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
Here, we directly compare the meaning of “some,” “most,” and “all” in order to see how some quantifiers can imply others. “All” is a stronger claim than “Most.”
All dogs like bacon → Most dogs like bacon dog → bacon dog - m → bacon
Let’s look at the relationship between “most” and “some.” “Most” is a stronger word than some and therefore a much stronger claim. What can be inferred from “most” is “some.”
Most cats are funny → Some cats are furry. [The word “most” implies “some”] cat - m → furry cat ← s → furry
So now we know “all” implies “most” and “most” implies “some.” That additionally means that “all” also implies “some”. This is because “all” implies “most,” and “most” implies “some.” So, therefore, “all” must also imply “some,” which means if we know that “all dogs like bacon,” it must also be true that “some dogs like bacon.” [dog ← S → bacon]. If the “all” claim is strong enough to support a “most” claim, it must also be strong enough to support an even weaker “some” claim.
RECAP:
“All” implies “most,” which implies “some.”
If “all X are Y," then “most X are Y” and “some X are Y.”
@RyanPrioreschi From the section Quantifier "Many" - On the LSAT, you can translate "many" to "some." Technically, this is false. Technically, "many" only implies "some." But I've never seen the test exploit this technicality to our disadvantage. Instead, it helps us avoid an actual trap of confusing "many" for "most."
@allyldh i'm literally going to answer myself here one second after I posted this because this is the lawgic chain showed up in the video as I accessed the next lesson.
If a some dogs like bacon is MBT, then given that some is reversible, cant some bacon like dogs? This is a stupid sentence, but I am just wondering in terms of our logic rules.
Some has no contrapositive so it would work in Logic like so... dog likes bacon. Two lessons ago they said that some and most for intersecting sets do not have contrapositives because they do not make sense. This circumstance is an example of that.
Yes true there are no contrapositives, but they said some is the only one that is reversible, meaning the main concepts can swap sides of the some arrow.
You can infer some from most, but not most from some. They are never "equal" but if you know most, you also know there are at least some.
It's just what you can guarantee to be true from the given quantifier. If you are given "some X are Y" you know nothing specific about the amount other than its at least 1 and not zero. You cannot guarantee that most X are Y from this because all you know for certain is at least 1 (but it very well could be most you just don't know). But, If you are given "most X are Y", then you know that more than half of X are Y. In this case you CAN say that some X are Y because at least one but not zero fits under the umbrella of more than half.
Basically you cannot go up from a weaker claim to a stronger one, but you can infer a stronger claim from a weaker one.
All→Most→Some
Chain only goes forward not back
Also some does not equal many (many is more than some) but he said you won't be penalized on the lsat for thinking of them as equivalent. However many is never equal to most.
I don't understand all implying some. If all dogs like bacon, and this implies that some dogs like bacon, doesn't that mean that all dogs like bacon implies some dogs do not like bacon?
"Some dogs like bacon" gives no info about dogs not liking bacon. It doesn't mean that some don't, only that some do. With "some" statements, there is a gray area as to what the rest of the set is, but don't conflate the unknown part of the set with the opposite of the "some" statement.
@Jedi1233Late, but if anyone knows the tweet "I like pancakes" 'So you hate waffles?' "No, that's an entirely different sentence!" kind of applies here. If you say you like pancakes, it doesn't mean you hate waffles. If you say some people like pancakes, it doesn't mean the rest hate waffles. This is just the first thing I thought of, lol
If this is the case, then how is the range of some and most include 100% of something? If All is a bigger claim then the other two, then the other two must not include all right?
The indicator words 'Some' and 'Most' are indicative of ranges.
The range of 'Some' goes from the lower boundary of 'at least 1' to the higher boundary of 'including all.'
The range of 'Most' goes from the lower boundary of '50%+1' to the higher boundary of 'including all.'
Because these indicators represent range, it is impossible to know for certain what specific value is being represented. We just know that the value falls within a certain range.
The indicator word 'All' does not indicate a range, and instead indicates an explicit and specific value. Seeing the indicator word 'All' tells us, with certainty, a numerical value.
Ex. There are 100 women at the hair salon. Some women have come for haircuts.
[There's no way to know how many women are their for haircuts, all I know is that the number of women who came for haircuts is within the range of 2-100.]
Ex. There are 100 women at the hair salon. Most women have come for haircuts.
[I am still not able to know the definitive number of women who are in the salon for hair cuts. All I know is that the number of women who came for haircuts are in the range of 51-100.]
Ex. There are 100 women at the hair salon. All of the women have come for haircuts.
