"Some" refers to 1-100. To deny the existence of "some", it has to be something that is not 1-100, which means 0. Therefore, the negation of some is none.
Saying "some" actually makes an existence claim, meaning that at least one of it exist but could be more. To negate that would be saying that none exist at all.
The negation of the intersections between parrots and clever is “some.” If we are looking for something not in the “some” range then all that is left is 0%
Some is 1-100% VS negation 0%
This is my understanding of the concepts
Whereas for "all" negation
we go from all which is 100% to the negation which can include 0-99%
After watching negating, all, conditional statements, some, and most. My question is, how can we spot that in the LSAT so we know when to start negating for LR?
I had to watch this a couple of times, and for me it was a great example of my intuitive thinking taking over my logical thinking. Even though it's so obviously simple when you draw the negation of "some" with circles, splitting the Venn diagram, thinking out loud to myself that "It is not true that some parrots are clever" is equivalent to saying "No parrots are clever" drove me nuts. I kept automatically thinking "well, if it is not true that some parrots are clever, maybe it's true that many/all/several parrots are clever" - always forgetting that you can't have all parrots being clever if some parrots aren't clever.
Whew!
All of which is to say - thank you for the diagrams!
The only way to safely negate the case that one parrot is clever or that all of them are clever (via "some") is simply by saying "none of these damn things are clever--not a single one!"
@gbeeven In these negation lessons, we are not talking about the claims themselves, but the relationship between them.
The contrapositive of a statement is the logical equivalent, and that's why we can't take the contrapositive of "some" and "most" statements.
For example, take this claim: Some cats that belong to George are shy.
"some" means that at least one cat is shy (maybe all, maybe half, etc.)
In lawgic this looks like: GC ← s → shy
If we were to try to take the contrapositive, it would look like this: /GC ← s → /shy
This means that some of George's non-cats are not shy, but this has nothing to do with the original claim. We are not talking about George's hypothetical non-cats, and we never said anything about the non-shy ones. All we are saying is that some cats are shy. That's all we know.
In the case of a simple if, then statement, we can take the contrapositive.
If a cat is shy, then it belongs to George. ... S --> G
If a does not belong to George, then it is not shy. /G --> /S
Now, what if we were to negate this relationship?
OG statement: Some cats that belong to George are shy.
Negation: It is not the case that some cats that belong to George are shy.
Lawgic: GC ← s → S .... GC --> /S
here we are negating the intersection between George's cats and being shy. If there is no intersection, then it is impossible for George's cats to be shy.
Hope this helps to clear up how negation is different than contrapositive.
@Isabella! A contrapositive just creates a new conditional statement, but it means the exact same thing as the original conditional statement.
If I say all cats are mammals, then the contrapositive is: if it is not a mammal, then it is not a cat. Although the conditional statements are different word for word, the claim itself has not changed. In my contrapositive example, even though the words have changed, I am still guaranteeing the truth of the original claim.
For negation, you are refuting the statement completely, so when I say all cats are mammals, if someone is able to demonstrate that just one cat is not a mammal, then my entire argument is now refuted. The original claim now no longer holds up the way it does when I use the contrapositive of the original statement.
@16dnholli Thinking of this like math, you'd think that the slash would distribute, but this isn't math since we're only using arrows and symbols to represent language and short-hand logic.
/P <-s-> /C would mean "Some non-parrots are not clever". This really isn't the opposite of the original statement "Some parrots are clever".
Since "some" means anywhere from 1% to 100% of the parrots are clever, the opposite of a "some" relationship means that we have to be outside those boundaries to negate -- at 0%. So the opposite of "Some parrots are clever" is that "No parrots are clever". And if we make that statement into Lawgic it becomes P-->/C.
I don't think "negating" itself is the point of this, it's more that practicing how to negate and learning the meaning of lawgic negations is what we learn through this.
For example a most seriously weakens questions:
"According to statistics on drunk driving some drivers with an alcohol permillage 5 times the legal limit are not a danger to other drivers"
Lets say the following are the two answers you're oscillating between:
a) no drivers with an alcohol permillage 5 times the legal limit are not a danger to other drivers
b) some drivers with an alcohol permillage 5 times the legal limit are a danger to other drivers
Do you see the difference between the answers? a) completely undermines the statement above. b), however, sort of agrees with the statement above. It says that there are some drivers who are a danger, which also implies that there are some drivers who are not a danger. Even though b) may seem like a 'negation' of the statement. (I know here it might be fairly obvious which answer is correct)
So learning how to recognize the correct negation of the different types of statements will help you recognize the right answer quicker and more accurately. Saving you time to focus on question you find more challenging.
