Here, we examine another formal argument pattern: if “most A are B” and “most A are C,” then we can conclude that “some B are C.” Last formal argument needed to be familiar with. Formal argument #6. This is called “two split most". This has to do with the generic form of the argument where it has to do with the generic form of the argument. Has to do with the shape of the argument. This is an argument where you get two premises that say [A most B]. So most [As are Bs]. Another premise that says [A most C.] So most [As are Cs.] When you have two premises that look like that, what must be true is that [some Bs are Cs.] There has to be a B and C intersection.
Premise: A - m → B - m → C
Conclusion: B ← s → C
A B, A B, A B C, A C, A C
Example #1: “Most almonds grown in California are produced for domestic consumption. Most almonds grown in California require intense irrigation. Therefore, some almonds produced for domestic consumption require intense irrigation.”
The conclusion follows logically!
If it’s true that most almonds are produced for domestic consumption, and it simultaneously is true that most almonds also require intense irrigation, well, there’s at least one intersecting almond that requires both intense irrigation and is produced for domestic consumption.
Premise/Chain Link: [Almonds - m → Domestic] [ - m → Irrigation]
Conclusion: [Domestic ← S → Irrigation]
RECAP:
Formal argument #6 “two split mosts”: Most As are Bs. Most As are Cs. Therefore, some Bs are Cs.
All arguments that instantiate this form are valid. We can substitute any concept for A, B, and C and the argument will still be valid.
This is made easier if one uses Venn diagrams. Put "almonds grown in CA" in the center circle. Then each of the concepts "grown for domestic consumption" and "requiring intense irrigation" have to overlap that circle more than 50%. There necessarily has to be some almonds in the overlapping zone.
Most almonds grown in California are produced for domestic consumption. Most almonds grown in California require intense irrigation to produce. Therefore, some almonds produced (forgot to put "in California" here) for domestic consumption require intense irrigation to produce.
Would an invalid version of this argument go something like:
Most oranges grown in Florida are delicious. Most oranges grown in Florida grow in the summer and spring. Therefore, all oranges grown in Florida are delicious and grow in the summer and spring.
Or something like: most oranges grown in Florida are delicious. Most oranges grown in Florida grow in the summer and spring. Therefore, no oranges grown in South Carolina are delicious and grow in the summer/spring.
#feedback would it be possible to provide an example of an invalid argument in these videos?
@breezyprabahar944 I learned in a one of the classes that you can most definitely conclude "some" between two alls, not sure about most, though. Would love to know someone elses thoughts!
One way I found to visualize this is by using your hands. On one hand, hold up three fingers to represent 'most A are B.' On the other hand, hold up three fingers for 'most A are C,' then bring your hands together. No matter what you do, at least one of your fingers will overlap. Hopefully, that helps someone!
That is premised on an assumption that there are only two contrasting phenomenon. What happens when there are three? For example, most girls like pink. There may be more than two colors involved and therefore, 50% may not necessary mean most. e.g 4 like pink, 3 like gold, 3 like yellow. To me, then "most" in this case means more than 33.33%. And, that negates this assertion. For purposes of LSAT, I will however assume that there are only two contrasting phenomenon and assume that most means more than 50% unless otherwise stated.
The other issue arises in such a case: Most men like black, Most women like black. Does that mean that some women are men or most men are women? No, it doesn't.
Most means greater than 50% regardless of how many options there are. So if most girls like pink, that means at least >50% of the girls like pink.
It may help you to view most as effectively the “majority.” 4 girls liking pink, 3 liking gold, and 3 liking yellow would mean that a plurality of girls like pink—not a majority (so not most).
For your other one, your argument is this:
Most men like black. Most women like black. Therefore, some women are men.
It would be visualized as this:
Men —m—> black
Women —m—>black
Women ←s→ Men
That’s an invalid argument, and it is not the format of formal argument #6.
The proper format results in you ending up with the same subject being the “most” twice.
