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## Question

puzzle in book:all but 2 of my cars a fords all but 2 of my cars are toyotas and all but 2 of my cars are hondas how many cars do i have

3 cars

couldnt he have 2 cars and none of them be a honda toyota or ford?

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English is a means of conveying information, but you need some other mechanism for getting meaning from the statement. In this case, the statement is such that it is fairly easy to convert to logic, and thus to me (and others) it seems appropriate to do the conversion.

Sure, but before we make the conversion, we have to make sure everything is being interpreted correctly.

"all but 2 of my cars are fords"

1. Zero amount of cars is not a different make from cars that exist. You can't make that statement if you have two cars which are Camaros because you have specifically said that cars that you own are Fords. This is even a more illogical when owning no Fords when coupled with the next point:

2. If all 'but" two of his cars are Fords, can it be logically possible to have only two cars? No. He is separating two cars from an amount of cars. That amount must logically be greater than 2 or no separation can be made. Claiming that he has 2 cars plus an amount of cars equaling zero that are Fords is illogical.

Edited by Scraff

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Sure, but before we make the conversion, we have to make sure everything is being interpreted correctly.

"all but 2 of my cars are fords"

1. Zero amount of cars is not a different make from cars that exist. You can't make that statement if you have two cars which are Camaros because you have specifically said that cars that you own are Fords. This is even a more illogical when owning no Fords when coupled with the next point:

2. If all 'but" two of his cars are Fords, can it be logically possible to have only two cars? No. He is separating two cars from an amount of cars. That amount must logically be greater than 2 or no separation can be made. Claiming that he has 2 cars plus an amount of cars equaling zero that are Fords is illogical.

You may argue that this is not directly related to the discussion, but I think this scenario shows that there are multiple uses for the word "all," some that refer to something as "all of it" even if there are none.

----

Smoke was billowing from the town hall when two paramedics, Johnson and Bates, arrived on the scene. A gas main had exploded, causing part of the building to collapse. Bates turned to Johnson and said, "You treat all of the burn victims and I'll deal with the lacerations and broken bones." Johnson did as he was told.

If no one got burned, is there a problem with Bates' statement?

----

Of course, in this situation, there is no foreknowledge of events, so Bates' can't know that there are no burn victims when he made the statement. Maybe this implies that the problem with the cars comes from the expectation of foreknowledge based on the fact that "I" am telling you about "my" cars.

So I'm not sure if this changes anything or not, but what if we take "me" out of the equation?

"There are x number of cars in a garage. If all but 2 of those cars are Fords, all but 2 of those cars are Toyotas and all but 2 of those cars are Hondas, what is x?"

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I would say that if you think that the only possible answer is 3, then you are probably applying Aristotelian logic, even if you don't realize it.

As such, I believe that 2 is a perfectly valid answer, but that's because:

All my cows are purple! (BTW, I don't own any cows. )

Nope I am not applying Aristotelian logic since I think you are not being honest (you're kidding me!) when you say all your cows are purple. "All my cows are purple" implies that you own more than one purple cow. You cannot apply logic until you reduce the sentence to have an unambiguous meaning. That meaning comes from the English language. My semantic interpretation of the sentence leads me to believe that there are 3 items involved when you use the phrase "All but two of ....". It has nothing to do with the school of logic I follow. Here are the two possible mathematical representations of the sentence:

The statement "All but two of my cars are Fords" can either mean

The cars in the set U - A are all Fords where the cardinality of A is 2

OR it can mean

The cars in the set U - A are all Fords where the cardinality of A is 2 and the cardinality of U is >= 3

The meaning of the original sentence is likely to be interpreted fairly uniformly amongst speakers of English as meaning the second due to the implicit connotations of the wording of the sentence. This is a purely semantic phenomenon and has nothing to do with the school of logic involved.

Cheers!

--

Vig

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You may argue that this is not directly related to the discussion, but I think this scenario shows that there are multiple uses for the word "all," some that refer to something as "all of it" even if there are none.

I never had a problem with any definition of the word "all".

Smoke was billowing from the town hall when two paramedics, Johnson and Bates, arrived on the scene. A gas main had exploded, causing part of the building to collapse. Bates turned to Johnson and said, "You treat all of the burn victims and I'll deal with the lacerations and broken bones." Johnson did as he was told.

If no one got burned, is there a problem with Bates' statement?

What sort of problem? Sure, there technically is a problem as Bates implied that there were burn victims to treat without being sure there were any, but this is not a very good analogy regarding the problems I found with there being only two cars in the riddle. BTW, I have no problem with zero being a number of burn victims.

