Hi all, I made another flashcard set. This time for memorizing Quantifiers. Flashcards are what really helped me in undergrad and so I decided to make them to companion my 7sage studies. Thought I'd share to help others who would benefit :) made a folder that I will most likely add more sets to as I go. Much Love and happy studying! https://quizlet.com/user/ehoffmanwallace/folders/lsat-7sage-flashcards
@AkshayaAnnampedu I think the focus is the meaning of the quantifier "some" although yes "students" is plural, "some" infers the range of the lower boundary of students is one. Just like how in the next lesson he discusses that even if all the students could read, that would not exclude it from the quantifier of "some." So just as "some" can mean all, "some" can mean one student even in the grammatical context of a plural "students"
during these ambiguous questions, is it safe to assume that when they say some student(s) (plural) that it is 2 or greater? (rather than just 1 or greater)?
Hmm, I think the example of "a heap of sand" slightly confuses me. If there is "a heap" of sand, then there must be at least 1 sand grain. Then how is it different from "some" in this regard?
@AcuteMeow because the definition of heap is 'a great number or large quantity' so it is not reasonable to interpret one grain of sand as a heap. So, the question becomes how many grains of sand until you have a large enough quantity for it to become a heap?
@henrymandenmuller heap implies a large quantity of sand (definition of heap is 'a great number or large quantity'). There is a lower limit of what can be a heap but we do not know what it is, hence why it is a vague amount.
Am i correct in that words like "all, every, etc" like we have learned in past conditional lessons, would make the intersecting sets become a conditional relationship? one of the differences with intersecting sets from subset/superset is that in the latter, all members of a group are within an umbrella of the necessary condition, not some of them, or most of them. So once it's only a bit of overlap, then we have to switch our perspective to the quantifier intersecting set language because it's explaining a different relationship in which there is non-guaranteed 100% overlap as there is in the conditional relationship. Is that right?
is the reason why we have a small circle and a bigger circle because the small one is a subset and the bigger one is the superset? can anyone please clarify
My understanding is that 'subset' and 'superset' apply when one group (set) is entirely encompassed by another group (set), which isn't the case here. Returning to the earlier example in the curriculum: all cats are mammals. Cats would be the subset, while mammals would be the superset, as it is impossible to be a cat without also being a mammal.
In this section, the size of the groups does not really matter. My understanding is that the larger circle demonstrates it contains members who do not belong to the intersecting group (the smaller circle). For example, while some students in Mrs. Stoops' class can read (represented by the intersecting area), others cannot (represented by the non-intersecting area of the smaller circle). If you look to the larger circle, you can see the opposite. Of all the people who can read (the larger circle), some are in Mrs. Stoop's class (the intersecting area), while others are not (the non-intersecting area of the larger circle). We can't say that the small circle is the subset of the large circle because the smaller one is not entirely engulfed by the larger, they just happen to share some common members.
Am I mistaken or does "some students" imply at least 2, not at least 1 students? The quantifier does NOT say "some of the students", it says some students, as in a plural amount. If it was one student it wouldn't fit, right?
What might be the sharp boundaries for a quantifier like "several", or "many"? "Some" is explained above, and most entails a majority, but what of these other ones? Thank you.
Doesn't vagueness have some, or at least one, sharp boundary condition even if it's fuzzy? For example, one grain of sand isn't a heap ("a disorderly collection of objects placed haphazardly on top of each other") while two grains of sand seems unclear if it is. Therefore, one gain of sand is a sharp boundary for 'heap,' namely that it isn't a heap.
The lesson takes lower boundaries of ambiguities like 'some' and says it has a sharp boundary and vague concepts like 'heap' as having a fuzzy boundary, but thinking of what would constitute a lower boundary for 'heap' seems like it would be a sharp boundary in the same way as 'some' has a sharp boundary. For upper boundaries, this distinction between sharp boundaries and fuzzy boundaries might be different and I think probably is. Yet, in this lesson alone, given the two examples, lower boundaries between the two aren't exactly different in the way it's making it out to be .
