The Bellhop Problem

[Edwardo Piet]

Try this one, it seems simple at first blush

A man walks into a store and steals a $100 bill. 5 minutes later, he returns to the store and buys stuff worth $70. He pays with the bill that he had stolen, so the owner of the store returns him $30. How many dollars did the store owner lose?

 but I have in the past seen flame wars over it.

$30?

He paid $70 because he was given $30 back but overall the store owner has lost $30 because the man stole $100 and the store owner gave $30 back?

---

I'll have to agree with Whatevs on this, but I'm open to being corrected.

As for Monty Hall problem, according to this author, the now accepted answer is questionable actually. At first, I was very skeptical about that, but after reading his post, I'm less skeptical of his view. It all lies in the assumptions being made.

https://ima.org.uk/4552/dont-switch-math...lem-wrong/

I already knew it relied on assumptions though

These things do! It's why there are no true paradoxes and why I believe the Liar's Paradox was already solved just not many people have accepted the solution because it's so confusing or far fetched to most people (although it makes perfect logical sense to me. P.S. If I was to write a book about logical argumentation it would be called "Contradictions and Equivocations" and it would fundamentally be about how equivocations are just as illogical and commonplace as contradictions and they simply both violate the law of identity in different ways... one does explicitly and the other does implicitly).

I haven't even read the article yet but one of the assumptions is that the owner shows the guest one of the incorrect doors (one of the goats) right? But in a real game show it would be nothing like that?

Of course, but that's one of the very premises of the problem. And yet people still feel they shouldn't switch because they are applying their realistic intuitions to an unrealistic problem. But however unrealistic the problem is... the solution to the problem follows logically once you accept the premises. If it were a valid argument it would be sound but not valid... but it works kind of like a mathematical thought experiment anyway and thought experiments are rarely sound. For example: The thought experiment of the experience machine. There's absolutely no reason to believe that the experience machine exists. The purpose is to illustrate that if it did exist then that would mean hedonism was false so hedonism can't be correct in all possible worlds (For the record, I think the experience machine argument against hedonism fails spectacularly and ultimately relies on personal incredulity about what one couldn't possibly want even though one would in fact want it).

[GrandizerII]

Try this one, it seems simple at first blush
 but I have in the past seen flame wars over it.

$30?

He paid $70 because he was given $30 back but overall the store owner has lost $30 because the man stole $100 and the store owner gave $30 back?

Depends on how "lost" is being defined. If by loss, we're also including loss in terms of the item value, then the store owner also loses 70$ because instead of earning 70$, they get 70% of the stolen money back (not new money).

I already knew it relied on assumptions though

The issue is that some of the standard assumptions, according to the author, are not warranted per the wording/description of the original problem that was answered by Marilyn vos Savant. For example, the assumption that Monty would choose a door randomly (given a choice, that is). Not saying I'm not iffy about his reasoning, but I can't help but think he may have a point being "nitpicky" about the assumptions.

These things do! It's why there are no true paradoxes and why I believe the Liar's Paradox was already solved just not many people have accepted the solution because it's so confusing or far fetched to most people (although it makes perfect logical sense to me. P.S. If I was to write a book about logical argumentation it would be called "Contradictions and Equivocations" and it would fundamentally be about how equivocations are just as illogical and commonplace as contradictions and they simply both violate the law of identity in different ways... one does explicitly and the other does implicitly).

What's your proposed solution to the Liar Paradox? How I see it is there is no such thing as an absolute unconditional liar who makes such statements. The best liar can still potentially say truths here and there, even if to this point they have yet to tell a truth. But I admit I have not really thought this through seriously, so perhaps I'm missing something here.

EDIT: Scratch it. I like what Alfred Tarski has to say about this.

Alfred Tarski diagnosed the paradox as arising only in languages that are "semantically closed", by which he meant a language in which it is possible for one sentence to predicate truth (or falsehood) of another sentence in the same language (or even of itself). To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the "object language", while the referring sentence is considered to be a part of a "meta-language" with respect to the object language. It is legitimate for sentences in "languages" higher on the semantic hierarchy to refer to sentences lower in the "language" hierarchy, but not the other way around. This prevents a system from becoming self-referential.

