A fun little probability puzzle for you.

[Skepdick]

I said

I think the $100 does give us (incomplete, imperfect, probabilistic) information about which box we initially selected

Well. That's just a silly tautology.

Having chosen any box gives me incomplete/imperfect/probabilistic information about which box I have initially selected.

It's the one with 2x 100 notes (with 50% probability)

You are getting mixed up between the internal and the external languages of the theory.

Does the $100 give you 1 bit of information to the question "Have I chosen the box with 2x100 notes?". No.
Does the 2nd draw give you 1 bit of information to answer the qustion "Have I chosen the box with 2x100 notes?". Yes.

[Flannel Jesus]

but... it's not 50% probability anymore. We've selected the first bill. It's a $100. It started out as 50% probability, before we picked the first bill out of it. It's 66%, by your own logic, by your own code that experimentally proves it, after we select the first bill and find that it's $100. After that moment, we know we have a 66% chance that we've selected the box with 2x $100.

It's not a silly tautology, it's probabilities mate.

Having chosen any box gives me incomplete/imperfect/probabilistic information about which box I have initially selected.

No, I still don't think you're getting it. Choosing the box doesn't give you any information AT ALL about which box you've selected. The moment after you've chosen the box, you don't know which box you've chosen. Selecting the first bill - that's what we're talking about. That's what gives you information, probabilistic information, about which box you've selected.

[Skepdick]

but... it's not 50% probability anymore. We've selected the first bill. It's a $100. It started out as 50% probability, before we picked the first bill out of it. It's 66%, by your own logic, by your own code that experimentally proves it, after we select the first bill and find that it's $100.

It's not a silly tautology, it's probabilities mate.

Having chosen any box gives me incomplete/imperfect/probabilistic information about which box I have initially selected.

No, I still don't think you're getting it. Choosing the box doesn't give you any information AT ALL about which box you've selected. The moment after you've chosen the box, you don't know which box you've chosen. Selecting the first bill - that's what we're talking about. That's what gives you information, probabilistic information, about which box you've selected.

You are not getting it.

There is information pertinent to the theory (internal language of the theory)
And information pertinent to the meta-theory (external language of the theory).

You ALWAYS have a meta-theory.

Does the first $100 give you 1 bit of information to the question "Have I chosen the box with 2x100 notes?". No!
Does the 2nd draw give you 1 bit of information to answer the qustion "Have I chosen the box with 2x100 notes?". Yes!

[Flannel Jesus]

But it's your logic, your code, that's proving to me that we do have information after the first draw. Specifically, information that increases our likelihood of having selected the box with 2x $100. When we first select a box, we have a 50% chance of having selected the box with 2x $100. When we pick out the first note, and see that it's a $100, by your own code we can show that we now have a 66% chance of having selected the box with 2x $100.

How does the probability change from 50% to 66% if we haven't gained any information about it?

[Skepdick]

But it's your logic, your code, that's proving to me that we do have information after the first draw. Specifically, information that increases our likelihood of having selected the box with 2x $100.

You still aren't getting it. ALL formal systems are incomplete!

I am using an imperative programming language so what's explicit is the control-flow (epistemology), not the data-flow (ontology).

The logic only represents my epistemic state of mind.
The logic does NOT represent my ontological state of mind.

These paradoxes arise because you can't resolve the instant shift in perspective which happens the moment one hypothesis lapses.
Epistemology returns only 1 result (the winner) and ontology walks away with the knowledge of exactly which box was chosen.

[Flannel Jesus]

That's a very nice poem.

[Skepdick]

That's a very nice poem.

Par for the course for humans then.

Effective poetry is all we have.

[Skepdick]

How does the probability change from 50% to 66% if we haven't gained any information about it?

There is absolutely no way for you to determine whether you have learned something from the first $100 note; or whether the rules of the game called probability theory make you change your mind in a way that makes you believe you've learned something.

You pull out the $100. You follow the rules and you update your probability from 0.5 to 0.666...

But that's an expected, not a true value for the variable that you are measuring! The true value of the variable is either 0 or 1!

Either: You pull out the $1. You follow the rules and you update your probability from 0.666... to 0 (you realise you over-estimated the initial learning).
OR: You pull out the $100. You follow the rules and you update your probability from 0.666... to 1 (you realise under-estimated the initial learning).

This is what happens when you are using continuous values (real numbers) to represent a discrete variable.

[Flannel Jesus]

Sounds like you just don't use the word 'information' the same way I do, really.

If you agree that seeing the first $100 should update your probability that you've selected the 2x100 box from 50% to 66.66...% , then you fundamentally agree with me, you're just using words differently.

[Skepdick]

Sounds like you just don't use the word 'information' the same way I do, really.

You don't use the word "information" in one particular way. You are overloading it.

Like we all do.

If you agree that seeing the first $100 should update your probability that you've selected the 2x100 box from 50% to 66.66...% , then you fundamentally agree with me, you're just using words differently.

Duh! Welcome to philosophy.

Everyone agrees. We agree to disagree.

[Flannel Jesus]

Okay, I'll try to remember when I see your username in the future that you like to spend pages and pages on word games. This is the second time in a row that's happened lmao.

[Skepdick]

Okay, I'll try to remember when I see your username in the future that you like to spend pages and pages on word games. This is the second time in a row that's happened lmao.

If you want to avoid the word games you should totally turn to empiricism.

What experiment could we perform (in silence) to determine whether we agree, or disagree on whether you actually learn something when you take the $100 from the box?

[Flannel Jesus]

There's literally no way to avoid word games with someone with your approach to language. You seem to at will start interpreting words in your unique way, and make no effort to understand what the other person is saying.

[Flannel Jesus]

You've already designed the experiment, and the results are in, you just spent 3 pages of a conversation disagreeing with the results of your own experiment.

[Skepdick]

There's literally no way to avoid word games with someone with your approach to language. You seem to at will start interpreting words in your unique way, and make no effort to understand what the other person is saying.

I interpret language the way I interpret language.
You interpret language the way you interpret language.

In so far as I can tell every person on Earth interprets language the way they interpret language.

In so far as "understanding" is goal-oriented task I'd be curious to know what its criteria for success are.
Could we test/falsify whether we understand each other?