aware of all of this, you end up in a very weird spot. Not knowing which day it is doesn't allow you to define "today" as this particular day. And knowing that you experience awakening on every day, doesn't allow you to define it as exclusive or between days of the week - there is no iteration of the experiment where event "Monday xor Tuesday xor Wednesday xor Thursday xor Friday xor Saturday xor Sunday" happens. You can define it as Monday or Tuesday or ... or Sunday" But then "today" refers to all the days at ones - not the way we usually use the term.
And so every statement about "today" is ill-defined in specifically this kind of scenarios. This is counterintuitive because we are not evolved for such situations, but the more you think about it in mathematical terms the clearer it becomes.
Of course, you can still make meaningful statements. Like "The time interval between Monday and Saturday is larger than the time interval of Sunday". Then you can use this information to coming up to correct betting odds. But this is not a statement about probability.
> Imagine that you’ll be put woken up every day for four years that contains a leap-day. Every day that isn’t a leap day, you’ll be given a cake. On the leap-day, however, you get something much worse than a cake—being stabbed in the eye. On AITC’s view, it’s incoherent to, upon waking up, think “probably I won’t be stabbed in the eye today, because it’s likely not a leap day.” This is just completely absurd and implausible.
On the contrary, what is absurd, is treating this situation as if you have merely 1/1460 chance to get stabbed in the eye on each day, as it deludes you into thinking that it's possible not to get stabbed in the eye at all, while participating in this experiment.
> Or imagine that you’ll be put to sleep and then awoken once during the time Biden is president and once during the time Trump is. When you wake during the time his successor is president, you have no memories of the previous day, and in neither case do you know who is president. On AITC’s, upon waking up, you can’t coherently wonder if Biden is currently president, as that doesn’t pick out a specific fact. Thus, on his view, the sentence “Biden is more likely to be the president than my shoe,” isn’t true.
That's incorrect. You can wonder as much as you like, but you can't assign any probability to the statement "Biden is president today".
You can also still make coherent statements like "During this experiment Biden is president", "During this experiment Trump is president" and "During this experiment my shoe is not a president". So you can say "During this experiment Biden is more likely to be president than my shoe" and it would be coherent. As long as you are not talking about ill-defined "today" everything is fine.
I'll post the second part later. But I hope that this already gives a good enough understanding of the reasoning and justifications behind my views.
Unsurprisingly, I disagree with many of your comments.
//This is not a correct description of my view. I agree that there are some valid usages of probability theory that fall under the categorization of "de se reasoning". But there are also a lot of invalid ones. What I'm saying is that this category is very unhelpful and so we better off simply stop using it all together. Instead, we need to think in terms of probability experiments://
Okay apparently you think you can have a de se credence in your place in the world but not on your time in a world. So you can have credence in "I'm right God rather than left God," but not in "I'm God on day 1 rather than day 2." Still, it's very hard to see what's supposed to distinguish those cases. Why can one be modeled by an experiment but not the other? In both cases, you don't learn anything about the state of the world, but you can do probability about your place in the world--whether you are now the first day person just as you can wonder whether you're the left God or right God.
//Here we can indeed construct a probability experiment with two mutually exclusive outcomes: being in place 1 or place 2, while not having any reason to think that being in place 1 is more likely than place 2 or vice versa. In every iteration of the experiment only one place will be occupied by the person in question. No problem with applying probability theory here.//
You can say the same thing about de se time cases. In sleeping beauty, you can say the mutually exclusive experiments are the present time slice being on day one and the present time slice being on day two. Anything about why that's invalid also applies to the two Gods case.
//Probability is a mathematical function and it's domain is event space. If your statement can not be interpreted as well-defined event of probability experiment, then you can't assign probability value to it. Easy as that.//
For anything that might be the case, you can assign a probability to it. You can always have some credence in an event. In SB, there is a well-defined event space--the coin came up heads and today is Monday, the coin came up tails and today is Monday, and the coin came up tails and today is Tuesday. Probability is just how likely something is to be true--it makes sense to have a likelihood to all and only propositions.