Unfortunately, this system is incomplete. One would like to be able to make statements such as "For every statement in level α of the hierarchy, there is a statement at level α+1 which asserts that the first statement is false." This is a true, meaningful statement about the hierarchy that Tarski defines, but it refers to statements at every level of the hierarchy, so it must be above every level of the hierarchy, and is therefore not possible within the hierarchy (although bounded versions of the sentence are possible).

https://en.wikipedia.org/wiki/Liar_paradox

[Abaddon_ire]

Yes, because Monty Hall is counterintuitive, no, because the bellhop problem is not. Bellhop relies on misdirection. It falsely adds the guests expenditure
($27) to the bellhops income ($2) for a total 0f $29 with one missing dollar. If that is allowed, the one could just as easily add the guests expenditure ($27) to the Hotel's income ($25) for a total of $52 which gives us $22 extra. Or one could add the guests income ($3) to the hotels income ($25) to give $28 with two dollars missing. To make it work as a problem, one must add inappropriate intermediate numbers while ignoring others.

Try this one, it seems simple at first blush
 but I have in the past seen flame wars over it.

Not familiar to me but ..

---

.. the store owner is out the $100 that was stolen and that is all.  He received the full $70 value of the merchandise (the $100 tendered minus the $30 he returned in change).

---

Ah, but

---

the store owner paid less than $70 for the goods

---

[GrandizerII]

Not familiar to me but ..

---

.. the store owner is out the $100 that was stolen and that is all.  He received the full $70 value of the merchandise (the $100 tendered minus the $30 he returned in change).

---

Ah, but

---

the store owner paid less than $70 for the goods

---

Realistically, this does need to be considered, so good point. But as a pure math problem/puzzle, do we need to make that assumption, or go with a simplistic one?

[Whateverist]

Ah, but

---

the store owner paid less than $70 for the goods

---

Realistically, this does need to be considered, so good point. But as a pure math problem/puzzle, do we need to make that assumption, or go with a simplistic one?

The solution is similar to that of the Bellhop puzzle.  

---

The calculation of how much of every sale goes to cover overhead, wholesale price of the goods and the margin required to stay in business is entirely separate from the question of how much money the shop owner is out owing to his interaction with the thief.  He is out $100 from the theft of the goods and that is all he is missing.  The later exchange of goods for cash involving the stolen money is status quo from the POV of the shop owner, and shouldn't be mixed up with loss from the theft.

---

[Angrboda]

I think I've got it. (And I haven't looked.)

---

The key quantity is not the $30 divided by three, which is what they originally paid, but the $25 divided by three which is what they owed.

They paid $27, but the hotel manager only kept $25 of that, leaving $2 extra that the bellhop kept for himself.

---

[Edwardo Piet]

Depends on how "lost" is being defined. If by loss, we're also including loss in terms of the item value, then the store owner also loses 70$ because instead of earning 70$, they get 70% of the stolen money back (not new money).

Yeah so that seems more like an equivocation than a problem.

The issue is that some of the standard assumptions, according to the author, are not warranted per the wording/description of the original problem that was answered by Marilyn vos Savant. For example, the assumption that Monty would choose a door randomly (given a choice, that is). Not saying I'm not iffy about his reasoning, but I can't help but think he may have a point being "nitpicky" about the assumptions.

Ha! Yes the assumptions are not warranted by the wording and it's only after the answer is revealed when you think "WTF? But no one told me about those assumptions" that it makes sense. It makes sense after you understand it because part of the understanding it is accepting the assumptions that were never presented.

Once again... seems like it's down to another equivocation. There are many ways of interpreting what was said. Or in this case perhaps it wasn't even expressed?

But all apparent paradoxes I am convinced are down to semantic errors (commonly equvocation) and the limits of language... rather than down to any so-called real paradox. The so-called real ones are one that are supposedly unanswered. And many of them are but that doesn't mean they always will be... and even if they will always be... again I think that's down to the limits of our language.

Some of them like the Liar's Paradox I believe have actually already been solved.

If you're curious this is the solution to the Liar's Paradox that I 100% accept:

Arthur Prior asserts that there is nothing paradoxical about the liar paradox. His claim (which he attributes to Charles Sanders Peirce and John Buridan) is that every statement includes an implicit assertion of its own truth. Thus, for example, the statement "It is true that two plus two equals four" contains no more information than the statement "two plus two equals four", because the phrase "it is true that..." is always implicitly there. And in the self-referential spirit of the Liar Paradox, the phrase "it is true that..." is equivalent to "this whole statement is true and ...".

Thus the following two statements are equivalent:

This statement is false.