[Only in this example am I able to know, with certainty, the number of women who have come for haircuts. I know there are 100 women. I know that 'all' is a definitive indicator that gives me a specific numerical value. I know that all 100 women came for haircuts]
The reason that 'All' is considered a stronger claim than 'most' or 'some' is because 'All' guarantees a specific value (in this case 100), while 'most' and 'some' only show that any numerical value within a certain range (2-100 or 51-100 depending on the example) can be considered logically valid.
#feedback https://7sage.com/lesson/quantifier-inferences/?ss_completed_lesson=25459 says: "All" is a subset of "most." That's why it implies "most." Similarly, "most" is a subset of "some." That's why it implies "some." Is this a typo?
BC While all implies most, most does not necessarily imply all—which means all is not a subset of most, but rather most is a subset of all in terms of quantity.
If most implies some, for example, we know that most is the sufficient condition for some M→S". Some would be the necessary condition for most "/S → /M". we know that necessary conditions are the superset, and its sufficient condition as a subset. Although that may initially seem counterintuitive by our standards that some is included in all (all is just some combination of a bunch of "somes"), by a strictly logical definition we can determine that the greater value is indeed a subset of the lesser value, given that it must fulfill that value but there are cases where it doesn't (some dogs like bacon, but not most). Hope this helps!
The way I like to see this to help me understand is reflecting it off of criminal law burdens to meet.
Beyond every reasonable doubt is ALL. Preponderance of Evidence is Most Probable Cause is Some.
Beyond a reasonable doubt is the highest burden a jury can find. So, if the jury has believed a defendant is beyond a reasonable doubt of lets say Grand Theft, then we can also infer that they have met preponderance of evidence, and probable cause because they are lower burdens to meet and those have to be met already to get to the highest burden.
This may help some, for me in my criminal brain, it helps to visualize it like this.
Kevin Lin has a GREATTT video on Youtube about this as well.
https://www.youtube.com/watch?v=JkQ2ZRizWIE
An example I came across that helps me a bit:
In the context of combining statements containing quantifiers, you can remember the golden rule by thinking of the “4 S Rule”: The Sufficient of the Stronger statement must be Shared, and the conclusion is a “Some” statement.
Here’s how it works in practice – take the following example:
1. Everyone who is studying for the LSAT is stressed.
2. Some people who are studying for the LSAT are abstaining from alcohol.
In this case, we have an “all” statement (sentence #1) and a “some” statement (sentence #2). The stronger of those two statements is the first one, the “all” statement. The sufficient condition of the stronger statement is “studying for the LSAT”; in order for us to be able to combine these statements, they must both share that condition. Fortunately, the “some” statement also references people who are studying for the LSAT, so we’re good to go.
The conclusion, as we know from our golden rule, is going to be a “some” statement — in this case, that “Some people who are stressed are abstaining from alcohol.”
These combinations work because at least one of the premises is relatively strong. However, there are two combinations that will never, EVER work, simply because the premises are too weak to support a conclusion. You can never combine a “most” statement with a “some” statement, and you can never combine two “some” statements.
There are other variations that this video and Kevin's video breaks down that is super helpful.
So you can only ever combine an "all" statement with a "some" statement or an "all" statement with a "most" statement? That's what I am gathering from you saying "You can never combine a “most” statement with a “some” statement, and you can never combine two “some” statements."
If you are going to say:
1. Everyone who is studying for the LSAT is stressed.
2. Most people who are studying for the LSAT are abstaining from alcohol.
You can still only say "Some people who are stressed are abstaining from alcohol," correct, and not "Most people who are stressed are abstaining from alcohol"?
for the first claim, you can grant that most people who are studying for the LSAT are stressed. you can also infer that some people who are studying for the lsat are stressed.
for the second claim, you can only infer that some people who are studying for the LSAT are abstaining from alcohol.
also remember the some statement is biconditional which open up one most statement (just flip the sets are around).
To clarify, can I think of "some" as necessary for both "most" and "all?" Or do we consider all as /some b/c of the fact that some = implies at least 1 & all = means all?
I also found this very confusing, the only way I can make it clearer in my mind is to stop thinking about one circle inside of the other and instead think that "ALL" is Sufficient for "MOST" as a necessary, and "MOST" is Sufficient for "SOME" as a Necessary. By this logic they are subsets to supersets. Am I wrong in thinking this way?
Why is it that all→most = subset→superset? would "all" not be the superset and "most" be the subset? visually, "most" would be a circle contained within the larger "all" circle.
How is "all" a subset of "most" ? And how is "most" a subset of "some" ?