//The problem is in the word "today", that is ill-defined in any experiment where you experience amnesia and repetition of the same experience.//
Why think that? All you do is repeat that today doesn't have a fixed value in an experiment. So??? The fact that whether something is true changes and what it picks out change doesn't mean it isn't well-defined. In 2020 it was true that Trump was president, now it's true Biden is. That doesn't mean that being president isn't well defined! There are three possible outcomes in SB--three things might be the case--and you can totally use probability theory to assign credences to them. This just seems like bizarre conceptual confusion!
//On the contrary, what is absurd, is treating this situation as if you have merely 1/1460 chance to get stabbed in the eye on each day, as it deludes you into thinking that it's possible not to get stabbed in the eye at all, while participating in this experiment.//
No, what? X having a low probability of happening per day is compatible with X having a high probability of happening total over the course of many days.
//That's incorrect. You can wonder as much as you like, but you can't assign any probability to the statement "Biden is president today".//
If you can coherently wonder about something then it might be true. And if something might be true then you can have a credence in it. Your view implies crazily that the statement, in the above case, "it's likelier Trump is president than that my shoe is," is false, when it's obviously true.
> Okay apparently you think you can have a de se credence in your place in the world but not on your time in a world.
No, you are still missing the point. I even used an example where a person is awaken once on a random day of the week. In your terms this is "de se" about your time in the world. And we can perfectly formalize it with the notion of probability experiment. On every iteration of the experiment you have one awakening. So an awakening on a particular day can be treated as an idividual, mutually exclusive outcome of the experiment, and so probability theory can be applied to our uncertanity about which awakening is being experienced.
The core principle isn't whether something is "de dicto" or "de se", nor whether it's uncertanity about space or time. Math doesn't care about this. The core principle is whether the outcomes are mutually exclusive or not - whether in every iteration of the experiemnt as stated *one and only one* of the outcomes happens.
Read the wikipedia page that I linked, please. And try to *really* understand the mathematical structure behind the notion probability experiment. I'm afraid, you won't be able to move forward until then.
> In sleeping beauty, you can say the mutually exclusive experiments are the present time slice being on day one and the present time slice being on day two. Anything about why that's invalid also applies to the two Gods case.
You can't, until you formally define "present time". In two Gods case "my place" is pointing to the one particular places - it's univocally refers to either place 1 or place 2 in every iteration of probability experiment. So a statement "I'm in place 1" have a coherent value in every iteration of the probability experiment. Therefore it's a well defined event and can be used as an argument of probability function.
In Sleeping Beauty on Tails "present time" is simultaneously pointing to two different moments in the same iteration of the experiment. A statement "Today is Monday" doesn't have a coherent truth value throughout iteration of Sleeping Beauty probability experiment where the coin is Tails. This means that such event is ill-defined. Therefore it can't have coherent probability.
Do you really not see the difference?
> For anything that might be the case, you can assign a probability to it.
I can grant you that. But then not-well-defined statements are not something that "might be the case".
> In SB, there is a well-defined event space--the coin came up heads and today is Monday, the coin came up tails and today is Monday, and the coin came up tails and today is Tuesday.
"Today is Monday" and "Today is Tuesday" are not mutually exlusive outcomes in the Sleeping Beauty experiment. Therefore they can't be used to define a sample space for it, therefore they can't be used to define the event space for it.
There are *some different* probability experiments where such events are well-defined - the one where you awaken only on one particular day in every iteration of the experiment. All this time SIA and SSA have been reasoning about such experiments, while wrongly attributing their conclusions to Sleeping Beauty.
> All you do is repeat that today doesn't have a fixed value in an experiment. So???
Just that. You can't define a value of a function on a not well-defined event. Therefore the function is undefined. This is a mathematical fact. There is no clever semantic argument around that.
> The fact that whether something is true changes and what it picks out change doesn't mean it isn't well-defined.
Once again I encorage you to understand the mathematical concepts before trying to argue about them.
> In 2020 it was true that Trump was president, now it's true Biden is. That doesn't mean that being president isn't well defined!
It means that statement "Trump is president today " is not well defined in the context of singular experiment, lasting from 2020 to 08.11.2024.