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
"The negated claim actually means something much stronger: No parrots are clever."
but could it not also mean ALL parrots are clever? we already said some could mean all, so why isn't it all parrots are clever rather than none at all?
@GDatria715 my understanding is that some can be up to all. If all parrots are clever, than some are clever. You could say, its not only the case that some parrots are clever.
@brendan.curran.paul I'm not sure what you mean by negating "some" in the positive sense. It looks to me like that example involves interpreting "Some" as if it means "Less than all". Under this interpretation, the negation of "less than all" becomes "all."
But the problem is "some" doesn't have to imply "less than all." It just means "at least one." So negating "some" must mean 0 (your first example).
Note that informally people often use "some" to mean "some, but not all". But "some" just means "at least one" (without expressing an opinion about whether the quantity could include all).
@Kevin_Lin Thank you for this clarification. However could I trouble you for some additional insights into this, as your comment just sparked something in my brain as to whether it could be possible to understand the above to be in a parallel relationship with negating the "All" quantifier, namely:
[Negating "Some"]
1) Initial: Some A are B (A <-s-> B)
2) Negates to: No A are B (A -> /B)
[Negating "All"]
3) Initial: All A are B (A -> B)
4) Negates to: Some A are not B (A <-s-> /B)
But notice that the 1) and 4) above, don't they amount to saying the same thing? Namely, saying "some A are B" effectively the same thing as "some A are not B" (A <-s-> B = A <-s-> /B?)
IF it is the case that the above statements 1) and 4) mean the same thing, shouldn't their contrapositives 2) and 3) also effectively mean the same thing? Ah but here I realized I walked into a contradiction, as there's no way "No A are B" would mean the same as "All A are B"...!
Thanks for indulging me as I thought aloud here--but I guess my qualm with 1) and 4) above still stands, that whether defining "Some" to mean "at least one" rather than "less than all" would be considered more a matter of linguistic convention than it is a matter of strict logic.
The motive for asking this question also stems partly from the fact that I do not come from an English-speaking country, and could not help comparing the meaning of "some" in English to the meaning of "some" in certain other languages.
@brendan.curran.paul Usually, in a real life context, the only reason you'd say "some A are B" is because you're in a situation in which people recognize that some/most A are NOT B. So you're trying to emphasize that some of them are.
And in a real life context, often the only reason you'd say "some A are NOT B" is because you're in a situation in which people recognize that some/most A are B, and you're trying to emphasize that this isn't true for all As.
But when you examine the language itself -- "some A are B" -- it really does just mean "at least one A is B" without expressing any opinion about whether any As are NOT B. And "some A are NOT B" really does just mean "at least one A is NOT B" without expressing any opinion about whether any As are B.
It's a good idea to just translate "some" to "at least one."
"Some tests are difficult" = At least one test is difficult
Are there any tests that aren't difficult? We don't know based on that statement.
"Some people are friendly" = At least one person is friendly
Are there any people that aren't friendly? We don't know based on that statement.
"Some police officers are not corrupt." = At least one police officer is not corrupt.
Are there any police officers that are corrupt? We don't know based on that statement.
"No vegetables taste good." = If vegetable --> NOT taste good
"No dogs can fly." = If dogs --> NOT fly
"No" at the beginning of a statement can be confusing; you might think it applies to the beginning of the sentence. That's why you think "No A are B" starts with "If NOT A..." But that's not the right understanding.
we are denying the intersection of the relationship in its entirety. Because few= at least 1 we must go with the definitive no claim to establish that no vegans are in fact kind
#feedback, can I accurately negate "some" with "all/no" without concerning myself with the preface is it not the case"? Essentially whenever/wherever I am negating "some" just treat it as "all/no."
Could it be the case that negating a "some" claim is simply denying that it is merely some rather than most or all? For example, could "It is not the case that some parrots are clever" imply that, actually, most parrots are clever, or all parrots are clever? #help
Nope, because "Some parrots are clever" literally means "At least one parrot is clever." What's the negation of "at least one"? Zero. The negation couldn't be "most" or "all", because those are consistent with the idea of at least one parrot being clever. Those quantities wouldn't actually contradict the idea of "some."
@TimCrawford thats contrapositive! you can negate them. eg., [some tom cruise movies are boring]. TC-mov <- s -> boring. that is the same as saying [some boring things are tom cruise movies]. but you cannot say [no boring things are tom cruise movies] because the original suggests that "at least one" (remember, some means at least one) movie that stars tom cruise is boring.
however you can negate it. to negate "at least one" you have to argue "that there is not even one." so the negation would be no thats not true. there is no intersection between boring things and tom cruise movies. so, [none of tom cruise's movies are boring OR no tom cruise movies are boring].
the two statements cannot both be true, and they cannot both be false.