We're not thinking of it purely quantitatively. We're thinking of it in terms of logic. All is sufficient enough to guarantee that "most" occurs. "Most" is necessary for "all" to occur, but not sufficient enough to guarantee it.
Most mean at least above 50%, but it can also mean all.
Since most can mean all, all is sufficient for most. Whereas most is not sufficient for all because most means at least 50.1%, but not necessarily all. If you have all then you know it's most. If you have most maybe you have 50.1%.
Reduce your confusion about conditional relationships by thinking to yourself, "What is sufficient for another thing to be true?" In other words, "if something is true what would the truth of that statement result in?" I don't know this helps me, maybe its not helpful. Good luck!
I am also confused about this! Someone else was saying it has to do with whether or not they have an upper bound. Maybe because most does not have an upper bound? However, all also does not have an upper bound so there is still some confusion there for me.
Yeah I think this is a poor and misleading analogy by the creators of the course because of the usage of the set diagrams in both this lesson and the conditional logic lesson
Visually you are right, but remember in the conditional logic section that the subset → superset diagramming with the subsumed circles doesn't apply very easily to a lot of the more abstract concepts we will encounter, but rather I think the teacher used it to illustrate the concepts of sufficiency vs. necessity to help build our intuitive and fundamental understanding of these concepts
He's using it here to talk about what inferences we can draw
Consider the following chain of conditional relationships:
All roofers must bring a hammer to work (R → H)
All hammers must have a blue handle (H → bh)
Only those who have hammers with blue handles bring lunch to work (bh → L)
So we have the chain:
R → H
H → bh
bh → L
Contrapositives:
/H → /R
/bh → /H
/L → /bh
These chain together due to the directionality of the arrows which allow us to make valid inferences/conclusions:
R→H→bh→L
contrapositive: /L→/bh→/H→/R
So I can infer that if someone has a hammer with a blue handle, then they bring lunch to work. But I can't infer that if someone brings lunch to work, that they bring a blue-handled hammer.
Similarly with the quantifiers, if I say "all" cats are pets, that also means "most" cats are pets, and "some" cats are pets, because all triggers most and some. They aren't exclusionary.
Also, "most" triggers "some" because some falls under "most" and isn't exclusionary
"Some" doesn't trigger most or all because their minimum thresholds lie ABOVE the maximum range that some encompasses
So basically it has to do with directionality and which inferences can be properly drawn, which is why he invoked the subset → superset relationship
"Few" has an upper limit of less than most, because it means "most not." Few cannot be a superset of all, because it cannot mean all.
Some can include all because it has no upper limit. Some includes many, because it has a lower floor, but no upper limit. Some includes most, because it has a lower floor, but no upper limit. Some includes all for the same reason.
^^This would make "All" a subset of "Few". Few means less than 1/2. A subset is contained within a superset. How can "all" of a set be contained within less than 1/2 of a set ?
It has to do with logical relationships. If I have 100 people in a room, then, of course "most", which could mean 60 people, or "some", which could mean 10 people, are subsets of "all." But this is not what we mean here. Consider the following: If all people in the room can read, then some people in the room must be able to read. In this sense, "all" is a subset of "some." This does NOT imply "all" is a subset of "some" in absolute quantitative terms, it actually means that the statement "all people can read" is a subset of the statement "some people can read." Think about a smaller circle inside of a bigger circle. In the smaller circle, we have the statement "all people can read," and the bigger circle is the statement "some people can read." All possible instantiations of "all people can read" coincide with those of "some people can read." However, not all possible instantiations of "some people can read" (e.g. exactly 10 people can read or exactly 40 people can read, etc.) coincide with those of "all people can read," so we can be within the larger circle but outside the smaller circle.
TLDR: These quantifiers are subsets and supersets of one another in terms of the logical possibilities of the statements they modify NOT in absolute quantitative terms.
Hello! Most can definitely imply many, as most has a higher lower bound. But it wouldn’t work backwards. Many wouldn’t necessarily imply most. The stronger, higher quantifier implies the lower. Hope this helps.
Yes! Except, I would quibble with "Few → Some." What I'm about to say may not be relevant to anything you see until you get much further into LR, but I see a difference between "few" and "a few."
"A few of my friends are lawyers." This implies at least some of them are lawyers. So, I'd agree that "a few → some"
However, when the LSAT uses "few" (as opposed to A few), they intend to focus on the limiting aspect of "few." "Few cheaters will succeed" puts limits on the proportion of cheaters that will succeed. I interpret that as MOST cheaters will NOT succeed.