Thankfully, we can use more preciese statements like "Trump was president in 2020" or Trump is not president on 08.11.2024". Most of the time we are not restricted to reasoning about large chunks of physical time in the context of a probability experiment. Only when we are participating in an experiemnt which requires experiencing amnesia and the repetition of the same experience.
> No, what? X having a low probability of happening per day is compatible with X having a high probability of happening total over the course of many days.
Not just high. The experiment as stated, supposes certanity that participating in it will lead to you getting stabbed in the eye. But believeing that you merely has small chance of getting stabbed in the eye on any day will never get you there.
Consider the simple case.
On day one a coin is put Heads. On day two it's put Tails. You expirience amnesia between day one and two.
If you believe that on any day of the experiment the probability that the coin is Heads is 1/2, then it means that probability that the coin is Heads in at least one of two days is merely 3/4, while in reality it's 1.
> If you can coherently wonder about something then it might be true.
What do you mean by "coherently wonder"? For me wondering is just a state of curiocity and/or confusion, it's not a mathematical term, what should be "coherent" there? I think it's totally fair to say that a person might wonder about something logically inconsistent. That people who are confused about semantics are still legitimately can be curious.
Now, if you define "wonder" in such a manner that one can't wonder about something before making sure that the statement is well-defined and logically consistent, then sure, I'll agree that according to your definition, you indeed can't wonder whether "Biden is a president today", while participating in such experiment. But the counterintuitiveness of it is mostly due to the fact that you picked a bad definition of wonder, that doesn't actually capture the way humans use this word.
> Your view implies crazily that the statement, in the above case, "it's likelier Trump is president than that my shoe is," is false
Not so. I've explained how one can make a formally coherent statement like that without using the ill-defined term "today".
But in general you should stop trying to appeal to incredulity. We are dealing with complicated settings for which our intuitions are not prepared for. Naturally they may feel weird to us. Thankfully we have math with its truth preserving properties. And as soon as one is using math lawfully, they can't be led astray.
If you think you can construct some betting argument that my view behaves irrationally towards - feel free to try. I recommend you first read this, where I clear the obvious cases:
For anyone interested I have a whole series of posts on anthropic reasoning. You can read them on LessWrong. Here is the first one:
https://www.lesswrong.com/posts/HQFpRWGbJxjHvTjnw/anthropical-motte-and-bailey-in-two-versions-of-sleeping
Sadly, Mathew has accidentally misinterpreted my views quite a bit, so I'll be explaining them here in the comments.
> Now, the reason AITC doesn’t buy this is that he says that you can’t have credences in de se facts.
This is not a correct description of my view. I agree that there are some valid usages of probability theory that fall under the categorization of "de se reasoning". But there are also a lot of invalid ones. What I'm saying is that this category is very unhelpful and so we better off simply stop using it all together. Instead, we need to think in terms of probability experiments: https://en.wikipedia.org/wiki/Experiment_(probability_theory)
When we can formalize a particular setting as a probability experiment with mutually exclusive outcomes, we, therefore, can lawfully use probability theory to describe our knowledge state of this setting. When we can't - we do not have any reason to expect that probability theory is applicable. The whole confusion started with David Lewis simply assuming without any justification that we should be able to use probability theory in a more broad class of situations than we have reasons to assume.
> Imagine, to give a case from Lewis, that there are two identical gods located on opposite sides of the world.
Here we can indeed construct a probability experiment with two mutually exclusive outcomes: being in place 1 or place 2, while not having any reason to think that being in place 1 is more likely than place 2 or vice versa. In every iteration of the experiment only one place will be occupied by the person in question. No problem with applying probability theory here.
Now compare it with a Sleeping Beauty case where both Tails awakening happen in the same iteration of the experiment, and yet, proponents of "de se reasoning" attempt to treat them as different outcomes. Such application is invalid.
To correctly reason about Sleeping Beauty we can talk about mutually exclusive outcomes: Heads&Monday; Tails&Monday&Tuesday, where Monday/Tuesday means "Monday/Tuesday awakening happens in this iteration of the experiment"
P(Monday) = 1; P(Tuesday) = 1/2;
P(Heads&Monday) = P(Tails&Monday&Tuesday) = 1/2
> I think you can assign probabilities—perhaps imprecisely—to any proposition, any fact about the world.
