In this lesson, we start using our language of Lawgic when dealing with relationships involving the quantity “some”. “Some” must include at least “one” or could include up to “all”.
In Lawgic, the quantifier “some” is represented with this bi-direction arrow ( ←S→ ) with an S in the middle of it. The “S” in this arrow is to distinguish it from the bi-conditional arrow ( ← → ).
The quantifier some ( ← S → ) which expresses an intersection. Let’s look at an example…..
“Some students in Mrs. Stoops’s class can read.”
Step 1: Identify that this is a statement, this is a claim that is amenable to translation. The way to do that is to (first) identify/notice the quantifier. Or the intersection indicator. In this instance it’s the word “Some”.
Step 2: Identify the two concepts. Typically, they are sets. The [first concept] is “students in Mrs. Stoops’'s class” & the [second concept] is “student or people can read”.
Step 3: Assign Symbols to represent these two sets. (Student) to represent students in Mrs. Stoops’s class & (Read) to represent the student/people who can read.
Translation Into Lawgic : Add the “Some” arrow! (student) ← S → (read) This means you can read this arrow either from left to right or from right to left. What that means is that this statement here, “student ← S → read,” is identical to the statement “read ← S → student.” In Lawgic, these two claims are the same.
Translating Back Into English: What it means is that “Some students in Mrs. Stoops’s class can read” is identical in meaning to “Some student that can read are in Mrs. Stoops’s class.”
Example→ “Some cats are pets” is identical to the claim that “Some pets are cats.” Translate into Lawgic: (c ← S → p) (p ← S → c) All four of these expressions are getting at the exact same idea, which, again, is just the idea of an intersection between two sets.
RECAP:
To translate “some” claims to Lawgic, use the bi-directional “some” arrow (← S →). The arrow is bi-directional because the intersection relationship works both ways: “some A are B” is identical to “some B are A.”
I was confused at first but I think I know why some can include all:
Some students pass the LSAT only if they study everyday.
In order for the necessary condition to be true it has to include at least 1 student, but can also include all students, that is, even if we say all students have to study for passing LSAT does not fail the necessary. So it can include all!
Essentially we need to look for anything can cause the necessary condition to fail and exclude that interpretation.
i get how some can mean all, but how does this help with lsat questions. I feel like from my limited experience, rarely are lsat answers with 'All' correct because they are too extreme.
"Some" covers an intersect and that's why it can go both ways.
Some cats are pets and Some pets are cats
These mean the same thing because they both have the same intersection in the middle where pets and cats overlap. Once you are in that intersection part in the middle you are part of both groups so it doesn't matter.
The word some means "at least one" so once I am in that middle group then yah, at least one cat is a pet, and at least one pet is a cat. Some can include all, sure. BUT it doesn't have to. All some has to do is mean "At least one"
Ex. I am a cat who is also a pet. I am in that middle intersection group so that means I can also say I am a pet who is a cat. On the flip lets pretend I am not a cat, but I am a pet. This means I can't be part of the that intersection in the middle. I am only part of the "pets" circle.
When you break down the Stoops's sentence, it would be better not to use "students" as one of the main concepts, since when you bring it back together students is in both concepts. It would be better to break that first main concept into "Mrs. Stoops's class."
this might be answered in a future lesson, but i'm just curious. are there situations where "some" means "all" like the previous lesson said, and are used qualifiers as a sufficient condition? i know the example here is "some people who can read are students in the class, some students in the class are people who can read," but if we were provided more context before or after that statement that would include "all" as the meaning of "some" students in the class (the meaning being "all students can read in the class"), are there cases that the LSAT expects you to use this as a sufficient condition?
I can't figure out if this is more or less intuitive than just using (∃x) in the usual language of logic. Learning predicate logic is very different than what I'm learning here on 7sage.
If we used it in this class example it would be like:
S = Stoop's class
R = predicate (can read)
∃x = what is called the "existential quantifier"
(∃r)[Sr]
(∃r)[Sr] is true when Sr is true for at least one value of r
I'm not understanding how we introduce a biconditional symbol in there for this lesson.
I am still confused as to why the arrow can be in both directions. The example given is that some cats can be pets. But isn't cat a subset of pets? Isn't pets the larger (necessary) condition in order for cats to be part of it? Just like the example given earlier in the course where cats --> mammals? Could someone please explain and try to clarify the difference for me because I see it as the same thing. Thank you
What does "Because if there are some students in Mrs. Stoops' class who can read, then there must be some students who can read in Mrs. Stoops' class." mean?
Does the translation to Lawgic really matter on questions that require us to take just over a minute to answer? It seems like while training ourselves to translate to Lawgic is taking away the ability to do this kind of translation in our head without having to learn extra formatting.
could we translate this to Lawgic to "there exists at least one student in Ms. Stoops' class who can read" instead? seems more intuitive #help#feedback
Why isn't translating to the contrapositive a step in the lawgic translation process for this? Is it not needed?
#help (Added by Admin)
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Step by step breakdown:
In this lesson, we start using our language of Lawgic when dealing with relationships involving the quantity “some”. “Some” must include at least “one” or could include up to “all”.
In Lawgic, the quantifier “some” is represented with this bi-direction arrow ( ←S→ ) with an S in the middle of it. The “S” in this arrow is to distinguish it from the bi-conditional arrow ( ← → ).