Now, the question is, does "Few cheaters will succeed" imply that there are at least some cheaters who DO succeed? Many LSAT instructors would say, yes, it does. However, I do not recall any question which requires that interpretation. So, my practice is to treat "Few cheaters will succeed" as "Most cheaters will NOT succeed," and I don't read it as implying anything about whether some of the cheaters DO succeed.
So, I would not agree with "Few → Some". I would agree with "A few → Some."
#help! can someone re-explain why "all" is a subset of "most" instead of the other way around, "most" being a subset of "all"? So like going back to cats & mammals, cats is the subset for mammals bc being a cat would imply it is a mammal but mammal is the broader category so why isn't "all" the broader category/superset? Does my question make sense or am I rambling lol
I had an issue with this as well. Not sure if my thought process is correct but the reason is that "all" doesn't have a range of meaning.
Whereas, words like "some" and "most" could have low boundaries of at least 1 or 50% + 1......"All" means all. There is no low boundary for all. Although, "Some" could include "all", but it doesn't have too.
Thats why all is the subset because it is inside the circle of "most" which that quantifier is inside the circle of "some".
Because many implies some due to the fact that some means at least one, it must be true that some of a thing is a valid inference to many of that thing. Most also validly implies some for similar reasons. But many cannot validly imply most because of how subset and superset are viewed for their relationships. The idea that there is many of something means its lower boundary (at least 3 or more) falls outside of the superset of most where its lowest boundary encompasses 51% or more of that thing. If it was entirely encompassed by the superset, then it would be valid to imply many→most. Most implies many because it does encompass its entire range of boundaries.
It all has to do with the fact that you can equate many to some as a falsehood to better see the relationship between them. Just because you have a large amount of something "many," it doesn't imply that you have most of that thing. The most relationship indicates a proportion of something, whereas many only has to be 3 or more. Few as a quantifier means some, but not many and typically implies "most not." You can't equate many and few with each other.
I like to use the example: Many people love the color blue. If many people love blue, then it is valid to say that some people like blue because there is at least 1 person that likes blue. It wouldn't make sense to say that because many people like blue, then most people like blue too due to the fact that "most" would mean that 51% of people like blue. We only know that at least 3 people like blue with the "many" quantifier. It must be true that some people like blue, but it COULD BE TRUE that most people like blue. For your diagrams, I think the one on the right is correct. If you still need more help, I can try to come up with more examples that may work better for you. Quantifier relationships are difficult and super easy to miss little changes, so I completely understand not quite getting it. I still struggle with it at times. lol
Yes, it is related to sufficiency and necessity. The way I understand is that all implies most but most does not imply all, and most implies some but some does not imply most. However, I do not think you have to organize as subset/superset relationship, but know which word is stronger for future problems. I am not sure if that was your question.
"All" implies "Most" because if something is true for all, it is also true for most.
"Most" implies "Some" because if something is true for most, it is also true for at least some.
"Some" is not a subset of "Most":
"Some" and "Most" are not in a subset relationship. "Some" does not necessarily imply "Most" because "some" could be a small portion, and it doesn't have to encompass the majority.
If you say "Some," it means at least one but not necessarily a majority.
If you say "Most," it implies a majority but not necessarily all.
So, the key point is that while "Some" includes the possibility of "Most," it doesn't guarantee it. "Some" can indeed be most in certain cases, but it's not a strict implication. It depends on the specific context and proportions being referred to.
Subscribe to unlock everything that 7Sage has to offer.
Hold on there, stranger! You need a free account for that.
We love that you want to get going. Just create a free account below—it only takes a minute—and then you can continue!
Hold on there, stranger! You need a free account for that.
We love that you came here to read all the amazing posts from our 300,000+ members. They all have accounts too! Just create a free account below—it only takes a minute—and then you’re free to discuss anything!
Hold on there, stranger! You need a free account for that.
We love that you want to give us feedback! Just create a free account below—it only takes a minute—and then you’re free to vote on this!
Hold on there, you need to slow down.
We love that you want post in our discussion forum! Just come back in a bit to post again!
Subscribers can learn all the LSAT secrets.
Happens all the time: now that you've had a taste of the lessons, you just can't stop -- and you don't have to! Click the button.
64 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
Here, we directly compare the meaning of “some,” “most,” and “all” in order to see how some quantifiers can imply others. “All” is a stronger claim than “Most.”
All dogs like bacon → Most dogs like bacon dog → bacon dog - m → bacon
Let’s look at the relationship between “most” and “some.” “Most” is a stronger word than some and therefore a much stronger claim. What can be inferred from “most” is “some.”