A more accurate one would be:
Most men like black. Most men like women. Therefore, some men who like black also like women.
I think the problem with the logic that you're using here is you're using the word "most" as an idea of "the most". While in your example it is true that the color that women like "the most" is pink, it is not true that "most" women like pink. Also 2nd example, you're using 2 different "A" variables.
Using Lawgic, the argument you made is
Men -m-> Black
Women -m- Black
Therefore, Men Women
This is not a valid argument since the most is not an omnidirectional statement. It is not valid to say in that most people who like the color black are men or women given the information that we have
"Given that the definition of "most" is "half plus one," the two scoops from A must overlap by at least one member. That one member transfers to B and to C. That necessarily creates an overlap between B and C."
Using the example of almonds, I think it becomes clearer if numbers are used. If 55% of the almonds grown in California are produced for domestic consumption, and 51% of the almonds grown in California require intense irrigation to produce, there must be an overlap of at least 4%. Therefore, some (at least 4%) of the almonds grown in California are both produced for domestic consumption and require intense irrigation.
Wouldn't the argument only be valid if the last sentence said "Therefore, some almonds produced for domestic consumption require intense irrigation to produce." Because almonds is a subset of foods which is a massive set.
My guess is that if the conclusion had said something like "some fruits" or "some vegetables", that would be invalid because almonds are not fruits or vegetables. But since almonds are food, the argument remains logically sound when it generalizes to "foods."
since being an almond is sufficient for being a food, you can say that some foods are produced for domestic consumption. it's like saying some cats drink milk therefore some mammals drink milk. mammals is a massive set, but cats are still apart of it.
on a purely logical front, you're correct. all you can suppose is what is given in the argument. i think 7sage calls their use of logic intended for the lsat as "lawgic," which allows for some presupposition, because if they just said they were teaching propositional logic, then yeah the above argument wouldn't technically be valid and they'd hail a lot of similar criticism (even though that type of strict logicality isn't necessary or beneficial for/to the lsat).
So with the reversal not being applicable with most this would be valid
"Most cafes that serve decaf source coffee beans from Blue Mountain Roasters. All cafes that serve decaf also serve tea. Therefore, most cafes that serve tea source coffee beans from Blue Mountain Roasters."
but this would not
"Most cafes that serve decaf source coffee beans from Blue Mountain Roasters. All cafes that serve decaf also serve tea. Therefore, most cafes that serve tea source coffee beans from Blue Mountain Roasters. "
I think these would also all be valid conclusions as well:
A --m--> B
A --m--> C
---------------
A ←s→ B and C
so if we said a majority of (most) dogs have brown fur (BF), and a majority of dogs have green eyes (GE), we can conclude that at least one dog that has brown fur has green eyes too. Or put another way: some dogs have both brown fur and green eyes
Dogs ←s→ BF and GE
All A ----> B or C or B+C
we can also conclude that all dogs have either green eyes, or brown fur, or both
Dogs ----> BF or GE or BF+GE
/A ----> /B and /C
and we can say that no dogs have neither brown fur nor green eyes
/Dogs ----> /BF and /GE
(put another way: no dogs lack both brown fur and green eyes)
In thinking about this more, I guess we can really only conclude the first one:
Dogs ←s→ BF and GE
This is because if we take the least possible overlap of the two subsets of dogs, there is still 1% of overlap no matter what
But the last two I posted can't be concluded because we don't know the actual percentage of each, nor the true extent of overlap between green eyes and brown fur. That is: they could both be 100% and therefore entirely overlap and leave no room for other hair and eye colors, or they could both be only 51% but the exact same 51% (leaving room for dogs with neither BF nor GE), etc.
So the fact that "Most" (at least 51%) can also include "All" (100%) does allow the possibility that the second and third COULD BE TRUE, but not that they MUST BE TRUE like I thought, so I guess it's still useful to know for the test. Though I wish 7 Sage would let us delete our comments and repost instead of having to reply to our own comments...
