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's my clean version notes with step-by-step breakdown on the first example.
In this lesson, we start our language of Lawgic when dealing with relationships involving the quantity “most”. We first have to learn the element in Lawgic that represents “most,” and it’s this “most” arrow [ -m→ ] Notice how the “most” arrow is unidirectional. It is a one-way street, just like your standard conditional arrow. It does not point in both directions. That’s different from the “some” arrow, which is a bi-directional arrow meaning it points in both directions. You must read “most” in an unidirectional way. You cannot read it reversibly.
Example #1 Lawgic: “Most students in Mrs. Stoops’s class can read.”
Step 1: This is a claim in English that is amenable to translation into Lawgic. And to do that, we identify the indicator or quantifier. “The word “most.”
Step 2: We identify the two main concepts or sets. (first set) the set of students in Mrs. Stoops’s class. (second set) is the set of students or people who can read.
Step 3: Assign Symbols: (students) (read)
Step 4: Add the “most” arrow. The Lawgic says (students) -m→ (read). Need to emphasize that the arrow is unidirectional. This is the point of difference between the “most” arrow and the “some” arrow. You cannot read this the other way around.
RECAP:
To translate “most” claims to Lawgic, use the unidirectional “most” arrow: (-m→). The arrow is unidirectional because the “most” intersection relationship does not work both ways: “most A are B” is not identical to “most B are A.”
With the example of the students that can read, can it also be that the main set is the students in the class and the subset being the number of students who can read?
Can a "most" claim be theoretically bi-directional if the size of both groups is the same?
For instance, say that there are an equal number of dogs and pets. We combine the circles in such a way that they both overlap to account for more than half of the circle. In that instance, would we just do d <-m-> p?
@BullMatt16 I think hypothetically that's possible. I think the issue though is that I've never ran into a such a strict "most" conditional where we know specific instance to be true (both being equal in size and what not).
It's just a safer standard (from what ive seen) to simply have D -(m)> P instead. That way we know regardless of D or P size in the set that "most" of D is subsumed by P superset.
Good example of why the most arrow can only be one directional
The statement "Most A’s are B’s" is not identical to "Most B’s are A’s" because "most" implies a majority (more than 50%) but does not guarantee symmetry.
Suppose there are 100 A's and 200 B's. If most A's are B's, this means more than 50 of the A's are also B's (say, 80 A's are B's). However, that does not mean that most B's are A's. If there are 200 B's and only 80 of them are A's, then only 40% of B's are A's, which is not "most."
Contrast this to "some" where the arrow can be bidirectional
The statement "Some A's are B's" is identical to "Some B's are A's" because "some" means at least one element belongs to both sets. Unlike "most," "some" does not imply a majority or proportion—just existence.
Why Are They Identical?
If at least one A is a B, that means there exists an element in the intersection of sets A and B.
This automatically means that at least one B is an A, because the same element belongs to both sets.
I have taken a very familiar course in college called "Logic" or Philosophy 120. It goes over a lot of what's mentioned here and even goes into proofs, truth functions, and more. It seems to be a well known way to translate sentences using formulaic approaches.
I have also taken a logic class in which symbols are used, and I'm reviewing an LSAT prep book that uses symbols. I just don't think they call it "Lawgic," just logic. Same concepts.
I'm a bit confused. First these statements we're looking at are conclusions. right? We're making evaluations on the "truth" of these conclusions with these arrows or is it the validity?
Regarding validity, wouldn't "most students who can read are in Mrs. Stoops' class" still be valid? Because we're imaging a separate world where it's true.
From what I can tell usually these will be the premises on which a conclusion is drawn, or the intermediate conclusions.
So for example:
Some cats are pets (C ←s→ P)
Most pets are kept in homes (P‑m→H)
Therefore, most things kept in homes are cats (H‑m→C)
Based on the premises above, the conclusion (therefore, most things kept in homes are cats) is demonstrably false due to the conditions laid out (i.e., C←s→P and P‑m→H).