Most cats are funny → Some cats are furry. [The word “most” implies “some”] cat - m → furry cat ← s → furry
So now we know “all” implies “most” and “most” implies “some.” That additionally means that “all” also implies “some”. This is because “all” implies “most,” and “most” implies “some.” So, therefore, “all” must also imply “some,” which means if we know that “all dogs like bacon,” it must also be true that “some dogs like bacon.” [dog ← S → bacon]. If the “all” claim is strong enough to support a “most” claim, it must also be strong enough to support an even weaker “some” claim.
RECAP:
“All” implies “most,” which implies “some.”
If “all X are Y," then “most X are Y” and “some X are Y.”
Id “most A are B,” then “some A are B.”
This guy keeps on confusing me instead of making things clearer for me
@LauraBolivar LOL
Guys by the same logic Many implies Some right
@RyanPrioreschi From the section Quantifier "Many" - On the LSAT, you can translate "many" to "some." Technically, this is false. Technically, "many" only implies "some." But I've never seen the test exploit this technicality to our disadvantage. Instead, it helps us avoid an actual trap of confusing "many" for "most."
Can we add the quantifier word few in the lawgic chain and make it : all -> most -> some -> few?
@allyldh i'm literally going to answer myself here one second after I posted this because this is the lawgic chain showed up in the video as I accessed the next lesson.
@allyldh not few follows from most not some.
If a some dogs like bacon is MBT, then given that some is reversible, cant some bacon like dogs? This is a stupid sentence, but I am just wondering in terms of our logic rules.
Some has no contrapositive so it would work in Logic like so... dog likes bacon. Two lessons ago they said that some and most for intersecting sets do not have contrapositives because they do not make sense. This circumstance is an example of that.
it implies that of the things that like bacon (people animals or whatever we are talking about) are dogs
Yes true there are no contrapositives, but they said some is the only one that is reversible, meaning the main concepts can swap sides of the some arrow.
In theory yes, I think your grammar parsing is off.
some dogs like bacon = some of the beings that like bacon are dogs
hope that helps clarify!
wait but didn't the last lesson just say some does not equal most??
xY
but
x-m->/Y
is what the last lesson said no? was I confusing m as Most instead of Many???
You can infer some from most, but not most from some. They are never "equal" but if you know most, you also know there are at least some.
It's just what you can guarantee to be true from the given quantifier. If you are given "some X are Y" you know nothing specific about the amount other than its at least 1 and not zero. You cannot guarantee that most X are Y from this because all you know for certain is at least 1 (but it very well could be most you just don't know). But, If you are given "most X are Y", then you know that more than half of X are Y. In this case you CAN say that some X are Y because at least one but not zero fits under the umbrella of more than half.
Basically you cannot go up from a weaker claim to a stronger one, but you can infer a stronger claim from a weaker one.
All→Most→Some
Chain only goes forward not back
Also some does not equal many (many is more than some) but he said you won't be penalized on the lsat for thinking of them as equivalent. However many is never equal to most.
I don't understand all implying some. If all dogs like bacon, and this implies that some dogs like bacon, doesn't that mean that all dogs like bacon implies some dogs do not like bacon?
"Some dogs like bacon" gives no info about dogs not liking bacon. It doesn't mean that some don't, only that some do. With "some" statements, there is a gray area as to what the rest of the set is, but don't conflate the unknown part of the set with the opposite of the "some" statement.
@Jedi1233Late, but if anyone knows the tweet "I like pancakes" 'So you hate waffles?' "No, that's an entirely different sentence!" kind of applies here. If you say you like pancakes, it doesn't mean you hate waffles. If you say some people like pancakes, it doesn't mean the rest hate waffles. This is just the first thing I thought of, lol
If this is the case, then how is the range of some and most include 100% of something? If All is a bigger claim then the other two, then the other two must not include all right?
@aakash2003ch457
The indicator words 'Some' and 'Most' are indicative of ranges.
The range of 'Some' goes from the lower boundary of 'at least 1' to the higher boundary of 'including all.'
The range of 'Most' goes from the lower boundary of '50%+1' to the higher boundary of 'including all.'
Because these indicators represent range, it is impossible to know for certain what specific value is being represented. We just know that the value falls within a certain range.
The indicator word 'All' does not indicate a range, and instead indicates an explicit and specific value. Seeing the indicator word 'All' tells us, with certainty, a numerical value.
Ex. There are 100 women at the hair salon. Some women have come for haircuts.
[There's no way to know how many women are their for haircuts, all I know is that the number of women who came for haircuts is within the range of 2-100.]
