im confused. "Some students in Mrs. Stoops's class can read," can also mean "Some students that can read are in Mrs. Stoops's class." However, when you mapped it, you said that students and read are two different sets. However, the translation seems to merge these two and is now "students reading" and "Mrs. stoops's class." Am I wrong?
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
@JDMurphy If you see it virtually you will understand: let's say pets include dog cat and rat in a ratio of 40:38:22
so if some cats are pets its saying 38% of cats are pet why? and what?; well cats includes all tigers, lions and cats at a ratio of 40:40:20 see of those only 20 are pets means 20% of cats are pet but those 20% constitute as 38% in pets world,
Now, in the second sentence pets who are they dog, cat and rat and some of them aka 38% are still cats.
one more example
like some Asians are Indian
some Indian are Asian use the above concept you will understand.
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.
@EmilyMacaluso no it does not. The relationship is different. The if/then is a sufficiency and necessity (a guarantee if you have the sufficient) and the some is of intersection (more of an over lap).
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."
omg yes finally - the last few lessons in the last unit were just awful. Im scared to continue studying just bc how hare those were -- just impossible. I hope it gets more doable now
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 took some logic as well, so that existential modifier brought back a lot of memories haha. But I think it is just approaching the propositions from a different angle. Using the existential modifier is bringing it to a higher level of logic then is necessary for understanding the passage. Also it doesn't fit as well with the rest of the logical language we are using, as you pointed out. The 'some' biconditional that 7sage uses, doesn't fit into the syntax of first or second order logic, so that is why it isn't "translatable." However, the phrase you expressed is saying the same thing. Simply that there are two sets, and there is at least one object that shares the attributes of these two sets. Could be more, but we know for sure that there is at least one. Which is what the existential modifier is saying, as well as the 'some' biconditional.
@ColinErickson I agree this is going to take some getting used too lol, I have taught predicate logic to college freshman for 3 years and this new style threw me through a loop
Also I'm pretty sure using the existential as a logical connective inevitably leads to a contradiction within a logical system that includes it (from Russell's "On Denoting") however, it may still be practical?
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
I think it comes down to the indicator word used and understanding the context of the conditional. By saying '"some" cats are pets,' the main concept "cats" cannot be the sufficient condition or subset because there is a chance that "some" cats aren't pets. (Although, "some" can include "all," we can't ignore the possibility that that might not be the case in this scenario.)
For something to be a sufficient condition, it needs to be fully enclosed within the bigger circle of the necessary condition, but the word "some" makes it unknown if the circle of "cats" is fully within the circle of "pets." Instead, we would imagine it as more of a venn diagram with some overlap between "cats" and "pets." This way, we are making sure to not jump to conclusion on the idea that ALL cats are pets.
If the example had explicitly stated, "All cats are pets," then we can safely conclude cats --> pets.
Cats are animals regardless of if they’re pets. Like how men can be men without being dads. Some cats are wild, some are pets. So some pets are cats, and some cats are pets. Hope this helps :)
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?
No, because the point he's making is that you can reverse the grammar and the meaning of the sentence doesn't change, just as you an reverse which side each symbol is on and the meaning doesn't change.
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.
I think the point of it is so we can move faster through problems, cause once it becomes second nature you'll be able to naturally associate the different parts of the argument with letters instead of big chunks of words.
My understanding is that the harder the sentences and paragraphs get, the more you'll need to translate. It's just to get into the habit of doing it so when it gets harder, you'll be able to answer hard questions correctly and (almost) just as fast without having to do extra mental heavy lifting.
It's a good tool for understanding the concept in the beginning, but you can train yourself to do these in your head as you become more comfortable with the material :)
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
The reason is because "some" is a bi-conditional indicator, and bi-conditionals don't have contrapositives. You can take the converse (b ←s→ a) or inverse of the statement if you want (/a ←s→ /b), but a contrapositive can only be taken from a conditional statement going in one direction.
Almost right. "some" is not a bi-conditional. Bi-conditionals do have contrapositives. You can review that lesson here: https://7sage.com/lesson/bi-conditionals/
You can read a "some" claim reversibly, which I think is what you mean by "take the converse." b ←s→ a is identical to a ←s→ b. But even here, it's not a "converse" which is a technical term in conditional logic.
"Inverse" is also a technical term in conditional logic with no application in the quantifier logic. a ←s→ b is not identical to /a ←s→ /b.
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43 comments
im confused. "Some students in Mrs. Stoops's class can read," can also mean "Some students that can read are in Mrs. Stoops's class." However, when you mapped it, you said that students and read are two different sets. However, the translation seems to merge these two and is now "students reading" and "Mrs. stoops's class." Am I wrong?