Of course, in this situation, there is no foreknowledge of events, so Bates' can't know that there are no burn victims when he made the statement. Maybe this implies that the problem with the cars comes from the expectation of foreknowledge based on the fact that "I" am telling you about "my" cars.

That's one problem, but I gave you two specific problems that you didn't address.

So I'm not sure if this changes anything or not, but what if we take "me" out of the equation?

Let's focus on the riddle as it is.

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well, all of my cars are boats

great discussion everyone! here's my take: i think the book is correct. i don't see how 'all' followed by 'but' can ever equal an amount less than or equal to the amount following 'but': in this case (2). bonanova is correct to remove truthiness and proofs from the logic also because all we have to work with is a statement, not the speaker's lifestory. We must use only proper language interpretation. To interpret 'all' as possibly equaling zero, which i tried to do, is just not a logical deduction based on the language, in my opinion at least.

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That is not logically or grammatically correct. You can not separate all your cars from all your cars.

The statement was: "All of my cars, except all of my cars, are DeLoreans." Please explain how it is grammatically incorrect. Is it my punctuation? Verb/object agreement?

Also, please explain how the statement is logically incorrect. If it is incorrect does that mean that one of the following is correct?

"Apart from a number, which is different from the number of cars I own, all my cars are DeLoreans."

OR,

"Apart from all my cars, a number different from the number of cars I own are DeLoreans."

because both of these statements are contradictory.

Nothing happening is not an event. We're getting on another topic here, but let's look at what was said:

"If a tree falls in the forest... "

Starting with, "If a tree falls in the forest" is asking about what would happen if a tree did in fact fall in the forest. That tree is one tree.

You explicitly changed the question, in post 41, to: "If zero trees fall in the forest and nobody sees it happen, did it in fact happen?" to which I contend the answer is "Yes. The event where no tree fell in the forest, and nobody witnessed it, occurred."

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Sure, but before we make the conversion, we have to make sure everything is being interpreted correctly.

"all but 2 of my cars are fords"

1. Zero amount of cars is not a different make from cars that exist. You can't make that statement if you have two cars which are Camaros because you have specifically said that cars that you own are Fords. This is even a more illogical when owning no Fords when coupled with the next point:

2. If all 'but" two of his cars are Fords, can it be logically possible to have only two cars? No. He is separating two cars from an amount of cars. That amount must logically be greater than 2 or no separation can be made. Claiming that he has 2 cars plus an amount of cars equaling zero that are Fords is illogical.

OK, let's look at your statements. Let's start with #2 since you refer to it in #1.

Is this situation illogical?

Say that Greedy Gus has a box of 10 chocolates and says, "I'll share these with you in the following manner: I'll flip a coin, and for every heads, I take a chocolate and for every tails, you take one." Greedy Gus never shares, so you expect a trick, but you figure that there's no disadvantage to agreeing, so Gus takes a coin from his pocket and flips it 10 times, and every time it lands heads. Highly suspicious, you grab the coin from Gus and discover that the coin is in fact two-headed.

----

Gus separated the chocolates into two groups: the heads group and the tails group. The tails group did not exist, but that doesn't mean that it is illogical for him to claim that it exists, merely misleading. Similarly, referring to the cars "all but 2 of my cars are fords" when you have no Fords is misleading, but is that different from the claiming that tails is an option?

As for argument #1, consider this. If I have no cows, then I can say that "All the cows I own are purple." This translates to propositional logic something like this:

If I own a cow, then it is purple.

Since I don't own any cows, the first part of the implication is false, so the statement is true. I'm afraid that I'm a little tired at the moment and I'm not putting this in quite the way I want to <_< , but maybe someone can see what I'm trying to get at...

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You explicitly changed the question, in post 41, to: "If zero trees fall in the forest and nobody sees it happen, did it in fact happen?" to which I contend the answer is "Yes. The event where no tree fell in the forest, and nobody witnessed it, occurred."
i would say "no", because nothing happened. You cannot say that something has occurred when in fact and in your previous statement you are clearly describing 'nothing'.

edit: perhaps 'no' is the wrong answer as well. it's more like 'undefined'

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The statement was: "All of my cars, except all of my cars, are DeLoreans." Please explain how it is grammatically incorrect. Is it my punctuation? Verb/object agreement?

Also, please explain how the statement is logically incorrect. If it is incorrect does that mean that one of the following is correct?

The punctuation is fine; your logic is not. You can't give all of your cars an exception from all of your cars. They are are all one group.

Lets do this with 0, 1, and 2 DeLoreans

If you have 0 DeLoreans, can they all be Deloreans and not DeLoreans simultaneously? No. Your set of zero DeLoreans can only logically belong to one group.