Ex. There are 100 women at the hair salon. Most women have come for haircuts.
[I am still not able to know the definitive number of women who are in the salon for hair cuts. All I know is that the number of women who came for haircuts are in the range of 51-100.]
Ex. There are 100 women at the hair salon. All of the women have come for haircuts.
[Only in this example am I able to know, with certainty, the number of women who have come for haircuts. I know there are 100 women. I know that 'all' is a definitive indicator that gives me a specific numerical value. I know that all 100 women came for haircuts]
The reason that 'All' is considered a stronger claim than 'most' or 'some' is because 'All' guarantees a specific value (in this case 100), while 'most' and 'some' only show that any numerical value within a certain range (2-100 or 51-100 depending on the example) can be considered logically valid.
Hope this helps :)
#feedback https://7sage.com/lesson/quantifier-inferences/?ss_completed_lesson=25459 says: "All" is a subset of "most." That's why it implies "most." Similarly, "most" is a subset of "some." That's why it implies "some." Is this a typo?
BC While all implies most, most does not necessarily imply all—which means all is not a subset of most, but rather most is a subset of all in terms of quantity.
If most implies some, for example, we know that most is the sufficient condition for some M→S". Some would be the necessary condition for most "/S → /M". we know that necessary conditions are the superset, and its sufficient condition as a subset. Although that may initially seem counterintuitive by our standards that some is included in all (all is just some combination of a bunch of "somes"), by a strictly logical definition we can determine that the greater value is indeed a subset of the lesser value, given that it must fulfill that value but there are cases where it doesn't (some dogs like bacon, but not most). Hope this helps!
The way I like to see this to help me understand is reflecting it off of criminal law burdens to meet.
Beyond every reasonable doubt is ALL. Preponderance of Evidence is Most Probable Cause is Some.
Beyond a reasonable doubt is the highest burden a jury can find. So, if the jury has believed a defendant is beyond a reasonable doubt of lets say Grand Theft, then we can also infer that they have met preponderance of evidence, and probable cause because they are lower burdens to meet and those have to be met already to get to the highest burden.
This may help some, for me in my criminal brain, it helps to visualize it like this.
Kevin Lin has a GREATTT video on Youtube about this as well.
https://www.youtube.com/watch?v=JkQ2ZRizWIE
An example I came across that helps me a bit:
In the context of combining statements containing quantifiers, you can remember the golden rule by thinking of the “4 S Rule”: The Sufficient of the Stronger statement must be Shared, and the conclusion is a “Some” statement.
Here’s how it works in practice – take the following example:
1. Everyone who is studying for the LSAT is stressed.
2. Some people who are studying for the LSAT are abstaining from alcohol.
In this case, we have an “all” statement (sentence #1) and a “some” statement (sentence #2). The stronger of those two statements is the first one, the “all” statement. The sufficient condition of the stronger statement is “studying for the LSAT”; in order for us to be able to combine these statements, they must both share that condition. Fortunately, the “some” statement also references people who are studying for the LSAT, so we’re good to go.
The conclusion, as we know from our golden rule, is going to be a “some” statement — in this case, that “Some people who are stressed are abstaining from alcohol.”
These combinations work because at least one of the premises is relatively strong. However, there are two combinations that will never, EVER work, simply because the premises are too weak to support a conclusion. You can never combine a “most” statement with a “some” statement, and you can never combine two “some” statements.
There are other variations that this video and Kevin's video breaks down that is super helpful.
So you can only ever combine an "all" statement with a "some" statement or an "all" statement with a "most" statement? That's what I am gathering from you saying "You can never combine a “most” statement with a “some” statement, and you can never combine two “some” statements."
If you are going to say:
1. Everyone who is studying for the LSAT is stressed.
2. Most people who are studying for the LSAT are abstaining from alcohol.
You can still only say "Some people who are stressed are abstaining from alcohol," correct, and not "Most people who are stressed are abstaining from alcohol"?
for the first claim, you can grant that most people who are studying for the LSAT are stressed. you can also infer that some people who are studying for the lsat are stressed.
for the second claim, you can only infer that some people who are studying for the LSAT are abstaining from alcohol.
also remember the some statement is biconditional which open up one most statement (just flip the sets are around).
dog bacon
Some dogs like bacon.
bacon dog
Some that like bacon are dogs.
To clarify, can I think of "some" as necessary for both "most" and "all?" Or do we consider all as /some b/c of the fact that some = implies at least 1 & all = means all?
all → most → some
all is sufficient for most.
most is sufficient for some.
