I think I need you to explain to me how anyone could see ‘all Xes’ and think it could potentially mean ‘no Xes.’ That just seems illogical to me. I’m not saying you’re wrong - I just can’t understand that train of thought. It’s using the word ‘all’ to mean, well, the opposite of what ‘all’ means.
Language may not be highly formalized, but if I said potato and claimed it actually meant hippopotamus, you’d think I was crazy; words do have commonly accepted meanings. They have to, or we can’t communicate.
Well, for example, I’m happy to say that all unicorns have horns. This despite the fact that I’m also happy saying there aren’t any unicorns. So this is one case where I am perfectly content to claim something holds of “all Xes”, even though there aren’t actually any Xes.
The logical negation of “all Xes have property P” isn’t “no Xes have property P”. On this understanding, those two claims do not contradict; rather, the logical negation of “all Xes have property P” is “there’s at least one X which doesn’t have property P”. That is, in this manner of speaking, to say “all Xes have property P” is the same as to say “there aren’t any Xes which don’t have property P”.
My first thought (since I figured all roosters ARE chickens) was that the answer to number one was “No”. One of those trick questions where you are expected to read thoroughly to find the actual question, not assume you know what they are asking from the top. “Yes or no…given this set of information, can you determine the answer?”
And I had no idea at all about using Ctrl-A to reveal spoilers! Thank you so much!
On the other hand “parents can fuck up too” is one of those valuable life lessons that he does need to go through a few times, and this is a relatively painless way of experiencing that one…
This strikes me as one of those assignments designed to find out which kids do their homework themselves and which ones discuss it with their parents. I remember “helping” my early grade school children over-think a number of assignments.
Although I got the answers, I take great offense that such questions would be put to someone so young. I feel they are taking extreme liberty with, if not outright misusing, the construct “all are X except Y”.
I have never seen anyone use such a term when the number of X’s is less than or equal to Y. The implication has always been the exception clause is just that, an exception. Exceptions are not ever the majority or half of the whole; they are by definition the “odd ones out”, and hence fewer in number than the majority".
That’s how I was taught to describe things in numerical relationship to others.
But unicorns don’t exist, so that idea makes sense in context. These questions are written in a way where, contextually, you could only say x=0 if you were the most extreme solipsist ever. I think the teacher was wrong - you can figure out how many animals there are.
As has been mentioned, it depends on how you define the animals.
The only way we can know how many animals there are in the first question is if “chickens” and “roosters” don’t intersect.
If all roosters are chickens, there could be any number of roosters, and one animal that is not a chicken.
If some roosters are chickens, there could be one non-chicken rooster, one non-rooster chicken, and any number of chicken roosters.
Undefined – roosters are male chickens, so there could have been six roosters and one dog, thirty-five roosters and one iguana, ninety roosters and one chimpanzee…
Eh, there’s a genuine ambiguity here. Your way of interpreting the language of the question isn’t the only possible way, and the teacher’s way of interpreting the language isn’t the only possible way. There isn’t a crisp One True Interpretaton of ordinary natural language; it’s not that formalized, as evidenced by the fact that so many speakers are willing to interpret things in different ways.
The ambiguity is unfortunate. But let’s not pretend that the teacher’s interpretation is untenable. It’s one reading of the words which is natural to some and unnatural to others. It’s not the only possible reading, and not the one everyone will settle on. But it is one which many would take as their interpretation. It’s not some crazy, la-la land misunderstanding of English.
I have to agree with you. My interpretation of the language is such that “all,” in the context of question number two, would not mean zero unless one is being deliberately evasive or using it as a rhetorical flourish (for example, “Where the beer?” “All the beer I have is in the fridge.” opens fridge “There’s no beer here.” “Exactly. That’s all the beer I have.”) But, I can concede that others will have differing interpretations that “all” does not include the assumption that there is at least one, especially in a logic puzzle which is quite often filled with such “gotchas.” It sounds weird to me, I don’t agree with it, but, hey, if others claim they do read it this way, who am I to argue?
I would note that using “all” to mean “just one” also has similar, albeit perhaps to a lesser degree, pragmatics of evasiveness and/or rhetorical flourish, so either way, as concerns this puzzle, we’re going to get some of that.
Like I said, I need you (or someone) to explain it to me, then, because I really don’t see where the ambiguity lies in this particular logic game. I’m not claiming that all language is unambiguous (how stupid would anyone have to be to claim that?) but I am saying that, in the questions posed by the teacher, I don’t see the ambiguity. ‘All’ can be ambiguous when people are using it casually, and they really mean ‘nearly all,’ or when people are using it hyperbolically, but a logic quiz isn’t casual and isn’t the place for hyperbole. Obviously you and a couple of others can see some ambiguity there, and I believe that you see it - but it’s invisible to me.
Even if you say that there’s ambiguity, meaning that you could say that you can’t figure out how many animals there are, then there’s still ambiguity the other way too, isn’t there? So it would be wrong to only accept ‘no’ as the answer.
For what it’s worth, belladonna, having just read through this thread for the first time, my answer was the same as yours.
This is one of those situations where the language is so vauge that you could have three or four correct answers depending on which the teacher meant. She should have accepted any of them as long as the logic was explained. My answer was based on the version I thought that a second grader would use, but apparently these are special super-advanced second graders.
The word “all” is not being used ambiguously here. When Mr. Carter says “All are cows except two.”, “all” means “all of the animals, however many that is”. It’s just that “all … except two” means “two fewer than all of them”. If you have two animals, then “all” means “two”, and two fewer than all is zero.
The only ambiguity in the question is that you don’t know how many “all” is. Suppose Mr. Carter says “All the animals in my pasture have four legs. There are at least two animals. Can you figure out how many animals there are?” Obviously the correct answer is “no”.
Any ambiguity in what Mr. and Mrs. carter mean just makes it harder to figure out how many animals there are, so the answer to both questions is still “no”.
I think I see what you mean, but, to be honest, I still don’t really ‘get it’ because I’m stuck in a world where nobody would say ‘all’ and mean ‘zero’ and it still comes down to that even if you focus on ‘two fewer than all.’ Honestly, if I said to you ‘all the students are in the hall, except two,’ wouldn’t you just be a little surprised if you opened the door to the hall and no-one was there?
Actually, rereading, no, I still don’t get it at all (heh). I don’t see how it matters that two fewer than all could be zero because ‘all’ can never be ‘zero.’ They cancel each other out. Oh well - I can live with failing at second-grade logic.
You don’t actually need “all” to encompass zero, as such. If “all” can be two, then “two fewer than all” can be two fewer than two, which is zero (so goes the argument).
Yes, because that implies to me there are at least two students in the hall. But no one is arguing that zero animals is a possibility for either question.
I remember “helping” my early grade school children over-think a number of assignments.