Probability is a mathematical function and it's domain is event space. If your statement can not be interpreted as well-defined event of probability experiment, then you can't assign probability value to it. Easy as that. This is not a matter of philosophical disagreement. It's the way math works as a truth preserving mechanism. Now, you are free to define a different function so that it had a coherent value for you statement. But that wouldn't be "probability" anymore.
> Imagine that you’ll be woken up each day of a week. Each day, you have no memories of previous days. On AITC’s view, you can’t coherently say “it’s more likely to be Monday through Saturday today than to be Sunday.”
The problem is in the word "today", that is ill-defined in any experiment where you experience amnesia and repetition of the same experience.
Suppose you were awakened on a random day of a week, now you could formalize "today" as "Monday xor Tuesday xor Wednesday xor Thursday xor Friday xor Saturday xor Sunday". In any iteration of a probability experiment there is exactly one awakening and therefore "today" refers a one particular day from 7 possibilities.
Suppose you were awakened on every day but remembered all your awakenings. Now you can treat this sequence of awakenings as a chain seven different experiments. In first experiment you are always awakened on Monday. In the second you are always awakened on Tuesday. In the third you are always awakened on Thursday and so on. And you have a way to formally define "today" in any of these experiments. In the first experiment "today" means Monday. In the second experiment "today" means Tuesday and so on. Once again today" refers a one particular day from 7 possibilities.
But when you are awakened on every day and experience amnesia and is aware of all of this, you end up in a very weird spot. Not knowing which day it is doesn't allow you to define "today" as this particular day. And knowing that you experience awakening on every day, doesn't allow you to define it as exclusive or between days of the week - there is no iteration of the experiment where event "Monday xor Tuesday xor Wednesday xor Thursday xor Friday xor Saturday xor Sunday" happens. You can define it as Monday or Tuesday or ... or Sunday" But then "today" refers to all the days at ones - not the way we usually use the term.
And so every statement about "today" is ill-defined in specifically this kind of scenarios. This is counterintuitive because we are not evolved for such situations, but the more you think about it in mathematical terms the clearer it becomes.
Of course, you can still make meaningful statements. Like "The time interval between Monday and Saturday is larger than the time interval of Sunday". Then you can use this information to coming up to correct betting odds. But this is not a statement about probability.
> Imagine that you’ll be put woken up every day for four years that contains a leap-day. Every day that isn’t a leap day, you’ll be given a cake. On the leap-day, however, you get something much worse than a cake—being stabbed in the eye. On AITC’s view, it’s incoherent to, upon waking up, think “probably I won’t be stabbed in the eye today, because it’s likely not a leap day.” This is just completely absurd and implausible.
On the contrary, what is absurd, is treating this situation as if you have merely 1/1460 chance to get stabbed in the eye on each day, as it deludes you into thinking that it's possible not to get stabbed in the eye at all, while participating in this experiment.
> Or imagine that you’ll be put to sleep and then awoken once during the time Biden is president and once during the time Trump is. When you wake during the time his successor is president, you have no memories of the previous day, and in neither case do you know who is president. On AITC’s, upon waking up, you can’t coherently wonder if Biden is currently president, as that doesn’t pick out a specific fact. Thus, on his view, the sentence “Biden is more likely to be the president than my shoe,” isn’t true.
That's incorrect. You can wonder as much as you like, but you can't assign any probability to the statement "Biden is president today".
You can also still make coherent statements like "During this experiment Biden is president", "During this experiment Trump is president" and "During this experiment my shoe is not a president". So you can say "During this experiment Biden is more likely to be president than my shoe" and it would be coherent. As long as you are not talking about ill-defined "today" everything is fine.
I'll post the second part later. But I hope that this already gives a good enough understanding of the reasoning and justifications behind my views.
Unsurprisingly, I disagree with many of your comments.