I also found this very confusing, the only way I can make it clearer in my mind is to stop thinking about one circle inside of the other and instead think that "ALL" is Sufficient for "MOST" as a necessary, and "MOST" is Sufficient for "SOME" as a Necessary. By this logic they are subsets to supersets. Am I wrong in thinking this way?
No, you are not wrong in thinking this way. Makes sense!
Why is it that all→most = subset→superset? would "all" not be the superset and "most" be the subset? visually, "most" would be a circle contained within the larger "all" circle.
How is "all" a subset of "most" ? And how is "most" a subset of "some" ?
We're not thinking of it purely quantitatively. We're thinking of it in terms of logic. All is sufficient enough to guarantee that "most" occurs. "Most" is necessary for "all" to occur, but not sufficient enough to guarantee it.
Most mean at least above 50%, but it can also mean all.
Since most can mean all, all is sufficient for most. Whereas most is not sufficient for all because most means at least 50.1%, but not necessarily all. If you have all then you know it's most. If you have most maybe you have 50.1%.
Reduce your confusion about conditional relationships by thinking to yourself, "What is sufficient for another thing to be true?" In other words, "if something is true what would the truth of that statement result in?" I don't know this helps me, maybe its not helpful. Good luck!
I am also confused about this! Someone else was saying it has to do with whether or not they have an upper bound. Maybe because most does not have an upper bound? However, all also does not have an upper bound so there is still some confusion there for me.
Yeah I think this is a poor and misleading analogy by the creators of the course because of the usage of the set diagrams in both this lesson and the conditional logic lesson
Visually you are right, but remember in the conditional logic section that the subset → superset diagramming with the subsumed circles doesn't apply very easily to a lot of the more abstract concepts we will encounter, but rather I think the teacher used it to illustrate the concepts of sufficiency vs. necessity to help build our intuitive and fundamental understanding of these concepts
He's using it here to talk about what inferences we can draw
Consider the following chain of conditional relationships:
All roofers must bring a hammer to work (R → H)
All hammers must have a blue handle (H → bh)
Only those who have hammers with blue handles bring lunch to work (bh → L)
So we have the chain:
R → H
H → bh
bh → L
Contrapositives:
/H → /R
/bh → /H
/L → /bh
These chain together due to the directionality of the arrows which allow us to make valid inferences/conclusions:
R→H→bh→L
contrapositive: /L→/bh→/H→/R
So I can infer that if someone has a hammer with a blue handle, then they bring lunch to work. But I can't infer that if someone brings lunch to work, that they bring a blue-handled hammer.
Similarly with the quantifiers, if I say "all" cats are pets, that also means "most" cats are pets, and "some" cats are pets, because all triggers most and some. They aren't exclusionary.
Also, "most" triggers "some" because some falls under "most" and isn't exclusionary
"Some" doesn't trigger most or all because their minimum thresholds lie ABOVE the maximum range that some encompasses
So basically it has to do with directionality and which inferences can be properly drawn, which is why he invoked the subset → superset relationship
"Few" has an upper limit of less than most, because it means "most not." Few cannot be a superset of all, because it cannot mean all.
Some can include all because it has no upper limit. Some includes many, because it has a lower floor, but no upper limit. Some includes most, because it has a lower floor, but no upper limit. Some includes all for the same reason.
^^This would make "All" a subset of "Few". Few means less than 1/2. A subset is contained within a superset. How can "all" of a set be contained within less than 1/2 of a set ?
It has to do with logical relationships. If I have 100 people in a room, then, of course "most", which could mean 60 people, or "some", which could mean 10 people, are subsets of "all." But this is not what we mean here. Consider the following: If all people in the room can read, then some people in the room must be able to read. In this sense, "all" is a subset of "some." This does NOT imply "all" is a subset of "some" in absolute quantitative terms, it actually means that the statement "all people can read" is a subset of the statement "some people can read." Think about a smaller circle inside of a bigger circle. In the smaller circle, we have the statement "all people can read," and the bigger circle is the statement "some people can read." All possible instantiations of "all people can read" coincide with those of "some people can read." However, not all possible instantiations of "some people can read" (e.g. exactly 10 people can read or exactly 40 people can read, etc.) coincide with those of "all people can read," so we can be within the larger circle but outside the smaller circle.
TLDR: These quantifiers are subsets and supersets of one another in terms of the logical possibilities of the statements they modify NOT in absolute quantitative terms.
Is it wrong to think of the quantifiers in this order from least to greatest?
some→few→many→most→overwhelming majority→all
I think the order disregards the overlapping bounds of some of the quantifiers.