If you have 1 DeLorean, can it be a Delorean and not a DeLorean simultaneously? No.

If you have 2 DeLoreans, can they both be Deloreans and not DeLoreans simultaneously? No.

You run into the same logical problem with your statement regardless of the number of DeLoreans you have.

"Apart from a number, which is different from the number of cars I own, all my cars are DeLoreans."

OR,

"Apart from all my cars, a number different from the number of cars I own are DeLoreans."

because both of these statements are contradictory.

They're both incorrect.

The number can't be different than the number of cars you own. "All your cars" can't be a different number than "all of your" cars.

You explicitly changed the question, in post 41, to: "If zero trees fall in the forest and nobody sees it happen, did it in fact happen?" to which I contend the answer is "Yes. The event where no tree fell in the forest, and nobody witnessed it, occurred."

Contend all you want. You're taking what I said out of context and not taking into account why I answered as I did. If zero trees fall, there is no it to see happen as "it" is referring to the falling of a tree. So no, it did not happen.

If a tree falls in the forest and no one sees it happen, did it in fact happen?

I replied:

Did it happen if zero amount of trees fell in the forest?

Yes, the event where zero trees fell in the forest occurred.

Nothing happening is not an event. We're getting on another topic here, but let's look at what was said:

"If a tree falls in the forest... "

Starting with, "If a tree falls in the forest" is asking about what would happen if a tree did in fact fall in the forest. That tree is one tree.

Now you're claiming I changed the question. No, I didn't. The question was "If a tree falls in the forest and no one sees it happen, did it in fact happen?"

Well, yes, of course it did. The reason I asked "Did it happen if zero amount of trees fell in the forest?" is because pw0nzd was attempting to make the statement analogous to the riddle. It's only analogous if there are no trees as there are no Fords in his scenario. ""If a tree falls in the forest..." is asking about what would happen if a tree did in fact fall in the forest. That tree is one tree.

None of this has anything to do with discussing specific problems with the answer to the riddle being 2, btw. Can we stop talking about trees and DeLoreans and get back to the riddle?

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Say that Greedy Gus has a box of 10 chocolates...

This is your second analogy. I'm not going to break down every analogy and show how they're different than the riddle.

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This is your second analogy. I'm not going to break down every analogy and show how they're different than the riddle.

It's just that when I have trouble solving one problem, I often find that solving a similar puzzle makes the first problem clearer. Plus, I was trying to address your points specifically and the analogies seemed to me to cover the issues you had with the reasoning behind the alternate answer.

You complained that it was illogical to separate objects into non-existent groups, so I created a non-existent group in a way that seemed quite logical. As such, I think that it is a valid counterpoint to your assertion.

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It's just that when I have trouble solving one problem, I often find that solving a similar puzzle makes the first problem clearer. Plus, I was trying to address your points specifically and the analogies seemed to me to cover the issues you had with the reasoning behind the alternate answer.

They don't suffer the same flaws.

You complained that it was illogical to separate objects into non-existent groups

No, I didn't. My objection to 2 being an answer goes beyond that and takes the entire statement into question.

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I immediately came up with the OP's answer, but the book's answer is also correct. I think the problem here is that English is an imprecise language which leads to misunderstandings between different people depending on their backgrounds, and there are many different backgrounds on this forum. I have a technical background and have taken multiple courses on Boolean logic, so I have no problem with the statement "All of my cars are Fords except two," when the speaker has two cars that are not Fords. It is absolutely a true statement logically. Potentially misleading, but true. Someone with a less technical background might have a problem with the statement and say it is false because they have a different understanding of what it means. But if this question were asked on a logic exam, both answers would be acceptable, and if the teacher tried to say otherwise, he or she would have some angry students to deal with.

I could show how the problem has two answers using some Boolean logic and set theory, but I'll refrain unless someone asks since this isn't a technical forum and I think it might go over some heads. No offense meant to anyone, of course.

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Earlier post that turns on the question of existential import [which you can Google]

A set may be empty and still be discussed.

We can talk about the set of humans living on Mars [empty, so far as we know].

Some have called empty sets non-existent sets.

Probably hoping to imply that they can't be talked about.

It's more accurate to say the members of an empty set don't exist.

Assuming no humans now live on Mars, the ordinary English meaning of this statement is not clear:

All humans living on Mars are male.

1. Does the statement assert human life on Mars?
2. Does the statement assert only that if there are humans on Mars, they are male.
If the second interpretation is taken, the statement is equivalent to saying

No humans living on Mars are non-male.

The first is the Aristotelian meaning; the second is the Boolean meaning.