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68 comments
If anyone is a visual learner, this is the best way I helped myself understand.
Totally lost me here!
Can I also say:
therefore, some A are both B and C
A<S>(B and C)
Here, we examine another formal argument pattern: if “most A are B” and “most A are C,” then we can conclude that “some B are C.” Last formal argument needed to be familiar with. Formal argument #6. This is called “two split most". This has to do with the generic form of the argument where it has to do with the generic form of the argument. Has to do with the shape of the argument. This is an argument where you get two premises that say [A most B]. So most [As are Bs]. Another premise that says [A most C.] So most [As are Cs.] When you have two premises that look like that, what must be true is that [some Bs are Cs.] There has to be a B and C intersection.
Premise: A - m → B - m → C
Conclusion: B ← s → C
A B, A B, A B C, A C, A C
Example #1: “Most almonds grown in California are produced for domestic consumption. Most almonds grown in California require intense irrigation. Therefore, some almonds produced for domestic consumption require intense irrigation.”
The conclusion follows logically!
If it’s true that most almonds are produced for domestic consumption, and it simultaneously is true that most almonds also require intense irrigation, well, there’s at least one intersecting almond that requires both intense irrigation and is produced for domestic consumption.
Premise/Chain Link: [Almonds - m → Domestic] [ - m → Irrigation]
Conclusion: [Domestic ← S → Irrigation]
RECAP:
Formal argument #6 “two split mosts”: Most As are Bs. Most As are Cs. Therefore, some Bs are Cs.
All arguments that instantiate this form are valid. We can substitute any concept for A, B, and C and the argument will still be valid.
Premise: [A - m → B]
[ - m → C]
Conclusion: B ← S → C
I don't know if this is right but I thought of it like an algebraic equation: AMB+AMC=BSC.
This is made easier if one uses Venn diagrams. Put "almonds grown in CA" in the center circle. Then each of the concepts "grown for domestic consumption" and "requiring intense irrigation" have to overlap that circle more than 50%. There necessarily has to be some almonds in the overlapping zone.
Most almonds grown in California are produced for domestic consumption. Most almonds grown in California require intense irrigation to produce. Therefore, some almonds produced (forgot to put "in California" here) for domestic consumption require intense irrigation to produce.
Would an invalid version of this argument go something like:
Most oranges grown in Florida are delicious. Most oranges grown in Florida grow in the summer and spring. Therefore, all oranges grown in Florida are delicious and grow in the summer and spring.
Or something like: most oranges grown in Florida are delicious. Most oranges grown in Florida grow in the summer and spring. Therefore, no oranges grown in South Carolina are delicious and grow in the summer/spring.
#feedback would it be possible to provide an example of an invalid argument in these videos?
LOVE this bucket analogy. Makes thinks super clear in understanding the form of these types of arguments.
Is there anything like this on two split "alls"?
@breezyprabahar944 I learned in a one of the classes that you can most definitely conclude "some" between two alls, not sure about most, though. Would love to know someone elses thoughts!
Why is the lawgic not A and B <-s-> A and C?
@IsaacZerby I think it could, but that's slightly more difficult to grasp and easier to fall into trap thinking.
Using math helped me with this concept:
55% of almonds are consumed domestically
55% of almonds require intense irrigation
55+55 = 110.
Some (at least 10%) of almonds consumed domestically require intense irrigation
One way I found to visualize this is by using your hands. On one hand, hold up three fingers to represent 'most A are B.' On the other hand, hold up three fingers for 'most A are C,' then bring your hands together. No matter what you do, at least one of your fingers will overlap. Hopefully, that helps someone!
Just thought I'd add that all of this also holds true when we're dealing with two "all" statements.
For example:
All pizzas have tomato sauce.
A → B
All pizzas have cheese.
A → C
Therefore, some things that have tomato sauce also have cheese.