The quantifier some ( ← S → ) which expresses an intersection. Let’s look at an example…..
“Some students in Mrs. Stoops’s class can read.”
Step 1: Identify that this is a statement, this is a claim that is amenable to translation. The way to do that is to (first) identify/notice the quantifier. Or the intersection indicator. In this instance it’s the word “Some”.
Step 2: Identify the two concepts. Typically, they are sets. The [first concept] is “students in Mrs. Stoops’'s class” & the [second concept] is “student or people can read”.
Step 3: Assign Symbols to represent these two sets. (Student) to represent students in Mrs. Stoops’s class & (Read) to represent the student/people who can read.
Translation Into Lawgic : Add the “Some” arrow! (student) ← S → (read) This means you can read this arrow either from left to right or from right to left. What that means is that this statement here, “student ← S → read,” is identical to the statement “read ← S → student.” In Lawgic, these two claims are the same.
Translating Back Into English: What it means is that “Some students in Mrs. Stoops’s class can read” is identical in meaning to “Some student that can read are in Mrs. Stoops’s class.”
Example→ “Some cats are pets” is identical to the claim that “Some pets are cats.” Translate into Lawgic: (c ← S → p) (p ← S → c) All four of these expressions are getting at the exact same idea, which, again, is just the idea of an intersection between two sets.
RECAP:
To translate “some” claims to Lawgic, use the bi-directional “some” arrow (← S →). The arrow is bi-directional because the intersection relationship works both ways: “some A are B” is identical to “some B are A.”
This makes sense so far. I'm waiting for something to hit that throws away all that sense.
I was confused at first but I think I know why some can include all:
Some students pass the LSAT only if they study everyday.
In order for the necessary condition to be true it has to include at least 1 student, but can also include all students, that is, even if we say all students have to study for passing LSAT does not fail the necessary. So it can include all!
Essentially we need to look for anything can cause the necessary condition to fail and exclude that interpretation.
it takes me so long to draw the some symbol smh
'Some cats are pets' and 'some pets are cats' do not seem identical to me.
I wonder where he is going with this lesson
i get how some can mean all, but how does this help with lsat questions. I feel like from my limited experience, rarely are lsat answers with 'All' correct because they are too extreme.
"Some" covers an intersect and that's why it can go both ways.
Some cats are pets and Some pets are cats
These mean the same thing because they both have the same intersection in the middle where pets and cats overlap. Once you are in that intersection part in the middle you are part of both groups so it doesn't matter.
The word some means "at least one" so once I am in that middle group then yah, at least one cat is a pet, and at least one pet is a cat. Some can include all, sure. BUT it doesn't have to. All some has to do is mean "At least one"
Ex. I am a cat who is also a pet. I am in that middle intersection group so that means I can also say I am a pet who is a cat. On the flip lets pretend I am not a cat, but I am a pet. This means I can't be part of the that intersection in the middle. I am only part of the "pets" circle.
Doess this apply to 'several' or 'many' or a few'?
#help
so i just want to be clear that this does not mean p>c as well as c>p
if it is a pet, then it is a cat,
if it is a cat, then it is a pet.
the <s> does not translate into if/then?
are there cats that are not pets?!
When you break down the Stoops's sentence, it would be better not to use "students" as one of the main concepts, since when you bring it back together students is in both concepts. It would be better to break that first main concept into "Mrs. Stoops's class."
Finally a lesson that's intuitive in my brain!
this might be answered in a future lesson, but i'm just curious. are there situations where "some" means "all" like the previous lesson said, and are used qualifiers as a sufficient condition? i know the example here is "some people who can read are students in the class, some students in the class are people who can read," but if we were provided more context before or after that statement that would include "all" as the meaning of "some" students in the class (the meaning being "all students can read in the class"), are there cases that the LSAT expects you to use this as a sufficient condition?
i hope this question makes sense!
I can't figure out if this is more or less intuitive than just using (∃x) in the usual language of logic. Learning predicate logic is very different than what I'm learning here on 7sage.
If we used it in this class example it would be like:
S = Stoop's class
R = predicate (can read)
∃x = what is called the "existential quantifier"
(∃r)[Sr]
(∃r)[Sr] is true when Sr is true for at least one value of r
I'm not understanding how we introduce a biconditional symbol in there for this lesson.
I am still confused as to why the arrow can be in both directions. The example given is that some cats can be pets. But isn't cat a subset of pets? Isn't pets the larger (necessary) condition in order for cats to be part of it? Just like the example given earlier in the course where cats --> mammals? Could someone please explain and try to clarify the difference for me because I see it as the same thing. Thank you
#help
What does "Because if there are some students in Mrs. Stoops' class who can read, then there must be some students who can read in Mrs. Stoops' class." mean?
Should it say "can't" instead of can twice??
Does the translation to Lawgic really matter on questions that require us to take just over a minute to answer? It seems like while training ourselves to translate to Lawgic is taking away the ability to do this kind of translation in our head without having to learn extra formatting.
do we need to really do the diagrams? asking because I can't imagine there being enough time for it on test day
could we translate this to Lawgic to "there exists at least one student in Ms. Stoops' class who can read" instead? seems more intuitive #help#feedback
Why isn't translating to the contrapositive a step in the lawgic translation process for this? Is it not needed?
#help (Added by Admin)