Aristotelian logic holds that the Universal Quantifier [fancy word for "All"]

has Existential Import [fancy term meaning the set is not empty - it has at least one member.]

Boolean logic does not hold that All has EI.

Note that if Some is used: Some humans living on Mars are male,

then we assert existence of humans living on Mars.

Both logical systems call Some the Existential Quantifier.

For everyone, Some has existential import [EI].

In Aristotelian logic, All has EI; in Boolean logic it does not.

Some have argued that in "ordinary English" All uniformly has EI.

If you talk about a class of things, then at least one member of that class exists.

Or, ordinary English is always Aristotelian.

To see this is not the case consider a tract of private land posted with the sign:

All trespassers will be prosecuted.

We do not usually take this sign to have asserted the existence of trespassers.

It makes the attempt, rather, to ensure there are no trespassers.

It means, logically, no one who trespasses will go unprosecuted, or

No trespassers will be unprosecuted.

In this case, ordinary English takes the Boolean meaning.

So, in discussing the statement All my cars but two are Fords,

to say that ordinary English demands there must be at least one Ford overstates a point.

That's an assertion. Let me attempt a proof, closer to the language of the OP.

Assume the posted land has two owners: Jim and John. Anyone on that land,

other than Jim or John, would then be trespassing. The sign could then be worded:

All persons found on this land other than Jim and John will be prosecuted.

The meaning of the sign is clear.

The sign does not imply the presence of a person other than Jim or John.

The sign makes sense by the rules of ordinary English.

Now lets talk about collections of cars.

A man tells his partner to go out and buy him 100 cars.

His instructions are:

1. Buy me a Mercedes, then
2. Buy me a Rolls Royce, then
3. Make the rest of them Fords.

How many cars do you have? - I don't know, he replies, my partner is buying them now.

What kind are they? Well, I don't know that either, but he called to say he'd bought a Mercedes and a Rolls Royce.

A Mercedes and a Rolls? Isn't that a bit extravagant?

Not really, except for those two, all my cars are Fords.

At the time the man made that statement, it's possible no Fords had been bought.

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Now lets talk about collections of cars.

A man tells his partner to go out and buy him 100 cars.

His instructions are:

1. Buy me a Mercedes, then
2. Buy me a Rolls Royce, then
3. Make the rest of them Fords.

How many cars do you have? - I don't know, he replies, my partner is buying them now.

What kind are they? Well, I don't know that either, but he called to say he'd bought a Mercedes and a Rolls Royce.

A Mercedes and a Rolls? Isn't that a bit extravagant?

Not really, except for those two, my cars are all Fords.

At the time the man made that statement, it's possible no Fords had been bought.

Like the Irishman that went to buy a used car - "I only have \$50" he sold the salesman, who said he had a running but old mustang that was a bit tied inside and could do with a make over - "For only "500".... "I ONLY have \$50" replies paddy...... How about a beat up old chevy with need of new head lamps, tail lights and a hood that needs replacing for only \$200. "I ONLY HAVE \$50" paddy insists impatiently....... Ok! Says the salesman - we have an old ford just traded in you can take way for \$50 but it has no doors "Jeesus" exclaims paddy. "It's no good to me if it has no doors" - "how will i get in?"

Edited by Lost in space

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The meaning of the original sentence is likely to be interpreted fairly uniformly amongst speakers of English as meaning the second due to the implicit connotations of the wording of the sentence. This is a purely semantic phenomenon and has nothing to do with the school of logic involved.

If popular opinion is how this will be decided, I think the number of people supporting 2 as a valid answer makes 2 a valid answer.

Also, a lot of people have been arguing that, in normal English usage, someone wouldn't describe the situation of having two Chevy's like in the OP. This ignores the fact that no one would describe the other solution - having one Ford, one Honda and one Toyota - in those words (unless it was for the purposes of a Brain Teaser. The phrase "all but" does normally ipmply a non-empty set; in fact it usually implies majority of the whole (ie, most of my cars are Fords, I just have two that aren't). Why is one unusual usage more acceptable than the other?

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After reading the argument I can see how you could get 2.

But in normal English conversation (as we speak in NZ) my vote would have to go to 3.

All but one of my cars is a Ford and all but one of my cars is a Nissan.

Guess how many cars I have and what car Make they are?

You got it - 2. a Ford and a Nissan

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So knowing how many cars I have and their makes. The "2" supporters would then agree with this statement:

All but 2 of my cars are Holdens.

All Ford owners would know that I wouldn't own a Holden so the answer would be 2.

However, in normal English conversation I wouldn't say this, well not unless I was talking to a Holden owner.

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But in normal English conversation (as we speak in NZ) my vote would have to go to 3.