//This is not a correct description of my view. I agree that there are some valid usages of probability theory that fall under the categorization of "de se reasoning". But there are also a lot of invalid ones. What I'm saying is that this category is very unhelpful and so we better off simply stop using it all together. Instead, we need to think in terms of probability experiments://
Okay apparently you think you can have a de se credence in your place in the world but not on your time in a world. So you can have credence in "I'm right God rather than left God," but not in "I'm God on day 1 rather than day 2." Still, it's very hard to see what's supposed to distinguish those cases. Why can one be modeled by an experiment but not the other? In both cases, you don't learn anything about the state of the world, but you can do probability about your place in the world--whether you are now the first day person just as you can wonder whether you're the left God or right God.
//Here we can indeed construct a probability experiment with two mutually exclusive outcomes: being in place 1 or place 2, while not having any reason to think that being in place 1 is more likely than place 2 or vice versa. In every iteration of the experiment only one place will be occupied by the person in question. No problem with applying probability theory here.//
You can say the same thing about de se time cases. In sleeping beauty, you can say the mutually exclusive experiments are the present time slice being on day one and the present time slice being on day two. Anything about why that's invalid also applies to the two Gods case.
//Probability is a mathematical function and it's domain is event space. If your statement can not be interpreted as well-defined event of probability experiment, then you can't assign probability value to it. Easy as that.//
For anything that might be the case, you can assign a probability to it. You can always have some credence in an event. In SB, there is a well-defined event space--the coin came up heads and today is Monday, the coin came up tails and today is Monday, and the coin came up tails and today is Tuesday. Probability is just how likely something is to be true--it makes sense to have a likelihood to all and only propositions.
//The problem is in the word "today", that is ill-defined in any experiment where you experience amnesia and repetition of the same experience.//
Why think that? All you do is repeat that today doesn't have a fixed value in an experiment. So??? The fact that whether something is true changes and what it picks out change doesn't mean it isn't well-defined. In 2020 it was true that Trump was president, now it's true Biden is. That doesn't mean that being president isn't well defined! There are three possible outcomes in SB--three things might be the case--and you can totally use probability theory to assign credences to them. This just seems like bizarre conceptual confusion!
//On the contrary, what is absurd, is treating this situation as if you have merely 1/1460 chance to get stabbed in the eye on each day, as it deludes you into thinking that it's possible not to get stabbed in the eye at all, while participating in this experiment.//
No, what? X having a low probability of happening per day is compatible with X having a high probability of happening total over the course of many days.
//That's incorrect. You can wonder as much as you like, but you can't assign any probability to the statement "Biden is president today".//
If you can coherently wonder about something then it might be true. And if something might be true then you can have a credence in it. Your view implies crazily that the statement, in the above case, "it's likelier Trump is president than that my shoe is," is false, when it's obviously true.
> Okay apparently you think you can have a de se credence in your place in the world but not on your time in a world.
No, you are still missing the point. I even used an example where a person is awaken once on a random day of the week. In your terms this is "de se" about your time in the world. And we can perfectly formalize it with the notion of probability experiment. On every iteration of the experiment you have one awakening. So an awakening on a particular day can be treated as an idividual, mutually exclusive outcome of the experiment, and so probability theory can be applied to our uncertanity about which awakening is being experienced.
The core principle isn't whether something is "de dicto" or "de se", nor whether it's uncertanity about space or time. Math doesn't care about this. The core principle is whether the outcomes are mutually exclusive or not - whether in every iteration of the experiemnt as stated *one and only one* of the outcomes happens.
Read the wikipedia page that I linked, please. And try to *really* understand the mathematical structure behind the notion probability experiment. I'm afraid, you won't be able to move forward until then.
> In sleeping beauty, you can say the mutually exclusive experiments are the present time slice being on day one and the present time slice being on day two. Anything about why that's invalid also applies to the two Gods case.
You can't, until you formally define "present time". In two Gods case "my place" is pointing to the one particular places - it's univocally refers to either place 1 or place 2 in every iteration of probability experiment. So a statement "I'm in place 1" have a coherent value in every iteration of the probability experiment. Therefore it's a well defined event and can be used as an argument of probability function.
In Sleeping Beauty on Tails "present time" is simultaneously pointing to two different moments in the same iteration of the experiment. A statement "Today is Monday" doesn't have a coherent truth value throughout iteration of Sleeping Beauty probability experiment where the coin is Tails. This means that such event is ill-defined. Therefore it can't have coherent probability.