A valid conclusion drawn from these premises would be some cats are kept in homes (I think, could be wrong!) (C←s→H)
I don't think so... I am sure we will learn but if I was to wager a guess I think it is more likely to be less because I would imagine it would have a range of half minus 1 but some has a range of 1-99 (which could be more than 50.. Assuming also that 'all' is excluded)
I don't think so because while most is included under some, i.e., given the range of most fits within some, they can't be contrapositives, because it doesn't have the same truth.
Contrapositives need to be logically equivalent statements
Since some can include anything from at least 1 to up to half (and further to all), but most can include anything from at least more than a half to all, therefore, it isn't logically equivalent
I agree with Hannah and I'm sure we will learn in a later lesson but I will add that I am not sure the concept of a contrapositive translates to quantifiers, because while there is overlap, it is hard for these words to have specific logical equivalencies due to the ranges they include.
Sufficient condition and necessary condition is different because the sufficient condition coming about is enough to guarantee the necessary condition, and vice versa, the necessary condition being absent is enough to guarantee the sufficient condition not coming about.
If I go to school, then I will bring my books (S → B)
If I don't bring my books, then I don't go to school (/B → /S)
What if the most is in the necessary condition? Like with the environmental goals example from the last lesson. Meet env goal only if most cars are electric. How would that be diagrammed to account for that fact that it's not read as if most goals are met then cars are electric. MG -m-> CE ?
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34 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
@Elideebeep Thank you so much!
@brightblurr you bet!
I have a question about contrapositive. how can we use it in quantifiers? and can we even do that?
Here's my clean version notes with step-by-step breakdown on the first example.
In this lesson, we start our language of Lawgic when dealing with relationships involving the quantity “most”. We first have to learn the element in Lawgic that represents “most,” and it’s this “most” arrow [ -m→ ] Notice how the “most” arrow is unidirectional. It is a one-way street, just like your standard conditional arrow. It does not point in both directions. That’s different from the “some” arrow, which is a bi-directional arrow meaning it points in both directions. You must read “most” in an unidirectional way. You cannot read it reversibly.
Example #1 Lawgic: “Most students in Mrs. Stoops’s class can read.”
Step 1: This is a claim in English that is amenable to translation into Lawgic. And to do that, we identify the indicator or quantifier. “The word “most.”
Step 2: We identify the two main concepts or sets. (first set) the set of students in Mrs. Stoops’s class. (second set) is the set of students or people who can read.
Step 3: Assign Symbols: (students) (read)
Step 4: Add the “most” arrow. The Lawgic says (students) -m→ (read). Need to emphasize that the arrow is unidirectional. This is the point of difference between the “most” arrow and the “some” arrow. You cannot read this the other way around.
RECAP:
To translate “most” claims to Lawgic, use the unidirectional “most” arrow: (-m→). The arrow is unidirectional because the “most” intersection relationship does not work both ways: “most A are B” is not identical to “most B are A.”
"so many other pets: cats, rabbits, hamsters, fish, reptiles, etc." This is bird erasure
I guessed what the element of most would look like before it revealed and yelled in excitement when I was proven right.
Can I not make the domain "Ms Stoops' Class" and then that would make the arrow multi-directional?
@ChristopherPeterVlassis if the Domain was her class, that doesn't change the Most to Some
With the example of the students that can read, can it also be that the main set is the students in the class and the subset being the number of students who can read?
Can a "most" claim be theoretically bi-directional if the size of both groups is the same?
For instance, say that there are an equal number of dogs and pets. We combine the circles in such a way that they both overlap to account for more than half of the circle. In that instance, would we just do d <-m-> p?
@BullMatt16 I think hypothetically that's possible. I think the issue though is that I've never ran into a such a strict "most" conditional where we know specific instance to be true (both being equal in size and what not).
It's just a safer standard (from what ive seen) to simply have D -(m)> P instead. That way we know regardless of D or P size in the set that "most" of D is subsumed by P superset.
Can i say Most cats are pets is equal to few pets are cats?
@adunker620 no because few does not have the same lower boundary as most
Good example of why the most arrow can only be one directional
The statement "Most A’s are B’s" is not identical to "Most B’s are A’s" because "most" implies a majority (more than 50%) but does not guarantee symmetry.