All but one of my cars is a Ford and all but one of my cars is a Nissan.

Guess how many cars I have and what car Make they are?

My argument is that, in normal English conversation, you wouldn't use the phrase "All but one of my cars is a Ford and all but one of my cars is a Nissan" in either case. You would either say, 'I have two cars, a Ford and a Nissan," or, "I have two cars, and they are both Holdens."

If you're using the normal English conversation rules, then you have to reject both solutions, because people don't talk like that outside of brainteasers. Assuming that the puzzle has at least one solution, though, I think you need to allow either one (meaning two or three, for the OP).

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Earlier post that turns on the question of existential import [which you can Google]

A set may be empty and still be discussed.

We can talk about the set of humans living on Mars [empty, so far as we know].

Some have called empty sets non-existent sets.

Probably hoping to imply that they can't be talked about.

It's more accurate to say the members of an empty set don't exist.

Assuming no humans now live on Mars, the ordinary English meaning of this statement is not clear:

All humans living on Mars are male.

1. Does the statement assert human life on Mars?
2. Does the statement assert only that if there are humans on Mars, they are male.
If the second interpretation is taken, the statement is equivalent to saying

No humans living on Mars are non-male.

The first is the Aristotelian meaning; the second is the Boolean meaning.

Aristotelian logic holds that the Universal Quantifier [fancy word for "All"]

has Existential Import [fancy term meaning the set is not empty - it has at least one member.]

Boolean logic does not hold that All has EI.

Sorry bonanova (yikes, should i really start an argument with you ) but I'm not buying it. I would add a third possible assertion, which is the one i would agree with. [Does the statement assert the speaker has made a logical fallacy?] If one were to take claims such as these seriously then I could proclaim anything and have it be taken seriously. I believe there is a point in English language which, if crossed, will render the statement illogical.

In your final example about the 98 Fords, the guy didn't have them. So saying that he did was a logical fallacy. He needed the qualifier 'will' or 'hopefully will' be Fords. Me saying: 'All but 5 of my submarines are yellow' to be thought of by someone as me having zero submarines (which would be the logical deduction about me hopefully ) does not follow suit. Sure, I have zero submarines, but my statement 'all but 5 of my submarines are yellow' is meaningless if i have zero of them.

I say (again) that not only does the phrase 'all but' imply a non-empty set, it implies a set of greater value than the figure mentioned after it. In this case it's 2, so the set must be greater than 2. Given that, the answer must be 3.

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I say (again) that not only does the phrase 'all but' imply a non-empty set, it implies a set of greater value than the figure mentioned after it. In this case it's 2, so the set must be greater than 2. Given that, the answer must be 3.

But if you insist that 'all but' implies a greater value than the figure mentioned, then you have to accept that neither solution works, like Chuck Rampart said. With the 3 answer, there are only one of each variety of car, which is less than two, so by your definition, neither answer correctly uses "all but." So I would reiterate what was stated above that in the case of brain teasers, both solutions should be viable.

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But if you insist that 'all but' implies a greater value than the figure mentioned, then you have to accept that neither solution works, like Chuck Rampart said. With the 3 answer, there are only one of each variety of car, which is less than two, so by your definition, neither answer correctly uses "all but." So I would reiterate what was stated above that in the case of brain teasers, both solutions should be viable.

the answer calls for how many cars, not how many of each car.

edit: 'all' clearly refers to all cars, not all Fords, etc...

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Hi Itachi-san,

I agree that if you have 0 submarines "all but 5 of them" does not make sense.

If you have 7 submarines, all but 5 would refer to a set of 2.

If you have 6, all but 5 would refer to one of them.

If you have 5, all but 5 would be the empty set.

If you have 4, all but 5 would not make sense.

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In my 100 car analogy, I made sure that his partner had purchased the mercedes and the rolls

before the the man made the statement All but two of my cars are Fords.

The words "but two" eliminates two of his cars - the mercedes and the rolls - from consideration of being fords.

"All the others" then refers to a set of cars, numbering 0-98 - depending on his partner's buying activity to that point.

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puzzle in book:all but 2 of my cars a fords all but 2 of my cars are toyotas and all but 2 of my cars are hondas how many cars do i have

3 cars

couldnt he have 2 cars and none of them be a honda toyota or ford?

The book is correct. 3 is the answer. "All but..." refers to at least one. Therefore, "all but 2 of my cars are ____" translates into "At least 1 car is THIS, 2 are not". So, at least 1 car is a Ford, 2 are not. At least 1 car is a Toyota, 2 are not. At least 1 car is a Honda, 2 are not. Add em' up and viola! You have 3 cars. Simple ;-)

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