This is a great digression, I love it LOL! For fun, I think there are two independent(?) reasons why your argument that even fuzzy vagueness has at least one sharp boundary condition isn't completely true.
1) It is possible that "heap" COULD have at least one sharp boundary condition as you define it here such that a heap is not vague but rather ambiguous. But when you step back, this might only be because we took the liberty of defining "heap" basically as "some amount of grain of sands" when that might not really be what, linguistically speaking, we mean by heap. This lesson was probably thinking of "heap" in the more colloquial sense, especially from the Sorites Paradox which tries to define where the boundary of heap and not-heap is. Aside from the Paradox's one assumption that a FEATURE of a heap is such that a heap of n and a heap of n-1 are indistinguishable, the argument seems pretty successful in pointing out that there does seem to be a paradox in trying to define what a heap is such that it can be distinguished from a non-heap.
I feel like there's a flaw in your logic because you pre-empted a definition of what a heap is --- that is, a heap = "some amount of grains" which is basically the same as "some." If you walk into the Sorites Paradox argument with that pre-formed definition in mind, you would of course not find the argument compelling at all nor would you find a paradox. But the proponent of the Sorites Paradox argument would also not find your dismissal of the argument compelling either, since the entire point of the Sorites Paradox is all about finding a definition to a heap vs. non-heap --- so it defeats the purpose of the paradox if someone assumes a definition at the onset! I think this is a very long-winded way of saying that a) vagueness and ambiguity are in fact meaningfully different; and b) lower boundaries in either case, if they do exist, are probably different too. The only way this seems not to be true is by perhaps committing a category error: i.e., by equating the definition of an instance of vagueness [a heap] with an instance of ambiguity [some].
2) Even if your argument that "heaps" are not vague but rather ambiguous is valid, it does not follow that all cases of valid things are in fact really ambiguous. You suggest in your comment that "vagueness has some, or at least one, sharp boundary condition even if it's fuzzy" by using the heaps case as an example. In the case of heaps, it could be true that the vagueness of heaps in fact has sharp boundary condition(s) that make it closer to/indistinguishable with ambiguity. This unfortunately doesn't mean that vagueness in other cases would necessarily have sharp boundary conditions.
I would like to clarify one thing, the point of my comment (my last sentence in the original comment).
Firstly, I just want to say that agree with your comments about vagueness.
I agree that vagueness exists and isn't like ambiguity in the sense that 'some' or 'many' are ambiguous. Sorites paradoxes exist for vague concepts, like 'heap' or 'bald' and this doesn't apply to ambiguities.
The lesson uses the word 'heap' in contrast to the word 'some' by pointing to one lower boundary as being sharp or not sharp, but the contrast, as its presented, isn't as sharp itself in the following sense.
It's true that I'm defining 'heap' as a set of members that are organized in a given way, but I'm just appealing to what I think 'heap' means through its use by common speakers. In my definition of heap, I quoted a dictionary definition (with the assumption that a dictionary definition is what's commonly used to define words that common speakers use). Through this definition, I take it as an implication that it is a set that excludes one and only one member.
Of course, these vague concepts aren't sharply defined in a logical sense from how people use 'heap' or 'bald.' Yet, common use of vague concepts do have someprima facie sharp boundary intuitively even if there seems to be no logical sharp boundary.
Sorties cases result from a given starting point of F being a predicate and through a series of applying, in some formulations, some method (Fn-1 or Fn+1) results in a conclusion where F remains constant even though that conclusion is counterintuitive (and therefore paradoxical).
I'm pointing at our intuitions, given what I take we mean by a word through its common use, and saying "here's a boundary" where F is or isn't sharp. This reference is in comparison to what the lesson uses with one boundary for one ambiguous word.
In my original comment I take it to be the case that vague concepts are fuzzy (in opening question of my comment), but I don't think the distinction between ambiguity and vagueness is given through the one example of lower boundaries by way of introducing 'heap' and 'some.'