For instance, some and few have the same lower bound, and some, most, many, overwhelmingly majority have the same upper bound.
I understand all-> most-> some, but can most imply many? why or why not?
Hello! Most can definitely imply many, as most has a higher lower bound. But it wouldn’t work backwards. Many wouldn’t necessarily imply most. The stronger, higher quantifier implies the lower. Hope this helps.
Is it also accurate to say:
All → Most
Most → Some
as well as
Many → Some
Few → Some
Yes! Except, I would quibble with "Few → Some." What I'm about to say may not be relevant to anything you see until you get much further into LR, but I see a difference between "few" and "a few."
"A few of my friends are lawyers." This implies at least some of them are lawyers. So, I'd agree that "a few → some"
However, when the LSAT uses "few" (as opposed to A few), they intend to focus on the limiting aspect of "few." "Few cheaters will succeed" puts limits on the proportion of cheaters that will succeed. I interpret that as MOST cheaters will NOT succeed.
Now, the question is, does "Few cheaters will succeed" imply that there are at least some cheaters who DO succeed? Many LSAT instructors would say, yes, it does. However, I do not recall any question which requires that interpretation. So, my practice is to treat "Few cheaters will succeed" as "Most cheaters will NOT succeed," and I don't read it as implying anything about whether some of the cheaters DO succeed.
So, I would not agree with "Few → Some". I would agree with "A few → Some."
Can you take the contrapositive of "all → most → some" and assume that
"/some → /most→ all"?
#help! can someone re-explain why "all" is a subset of "most" instead of the other way around, "most" being a subset of "all"? So like going back to cats & mammals, cats is the subset for mammals bc being a cat would imply it is a mammal but mammal is the broader category so why isn't "all" the broader category/superset? Does my question make sense or am I rambling lol
I had an issue with this as well. Not sure if my thought process is correct but the reason is that "all" doesn't have a range of meaning.
Whereas, words like "some" and "most" could have low boundaries of at least 1 or 50% + 1......"All" means all. There is no low boundary for all. Although, "Some" could include "all", but it doesn't have too.
Thats why all is the subset because it is inside the circle of "most" which that quantifier is inside the circle of "some".
I hope I didn't ramble and helped lol
Because many implies some due to the fact that some means at least one, it must be true that some of a thing is a valid inference to many of that thing. Most also validly implies some for similar reasons. But many cannot validly imply most because of how subset and superset are viewed for their relationships. The idea that there is many of something means its lower boundary (at least 3 or more) falls outside of the superset of most where its lowest boundary encompasses 51% or more of that thing. If it was entirely encompassed by the superset, then it would be valid to imply many→most. Most implies many because it does encompass its entire range of boundaries.
It all has to do with the fact that you can equate many to some as a falsehood to better see the relationship between them. Just because you have a large amount of something "many," it doesn't imply that you have most of that thing. The most relationship indicates a proportion of something, whereas many only has to be 3 or more. Few as a quantifier means some, but not many and typically implies "most not." You can't equate many and few with each other.
I like to use the example: Many people love the color blue. If many people love blue, then it is valid to say that some people like blue because there is at least 1 person that likes blue. It wouldn't make sense to say that because many people like blue, then most people like blue too due to the fact that "most" would mean that 51% of people like blue. We only know that at least 3 people like blue with the "many" quantifier. It must be true that some people like blue, but it COULD BE TRUE that most people like blue. For your diagrams, I think the one on the right is correct. If you still need more help, I can try to come up with more examples that may work better for you. Quantifier relationships are difficult and super easy to miss little changes, so I completely understand not quite getting it. I still struggle with it at times. lol
#Help
This doesn't imply that this subset & superset relationship is a sufficient and necessary one does it?
Yes, it is related to sufficiency and necessity. The way I understand is that all implies most but most does not imply all, and most implies some but some does not imply most. However, I do not think you have to organize as subset/superset relationship, but know which word is stronger for future problems. I am not sure if that was your question.
"All" → "Most" → "Some":
"All" implies "Most" because if something is true for all, it is also true for most.
"Most" implies "Some" because if something is true for most, it is also true for at least some.
"Some" is not a subset of "Most":
"Some" and "Most" are not in a subset relationship. "Some" does not necessarily imply "Most" because "some" could be a small portion, and it doesn't have to encompass the majority.
If you say "Some," it means at least one but not necessarily a majority.
If you say "Most," it implies a majority but not necessarily all.
So, the key point is that while "Some" includes the possibility of "Most," it doesn't guarantee it. "Some" can indeed be most in certain cases, but it's not a strict implication. It depends on the specific context and proportions being referred to.