This statement is true and this statement is false.

The latter is a simple contradiction of the form "A and not A", and hence is false. There is therefore no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction. Eugene Mills and Neil Lefebvre and Melissa Schelein present similar answers.

What's your proposed solution to the Liar Paradox?

See above

How I see it is there is no such thing as an absolute unconditional liar who makes such statements.

I don't think that that resolves the paradox at all as it's not about lying or dishonesty or whether such a person could exist or not (whether such a person could exist or not is synthetic matter but the Liar's Paradox is an analytic one). The paradox is about how when you assume the sentence "this sentence is false" is true it becomes false and when you assume it is false it becomes true. But the reality of the matter is it only appears that way for the reason quoted above in the solution I accept.

The best liar can still potentially say truths here and there, even if to this point they have yet to tell a truth. But I admit I have not really thought this through seriously, so perhaps I'm missing something here.

That is more of an empirical matter than a logical matter... and even if the liar always lied the question is not whether they are lying but whether it's really true that the statement "this statement is false" leads to it becoming true if taken as false and leads to it becoming false if taken as true. It's called the liar's paradox... but it isn't really about lying... because people can lie about something that is actually true just as someone can be mistaken about something they honestly assert.

Alfred Tarski diagnosed the paradox as arising only in languages that are "semantically closed", by which he meant a language in which it is possible for one sentence to predicate truth (or falsehood) of another sentence in the same language (or even of itself). To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the "object language", while the referring sentence is considered to be a part of a "meta-language" with respect to the object language. It is legitimate for sentences in "languages" higher on the semantic hierarchy to refer to sentences lower in the "language" hierarchy, but not the other way around. This prevents a system from becoming self-referential.

Unfortunately, this system is incomplete. One would like to be able to make statements such as "For every statement in level α of the hierarchy, there is a statement at level α+1 which asserts that the first statement is false." This is a true, meaningful statement about the hierarchy that Tarski defines, but it refers to statements at every level of the hierarchy, so it must be above every level of the hierarchy, and is therefore not possible within the hierarchy (although bounded versions of the sentence are possible).

I don't like that response because 1. I think that the objection to Tarski's solution given at the end is valid rendering his solution false and 2. I much prefer Arthur Prior's answer and, at least to me, it is very clearly correct. I find it very curious that it's not accepted universally but of course that's why paradoxes stick around: because they're so counterintutiive that even when they've been solved many people disagree that they've been solved.

[Divinity]

As my son says: The real problem with this question isn't the fuzzy math where you don't count the Bellhop's salary as part of the $27 they pay. The REAL problem is the manager giving back $5 because he decides they overcharged. They already paid the fucking money, that manager isn't giving any money back, let alone $5.00.

[polymath257]

But all apparent paradoxes I am convinced are down to semantic errors (commonly equvocation) and the limits of language... rather than down to any so-called real paradox. The so-called real ones are one that are supposedly unanswered. And many of them are but that doesn't mean they always will be... and even if they will always be... again I think that's down to the limits of our language.

Some of them like the Liar's Paradox I believe have actually already been solved.

If you're curious this is the solution to the Liar's Paradox that I 100% accept:

Arthur Prior asserts that there is nothing paradoxical about the liar paradox. His claim (which he attributes to Charles Sanders Peirce and John Buridan) is that every statement includes an implicit assertion of its own truth. Thus, for example, the statement "It is true that two plus two equals four" contains no more information than the statement "two plus two equals four", because the phrase "it is true that..." is always implicitly there. And in the self-referential spirit of the Liar Paradox, the phrase "it is true that..." is equivalent to "this whole statement is true and ...".

Thus the following two statements are equivalent:

This statement is false.

This statement is true and this statement is false.

The latter is a simple contradiction of the form "A and not A", and hence is false. There is therefore no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction. Eugene Mills and Neil Lefebvre and Melissa Schelein present similar answers.

That is a very cute argument! It also resolves a more complex version of the Liar's Paradox:
Statement A: Statement B is true
Statement B: Statement A is false.

If these are interpreted as

Statement A: Statement A is true and Statement B is true
Statement B: Statement B is true and Statement A is false,

then the negations are

Neg(A): Statement A is false or statement B is false
Neg(B): Statement B is false or Statement A is true.

In this case, it is fairly easy to see that both Statement A and Statement B have to be false. No remaining paradox!

---