I think the distinction would have been given if the lesson would have taken upper boundaries as examples instead since it would fall outside of what I think is the common definition of 'heap' to mean ( "a disorderly collection of objects placed haphazardly on top of each other").
Great comment! I applaud you for getting into some of the nitty gritty of the sorties paradox!
Vagueness does not have a specific limit its limits are not sharp and fuzzy. If you say some or other quantifiers, you know "AT LEAST ONE" that means that there is lower limit if that makes sense.
In the full diagram, it says that ambiguous quantifiers have "sharp upper and lower boundaries." But in the next lesson (ie. lesson 3) it says that the ambiguous quantifier, 'Some,' "does not have an upper boundary."
This seems to me like either an error or improperly explained.
It becomes clear in later lessons that the 'sharp upper boundary' refers to something like 'all.' It is clear that this is different than vagueness. Still, though, it might be worth some clarification earlier on so as to avoid the apparent contradiction.
I got to give feedback on how well this V2 version is organized. It is so much better than V1. I started learning Quantifiers after like 30% of the actual LR section ON V1, I love how this is introduced way before actually starting LR. #feedback
As someone who didn't experience v1, this is good to hear. I take PTs weekly and already am seeing a huge difference in my LRs and RCs because of the grammar tools 7Sage gives in these foundation lessons. #feedback
If you dont mind me asking, when did you start taking PTs? I am at this point in the curriculum and have not taken one yet as I dont want to spoil them before I have a better foundation.
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31 comments
Hi all, I made another flashcard set. This time for memorizing Quantifiers. Flashcards are what really helped me in undergrad and so I decided to make them to companion my 7sage studies. Thought I'd share to help others who would benefit :) made a folder that I will most likely add more sets to as I go. Much Love and happy studying! https://quizlet.com/user/ehoffmanwallace/folders/lsat-7sage-flashcards
But doesn't "some students" imply that the lower bound should be 2, not 1? Since students is plural?
@AkshayaAnnampedu I think the focus is the meaning of the quantifier "some" although yes "students" is plural, "some" infers the range of the lower boundary of students is one. Just like how in the next lesson he discusses that even if all the students could read, that would not exclude it from the quantifier of "some." So just as "some" can mean all, "some" can mean one student even in the grammatical context of a plural "students"
during these ambiguous questions, is it safe to assume that when they say some student(s) (plural) that it is 2 or greater? (rather than just 1 or greater)?
@TONI.SCHIFFMAIER some in lsat language is "at least 1 or more"
Hmm, I think the example of "a heap of sand" slightly confuses me. If there is "a heap" of sand, then there must be at least 1 sand grain. Then how is it different from "some" in this regard?
@AcuteMeow because the definition of heap is 'a great number or large quantity' so it is not reasonable to interpret one grain of sand as a heap. So, the question becomes how many grains of sand until you have a large enough quantity for it to become a heap?
Some bare minimum is 1 but is upper limit just under 50%
@mattsenemar what is there are only 2 students in the class? Some would still be applicable.
A "heap of sand" seems to suggest a sharp lower bound of at least 1 grain of sand
@henrymandenmuller heap implies a large quantity of sand (definition of heap is 'a great number or large quantity'). There is a lower limit of what can be a heap but we do not know what it is, hence why it is a vague amount.
Am i correct in that words like "all, every, etc" like we have learned in past conditional lessons, would make the intersecting sets become a conditional relationship? one of the differences with intersecting sets from subset/superset is that in the latter, all members of a group are within an umbrella of the necessary condition, not some of them, or most of them. So once it's only a bit of overlap, then we have to switch our perspective to the quantifier intersecting set language because it's explaining a different relationship in which there is non-guaranteed 100% overlap as there is in the conditional relationship. Is that right?
is the reason why we have a small circle and a bigger circle because the small one is a subset and the bigger one is the superset? can anyone please clarify
To clarify: we don't know, in this case, that it is entirely engulfed like we do when we have a subset-superset relationship.