B←s→C
love the pfp
That is premised on an assumption that there are only two contrasting phenomenon. What happens when there are three? For example, most girls like pink. There may be more than two colors involved and therefore, 50% may not necessary mean most. e.g 4 like pink, 3 like gold, 3 like yellow. To me, then "most" in this case means more than 33.33%. And, that negates this assertion. For purposes of LSAT, I will however assume that there are only two contrasting phenomenon and assume that most means more than 50% unless otherwise stated.
The other issue arises in such a case: Most men like black, Most women like black. Does that mean that some women are men or most men are women? No, it doesn't.
Most means greater than 50% regardless of how many options there are. So if most girls like pink, that means at least >50% of the girls like pink.
It may help you to view most as effectively the “majority.” 4 girls liking pink, 3 liking gold, and 3 liking yellow would mean that a plurality of girls like pink—not a majority (so not most).
For your other one, your argument is this:
Most men like black. Most women like black. Therefore, some women are men.
It would be visualized as this:
Men —m—> black
Women —m—>black
Women ←s→ Men
That’s an invalid argument, and it is not the format of formal argument #6.
The proper format results in you ending up with the same subject being the “most” twice.
A more accurate one would be:
Most men like black. Most men like women. Therefore, some men who like black also like women.
Men —m—> black
Men —m—> women
black ←s→ women
I think the problem with the logic that you're using here is you're using the word "most" as an idea of "the most". While in your example it is true that the color that women like "the most" is pink, it is not true that "most" women like pink. Also 2nd example, you're using 2 different "A" variables.
Using Lawgic, the argument you made is
Men -m-> Black
Women -m- Black
Therefore, Men Women
This is not a valid argument since the most is not an omnidirectional statement. It is not valid to say in that most people who like the color black are men or women given the information that we have
Is there anyway that we can drill on each of the formal arguments?
#feedback this would be helpful
agreed, I would love to drill on the formal arguments!
"Given that the definition of "most" is "half plus one," the two scoops from A must overlap by at least one member. That one member transfers to B and to C. That necessarily creates an overlap between B and C."
What the heck does this mean?
Using the example of almonds, I think it becomes clearer if numbers are used. If 55% of the almonds grown in California are produced for domestic consumption, and 51% of the almonds grown in California require intense irrigation to produce, there must be an overlap of at least 4%. Therefore, some (at least 4%) of the almonds grown in California are both produced for domestic consumption and require intense irrigation.
Thank you!
Most almonds grown in California are produced for domestic consumption.
Almonds (5 pieces)
Domestic consumption of food in california (100 pieces)
Most almonds are for domestic consumption = atleast 3 out of 100 food items are almonds
Suppose, only 3 are for domestic consumption and balance 2 are for exports.
Most almonds grown in California require intense irrigation to produce.
Almonds (5 pieces)
Items that require intense irrigation in california (100 pieces)
Most almonds in C require intense irrigation = atleast 3 out of 100 items that require irrigation are almonds
Even if you assume that out of these 3 almonds, 2 are exported, atleast 1 almond is consumed domestically.
Therefore, some foods produced for domestic consumption require intense irrigation to produce.
Total food items in C = 100
Total almonds domestically consumed = 3, out of which atleast 1 almond was produced with intense irrigation
So, some food in California (that 1 fat almond which sucked all the waters in california) require intense irrigation.
some, all, some
Some water have minerals in it. All minerals give water flavor. Therefore, some water has flavor in it.
Water ←s→ minerals
Minerals → flavor
Water ←s→ flavor
Negated:
Water → minerals
Minerals ←s→ /flavor
Water → flavor
All or no water has minerals in it. Some minerals don’t give water flavor. All or no water has flavor in it.
most, all, most
Most water have minerals in it. All minerals give water flavor. Therefore, most water has flavor in it.
Water ‑m→ minerals
Minerals → flavor
Water ‑m→ flavor
Negated:
Water ‑m→ /minerals
Minerals ←s→ /flavor
Water ‑m→ /flavor
Most water doesn’t have minerals in it. Some minerals don’t give water flavor. Most water doesn’t have flavor.
most, most, some
Most water have minerals in it. Most minerals give water flavor. Therefore, some water has flavor in it.