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
This is how they get you — ‘it’s easy!’… 5 minutes later I’m solving advanced algebra with no numbers, just vibes and letters.
@JDMurphy If you see it virtually you will understand: let's say pets include dog cat and rat in a ratio of 40:38:22
so if some cats are pets its saying 38% of cats are pet why? and what?; well cats includes all tigers, lions and cats at a ratio of 40:40:20 see of those only 20 are pets means 20% of cats are pet but those 20% constitute as 38% in pets world,
Now, in the second sentence pets who are they dog, cat and rat and some of them aka 38% are still cats.
one more example
like some Asians are Indian
some Indian are Asian use the above concept you will understand.
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?
@EmilyMacaluso no it does not. The relationship is different. The if/then is a sufficiency and necessity (a guarantee if you have the sufficient) and the some is of intersection (more of an over lap).
@8M_M8 ^^^
are there cats that are not pets?!
@OrcaPark Yes. Some cats are feral street cats. Some are wild tigers. Not all are 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!
omg yes finally - the last few lessons in the last unit were just awful. Im scared to continue studying just bc how hare those were -- just impossible. I hope it gets more doable now
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 took some logic as well, so that existential modifier brought back a lot of memories haha. But I think it is just approaching the propositions from a different angle. Using the existential modifier is bringing it to a higher level of logic then is necessary for understanding the passage. Also it doesn't fit as well with the rest of the logical language we are using, as you pointed out. The 'some' biconditional that 7sage uses, doesn't fit into the syntax of first or second order logic, so that is why it isn't "translatable." However, the phrase you expressed is saying the same thing. Simply that there are two sets, and there is at least one object that shares the attributes of these two sets. Could be more, but we know for sure that there is at least one. Which is what the existential modifier is saying, as well as the 'some' biconditional.
@ColinErickson I agree this is going to take some getting used too lol, I have taught predicate logic to college freshman for 3 years and this new style threw me through a loop
Also I'm pretty sure using the existential as a logical connective inevitably leads to a contradiction within a logical system that includes it (from Russell's "On Denoting") however, it may still be practical?
@aldertree00644 I feel just taking ∩ from set theory could suffice alongside the existential quantifier.
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
I think it comes down to the indicator word used and understanding the context of the conditional. By saying '"some" cats are pets,' the main concept "cats" cannot be the sufficient condition or subset because there is a chance that "some" cats aren't pets. (Although, "some" can include "all," we can't ignore the possibility that that might not be the case in this scenario.)
For something to be a sufficient condition, it needs to be fully enclosed within the bigger circle of the necessary condition, but the word "some" makes it unknown if the circle of "cats" is fully within the circle of "pets." Instead, we would imagine it as more of a venn diagram with some overlap between "cats" and "pets." This way, we are making sure to not jump to conclusion on the idea that ALL cats are pets.
If the example had explicitly stated, "All cats are pets," then we can safely conclude cats --> pets.
Cats are animals regardless of if they’re pets. Like how men can be men without being dads. Some cats are wild, some are pets. So some pets are cats, and some cats are pets. Hope this helps :)
Aha, thank you for this explanation.
#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??
No, because the point he's making is that you can reverse the grammar and the meaning of the sentence doesn't change, just as you an reverse which side each symbol is on and the meaning doesn't change.
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.
I think the point of it is so we can move faster through problems, cause once it becomes second nature you'll be able to naturally associate the different parts of the argument with letters instead of big chunks of words.
My understanding is that the harder the sentences and paragraphs get, the more you'll need to translate. It's just to get into the habit of doing it so when it gets harder, you'll be able to answer hard questions correctly and (almost) just as fast without having to do extra mental heavy lifting.
do we need to really do the diagrams? asking because I can't imagine there being enough time for it on test day
It's a good tool for understanding the concept in the beginning, but you can train yourself to do these in your head as you become more comfortable with the material :)
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
+1 student who can read ←s→ Ms. Stoop's class
Why isn't translating to the contrapositive a step in the lawgic translation process for this? Is it not needed?
#help (Added by Admin)
Replying in case anyone else sees this:
The reason is because "some" is a bi-conditional indicator, and bi-conditionals don't have contrapositives. You can take the converse (b ←s→ a) or inverse of the statement if you want (/a ←s→ /b), but a contrapositive can only be taken from a conditional statement going in one direction.
Almost right. "some" is not a bi-conditional. Bi-conditionals do have contrapositives. You can review that lesson here: https://7sage.com/lesson/bi-conditionals/
You can read a "some" claim reversibly, which I think is what you mean by "take the converse." b ←s→ a is identical to a ←s→ b. But even here, it's not a "converse" which is a technical term in conditional logic.
"Inverse" is also a technical term in conditional logic with no application in the quantifier logic. a ←s→ b is not identical to /a ←s→ /b.