My understanding is that 'subset' and 'superset' apply when one group (set) is entirely encompassed by another group (set), which isn't the case here. Returning to the earlier example in the curriculum: all cats are mammals. Cats would be the subset, while mammals would be the superset, as it is impossible to be a cat without also being a mammal.
In this section, the size of the groups does not really matter. My understanding is that the larger circle demonstrates it contains members who do not belong to the intersecting group (the smaller circle). For example, while some students in Mrs. Stoops' class can read (represented by the intersecting area), others cannot (represented by the non-intersecting area of the smaller circle). If you look to the larger circle, you can see the opposite. Of all the people who can read (the larger circle), some are in Mrs. Stoop's class (the intersecting area), while others are not (the non-intersecting area of the larger circle). We can't say that the small circle is the subset of the large circle because the smaller one is not entirely engulfed by the larger, they just happen to share some common members.
Am I mistaken or does "some students" imply at least 2, not at least 1 students? The quantifier does NOT say "some of the students", it says some students, as in a plural amount. If it was one student it wouldn't fit, right?
I understand what you're saying. However, in terms of lawgic, some = at least 1.
A video has recently become available for this lesson.
What might be the sharp boundaries for a quantifier like "several", or "many"? "Some" is explained above, and most entails a majority, but what of these other ones? Thank you.
#philosophicaldigression
Doesn't vagueness have some, or at least one, sharp boundary condition even if it's fuzzy? For example, one grain of sand isn't a heap ("a disorderly collection of objects placed haphazardly on top of each other") while two grains of sand seems unclear if it is. Therefore, one gain of sand is a sharp boundary for 'heap,' namely that it isn't a heap.
The lesson takes lower boundaries of ambiguities like 'some' and says it has a sharp boundary and vague concepts like 'heap' as having a fuzzy boundary, but thinking of what would constitute a lower boundary for 'heap' seems like it would be a sharp boundary in the same way as 'some' has a sharp boundary. For upper boundaries, this distinction between sharp boundaries and fuzzy boundaries might be different and I think probably is. Yet, in this lesson alone, given the two examples, lower boundaries between the two aren't exactly different in the way it's making it out to be .
This is a great digression, I love it LOL! For fun, I think there are two independent(?) reasons why your argument that even fuzzy vagueness has at least one sharp boundary condition isn't completely true.
1) It is possible that "heap" COULD have at least one sharp boundary condition as you define it here such that a heap is not vague but rather ambiguous. But when you step back, this might only be because we took the liberty of defining "heap" basically as "some amount of grain of sands" when that might not really be what, linguistically speaking, we mean by heap. This lesson was probably thinking of "heap" in the more colloquial sense, especially from the Sorites Paradox which tries to define where the boundary of heap and not-heap is. Aside from the Paradox's one assumption that a FEATURE of a heap is such that a heap of n and a heap of n-1 are indistinguishable, the argument seems pretty successful in pointing out that there does seem to be a paradox in trying to define what a heap is such that it can be distinguished from a non-heap.
I feel like there's a flaw in your logic because you pre-empted a definition of what a heap is --- that is, a heap = "some amount of grains" which is basically the same as "some." If you walk into the Sorites Paradox argument with that pre-formed definition in mind, you would of course not find the argument compelling at all nor would you find a paradox. But the proponent of the Sorites Paradox argument would also not find your dismissal of the argument compelling either, since the entire point of the Sorites Paradox is all about finding a definition to a heap vs. non-heap --- so it defeats the purpose of the paradox if someone assumes a definition at the onset! I think this is a very long-winded way of saying that a) vagueness and ambiguity are in fact meaningfully different; and b) lower boundaries in either case, if they do exist, are probably different too. The only way this seems not to be true is by perhaps committing a category error: i.e., by equating the definition of an instance of vagueness [a heap] with an instance of ambiguity [some].