Water ‑m→ minerals
Minerals ‑m→ flavor
Water ←s→ flavor
Negated:
Water ‑m→ /minerals
Minerals ‑m→ /flavor
/Water → flavor
Most water doesn’t have minerals in it. Most minerals don’t give water flavor. No water has flavor.
Hey, remember you can't negate/do contrapositives of some statements or most.
If it says: Some water is in California
Then it will be: Water - some - California OR California - some - Water
Wouldn't the argument only be valid if the last sentence said "Therefore, some almonds produced for domestic consumption require intense irrigation to produce." Because almonds is a subset of foods which is a massive set.
My guess is that if the conclusion had said something like "some fruits" or "some vegetables", that would be invalid because almonds are not fruits or vegetables. But since almonds are food, the argument remains logically sound when it generalizes to "foods."
since being an almond is sufficient for being a food, you can say that some foods are produced for domestic consumption. it's like saying some cats drink milk therefore some mammals drink milk. mammals is a massive set, but cats are still apart of it.
I agree I'm also confused on that. I feel as though the use of 'foods' makes the argument invalid
on a purely logical front, you're correct. all you can suppose is what is given in the argument. i think 7sage calls their use of logic intended for the lsat as "lawgic," which allows for some presupposition, because if they just said they were teaching propositional logic, then yeah the above argument wouldn't technically be valid and they'd hail a lot of similar criticism (even though that type of strict logicality isn't necessary or beneficial for/to the lsat).
Okay, so:
Most cats are domesticated.
Most cats have fur.
Therefore, some domesticated cats have fur.
Does that work?
Looks right to me!
I don't really like how the formal arguments are split up into different sections. I think the ordering of lessons could be done better.
So with the reversal not being applicable with most this would be valid
"Most cafes that serve decaf source coffee beans from Blue Mountain Roasters. All cafes that serve decaf also serve tea. Therefore, most cafes that serve tea source coffee beans from Blue Mountain Roasters."
but this would not
"Most cafes that serve decaf source coffee beans from Blue Mountain Roasters. All cafes that serve decaf also serve tea. Therefore, most cafes that serve tea source coffee beans from Blue Mountain Roasters. "
Yes or no?
Most students who study daily do well
Most students who study daily have willpower
Therefore, some students who do well have willpower.
I think these would also all be valid conclusions as well:
A --m--> B
A --m--> C
---------------
A ←s→ B and C
so if we said a majority of (most) dogs have brown fur (BF), and a majority of dogs have green eyes (GE), we can conclude that at least one dog that has brown fur has green eyes too. Or put another way: some dogs have both brown fur and green eyes
Dogs ←s→ BF and GE
All A ----> B or C or B+C
we can also conclude that all dogs have either green eyes, or brown fur, or both
Dogs ----> BF or GE or BF+GE
/A ----> /B and /C
and we can say that no dogs have neither brown fur nor green eyes
/Dogs ----> /BF and /GE
(put another way: no dogs lack both brown fur and green eyes)
In thinking about this more, I guess we can really only conclude the first one:
Dogs ←s→ BF and GE
This is because if we take the least possible overlap of the two subsets of dogs, there is still 1% of overlap no matter what
But the last two I posted can't be concluded because we don't know the actual percentage of each, nor the true extent of overlap between green eyes and brown fur. That is: they could both be 100% and therefore entirely overlap and leave no room for other hair and eye colors, or they could both be only 51% but the exact same 51% (leaving room for dogs with neither BF nor GE), etc.
So the fact that "Most" (at least 51%) can also include "All" (100%) does allow the possibility that the second and third COULD BE TRUE, but not that they MUST BE TRUE like I thought, so I guess it's still useful to know for the test. Though I wish 7 Sage would let us delete our comments and repost instead of having to reply to our own comments...