Suppose there are 100 A's and 200 B's. If most A's are B's, this means more than 50 of the A's are also B's (say, 80 A's are B's). However, that does not mean that most B's are A's. If there are 200 B's and only 80 of them are A's, then only 40% of B's are A's, which is not "most."
Contrast this to "some" where the arrow can be bidirectional
The statement "Some A's are B's" is identical to "Some B's are A's" because "some" means at least one element belongs to both sets. Unlike "most," "some" does not imply a majority or proportion—just existence.
Why Are They Identical?
If at least one A is a B, that means there exists an element in the intersection of sets A and B.
This automatically means that at least one B is an A, because the same element belongs to both sets.
seeing this all come together when solving is mind blowing lol
how do you determine what set goes on which side of the arrow
The quantifier modifies the part that goes on the right side of the arrow.
look at what most is modifying.
Most "FISH" live in the ocean.
Most is modifying the fish not the ocean. Fish -m-> Ocean
Stoops's should be Stoops'
for proper nouns you actually do need the s after the apostrophe!
This has been bothering me too
I know this maybe a stupid question, but isn't Lawgic produced by 7Sagee, not a universal curriculum right?
I have taken a very familiar course in college called "Logic" or Philosophy 120. It goes over a lot of what's mentioned here and even goes into proofs, truth functions, and more. It seems to be a well known way to translate sentences using formulaic approaches.
I have also taken a logic class in which symbols are used, and I'm reviewing an LSAT prep book that uses symbols. I just don't think they call it "Lawgic," just logic. Same concepts.
#help
I'm a bit confused. First these statements we're looking at are conclusions. right? We're making evaluations on the "truth" of these conclusions with these arrows or is it the validity?
Regarding validity, wouldn't "most students who can read are in Mrs. Stoops' class" still be valid? Because we're imaging a separate world where it's true.
Unfortunately not, in my opinion we have to take it at face value and not assume anything unless need be assumed
From what I can tell usually these will be the premises on which a conclusion is drawn, or the intermediate conclusions.
So for example:
Some cats are pets (C ←s→ P)
Most pets are kept in homes (P‑m→H)
Therefore, most things kept in homes are cats (H‑m→C)
Based on the premises above, the conclusion (therefore, most things kept in homes are cats) is demonstrably false due to the conditions laid out (i.e., C←s→P and P‑m→H).
A valid conclusion drawn from these premises would be some cats are kept in homes (I think, could be wrong!) (C←s→H)
would the contrapositive of most be some?
student ----m ----> readers
readers students?
I don't think so... I am sure we will learn but if I was to wager a guess I think it is more likely to be less because I would imagine it would have a range of half minus 1 but some has a range of 1-99 (which could be more than 50.. Assuming also that 'all' is excluded)
I don't think so because while most is included under some, i.e., given the range of most fits within some, they can't be contrapositives, because it doesn't have the same truth.
Contrapositives need to be logically equivalent statements
Since some can include anything from at least 1 to up to half (and further to all), but most can include anything from at least more than a half to all, therefore, it isn't logically equivalent
I agree with Hannah and I'm sure we will learn in a later lesson but I will add that I am not sure the concept of a contrapositive translates to quantifiers, because while there is overlap, it is hard for these words to have specific logical equivalencies due to the ranges they include.
Sufficient condition and necessary condition is different because the sufficient condition coming about is enough to guarantee the necessary condition, and vice versa, the necessary condition being absent is enough to guarantee the sufficient condition not coming about.
If I go to school, then I will bring my books (S → B)
If I don't bring my books, then I don't go to school (/B → /S)
Logically equivalent statements
What if the most is in the necessary condition? Like with the environmental goals example from the last lesson. Meet env goal only if most cars are electric. How would that be diagrammed to account for that fact that it's not read as if most goals are met then cars are electric. MG -m-> CE ?
I think it would be something like this:
If goals are met, then most cars are electric.
G --> (C -m-> E)