Do you really not see the difference?
> For anything that might be the case, you can assign a probability to it.
I can grant you that. But then not-well-defined statements are not something that "might be the case".
> In SB, there is a well-defined event space--the coin came up heads and today is Monday, the coin came up tails and today is Monday, and the coin came up tails and today is Tuesday.
"Today is Monday" and "Today is Tuesday" are not mutually exlusive outcomes in the Sleeping Beauty experiment. Therefore they can't be used to define a sample space for it, therefore they can't be used to define the event space for it.
There are *some different* probability experiments where such events are well-defined - the one where you awaken only on one particular day in every iteration of the experiment. All this time SIA and SSA have been reasoning about such experiments, while wrongly attributing their conclusions to Sleeping Beauty.
I descibe these experiments here:
https://www.lesswrong.com/posts/SjoPCwmNKtFvQ3f2J/lessons-from-failed-attempts-to-model-sleeping-beauty#No_Coin_Toss_Problem
https://www.lesswrong.com/posts/SjoPCwmNKtFvQ3f2J/lessons-from-failed-attempts-to-model-sleeping-beauty#Single_Awakening_Problem
> All you do is repeat that today doesn't have a fixed value in an experiment. So???
Just that. You can't define a value of a function on a not well-defined event. Therefore the function is undefined. This is a mathematical fact. There is no clever semantic argument around that.
> The fact that whether something is true changes and what it picks out change doesn't mean it isn't well-defined.
It literally means that:
https://en.wikipedia.org/wiki/Well-defined_expression
Once again I encorage you to understand the mathematical concepts before trying to argue about them.
> In 2020 it was true that Trump was president, now it's true Biden is. That doesn't mean that being president isn't well defined!
It means that statement "Trump is president today " is not well defined in the context of singular experiment, lasting from 2020 to 08.11.2024.
Thankfully, we can use more preciese statements like "Trump was president in 2020" or Trump is not president on 08.11.2024". Most of the time we are not restricted to reasoning about large chunks of physical time in the context of a probability experiment. Only when we are participating in an experiemnt which requires experiencing amnesia and the repetition of the same experience.
> No, what? X having a low probability of happening per day is compatible with X having a high probability of happening total over the course of many days.
Not just high. The experiment as stated, supposes certanity that participating in it will lead to you getting stabbed in the eye. But believeing that you merely has small chance of getting stabbed in the eye on any day will never get you there.
Consider the simple case.
On day one a coin is put Heads. On day two it's put Tails. You expirience amnesia between day one and two.
If you believe that on any day of the experiment the probability that the coin is Heads is 1/2, then it means that probability that the coin is Heads in at least one of two days is merely 3/4, while in reality it's 1.
> If you can coherently wonder about something then it might be true.
What do you mean by "coherently wonder"? For me wondering is just a state of curiocity and/or confusion, it's not a mathematical term, what should be "coherent" there? I think it's totally fair to say that a person might wonder about something logically inconsistent. That people who are confused about semantics are still legitimately can be curious.
Now, if you define "wonder" in such a manner that one can't wonder about something before making sure that the statement is well-defined and logically consistent, then sure, I'll agree that according to your definition, you indeed can't wonder whether "Biden is a president today", while participating in such experiment. But the counterintuitiveness of it is mostly due to the fact that you picked a bad definition of wonder, that doesn't actually capture the way humans use this word.
> Your view implies crazily that the statement, in the above case, "it's likelier Trump is president than that my shoe is," is false
Not so. I've explained how one can make a formally coherent statement like that without using the ill-defined term "today".
But in general you should stop trying to appeal to incredulity. We are dealing with complicated settings for which our intuitions are not prepared for. Naturally they may feel weird to us. Thankfully we have math with its truth preserving properties. And as soon as one is using math lawfully, they can't be led astray.
If you think you can construct some betting argument that my view behaves irrationally towards - feel free to try. I recommend you first read this, where I clear the obvious cases:
https://www.lesswrong.com/posts/cvCQgFFmELuyord7a/beauty-and-the-bets