2) Even if your argument that "heaps" are not vague but rather ambiguous is valid, it does not follow that all cases of valid things are in fact really ambiguous. You suggest in your comment that "vagueness has some, or at least one, sharp boundary condition even if it's fuzzy" by using the heaps case as an example. In the case of heaps, it could be true that the vagueness of heaps in fact has sharp boundary condition(s) that make it closer to/indistinguishable with ambiguity. This unfortunately doesn't mean that vagueness in other cases would necessarily have sharp boundary conditions.
I was thinking the same.
Thanks for the reply!
I would like to clarify one thing, the point of my comment (my last sentence in the original comment).
Firstly, I just want to say that agree with your comments about vagueness.
I agree that vagueness exists and isn't like ambiguity in the sense that 'some' or 'many' are ambiguous. Sorites paradoxes exist for vague concepts, like 'heap' or 'bald' and this doesn't apply to ambiguities.
The lesson uses the word 'heap' in contrast to the word 'some' by pointing to one lower boundary as being sharp or not sharp, but the contrast, as its presented, isn't as sharp itself in the following sense.
It's true that I'm defining 'heap' as a set of members that are organized in a given way, but I'm just appealing to what I think 'heap' means through its use by common speakers. In my definition of heap, I quoted a dictionary definition (with the assumption that a dictionary definition is what's commonly used to define words that common speakers use). Through this definition, I take it as an implication that it is a set that excludes one and only one member.
Of course, these vague concepts aren't sharply defined in a logical sense from how people use 'heap' or 'bald.' Yet, common use of vague concepts do have some prima facie sharp boundary intuitively even if there seems to be no logical sharp boundary.
Sorties cases result from a given starting point of F being a predicate and through a series of applying, in some formulations, some method (Fn-1 or Fn+1) results in a conclusion where F remains constant even though that conclusion is counterintuitive (and therefore paradoxical).
I'm pointing at our intuitions, given what I take we mean by a word through its common use, and saying "here's a boundary" where F is or isn't sharp. This reference is in comparison to what the lesson uses with one boundary for one ambiguous word.
In my original comment I take it to be the case that vague concepts are fuzzy (in opening question of my comment), but I don't think the distinction between ambiguity and vagueness is given through the one example of lower boundaries by way of introducing 'heap' and 'some.'
I think the distinction would have been given if the lesson would have taken upper boundaries as examples instead since it would fall outside of what I think is the common definition of 'heap' to mean ( "a disorderly collection of objects placed haphazardly on top of each other").
Great comment! I applaud you for getting into some of the nitty gritty of the sorties paradox!
Hi! Can someone explain in more detail how this is different from vagueness?
Vagueness does not have a specific limit its limits are not sharp and fuzzy. If you say some or other quantifiers, you know "AT LEAST ONE" that means that there is lower limit if that makes sense.
#feedback
In the full diagram, it says that ambiguous quantifiers have "sharp upper and lower boundaries." But in the next lesson (ie. lesson 3) it says that the ambiguous quantifier, 'Some,' "does not have an upper boundary."
This seems to me like either an error or improperly explained.
It becomes clear in later lessons that the 'sharp upper boundary' refers to something like 'all.' It is clear that this is different than vagueness. Still, though, it might be worth some clarification earlier on so as to avoid the apparent contradiction.
I got to give feedback on how well this V2 version is organized. It is so much better than V1. I started learning Quantifiers after like 30% of the actual LR section ON V1, I love how this is introduced way before actually starting LR. #feedback
As someone who didn't experience v1, this is good to hear. I take PTs weekly and already am seeing a huge difference in my LRs and RCs because of the grammar tools 7Sage gives in these foundation lessons. #feedback
Curious as well!
If you dont mind me asking, when did you start taking PTs? I am at this point in the curriculum and have not taken one yet as I dont want to spoil them before I have a better foundation.
I am also wondering this! I thought the general rule of thumb was to not take scheduled PTs until after you'd already finished the core curriculum.