90 comments
i'm tired of this grandpa :(
Something that helped me understand:
"Some" refers to 1-100. To deny the existence of "some", it has to be something that is not 1-100, which means 0. Therefore, the negation of some is none.
Hope this helps :)
@leahlovescoco To add to that:
Saying "some" actually makes an existence claim, meaning that at least one of it exist but could be more. To negate that would be saying that none exist at all.
The negation of the intersections between parrots and clever is “some.” If we are looking for something not in the “some” range then all that is left is 0%
Some is 1-100% VS negation 0%
This is my understanding of the concepts
Whereas for "all" negation
we go from all which is 100% to the negation which can include 0-99%
After watching negating, all, conditional statements, some, and most. My question is, how can we spot that in the LSAT so we know when to start negating for LR?
is anyone getting confused about how these transitions to lawgic are working
I had to watch this a couple of times, and for me it was a great example of my intuitive thinking taking over my logical thinking. Even though it's so obviously simple when you draw the negation of "some" with circles, splitting the Venn diagram, thinking out loud to myself that "It is not true that some parrots are clever" is equivalent to saying "No parrots are clever" drove me nuts. I kept automatically thinking "well, if it is not true that some parrots are clever, maybe it's true that many/all/several parrots are clever" - always forgetting that you can't have all parrots being clever if some parrots aren't clever.
Whew!
All of which is to say - thank you for the diagrams!
Yep, remember that "some" can include "all"
The only way to safely negate the case that one parrot is clever or that all of them are clever (via "some") is simply by saying "none of these damn things are clever--not a single one!"
Good point Colin!
Just want to randomly pop in and wish everyone luck in their studying! I took a break from it, but I'm back!
I'm confused what the difference is between negation and contrapositive and when to use each of them?
@Isabella! yeah im also lost on this soooo if someone could clarify that'd be so awesome
@gbeeven In these negation lessons, we are not talking about the claims themselves, but the relationship between them.
The contrapositive of a statement is the logical equivalent, and that's why we can't take the contrapositive of "some" and "most" statements.
For example, take this claim: Some cats that belong to George are shy.
"some" means that at least one cat is shy (maybe all, maybe half, etc.)
In lawgic this looks like: GC ← s → shy
If we were to try to take the contrapositive, it would look like this: /GC ← s → /shy
This means that some of George's non-cats are not shy, but this has nothing to do with the original claim. We are not talking about George's hypothetical non-cats, and we never said anything about the non-shy ones. All we are saying is that some cats are shy. That's all we know.
In the case of a simple if, then statement, we can take the contrapositive.
If a cat is shy, then it belongs to George. ... S --> G
If a does not belong to George, then it is not shy. /G --> /S
Now, what if we were to negate this relationship?
OG statement: Some cats that belong to George are shy.
Negation: It is not the case that some cats that belong to George are shy.
Lawgic: GC ← s → S .... GC --> /S
here we are negating the intersection between George's cats and being shy. If there is no intersection, then it is impossible for George's cats to be shy.
Hope this helps to clear up how negation is different than contrapositive.
@Isabella! A contrapositive just creates a new conditional statement, but it means the exact same thing as the original conditional statement.
If I say all cats are mammals, then the contrapositive is: if it is not a mammal, then it is not a cat. Although the conditional statements are different word for word, the claim itself has not changed. In my contrapositive example, even though the words have changed, I am still guaranteeing the truth of the original claim.
For negation, you are refuting the statement completely, so when I say all cats are mammals, if someone is able to demonstrate that just one cat is not a mammal, then my entire argument is now refuted. The original claim now no longer holds up the way it does when I use the contrapositive of the original statement.
I think I'm confusing the difference between the contrapositive and negating...
I don't understand how:
/(P <-s-> C) does not equal /P <-s->/C
Why does the slash only distribute to C and how do we know to only distribute it to C?
@16dnholli Thinking of this like math, you'd think that the slash would distribute, but this isn't math since we're only using arrows and symbols to represent language and short-hand logic.
/P <-s-> /C would mean "Some non-parrots are not clever". This really isn't the opposite of the original statement "Some parrots are clever".
Since "some" means anywhere from 1% to 100% of the parrots are clever, the opposite of a "some" relationship means that we have to be outside those boundaries to negate -- at 0%. So the opposite of "Some parrots are clever" is that "No parrots are clever". And if we make that statement into Lawgic it becomes P-->/C.
#help
Can someone explain how the distribution of the "not" symbol is working here?
@bappel
Example: Some A are B.
Lawgic: A <-s-> B.
Negate: No A are B or All A are not B, which are equitant to if A, the not B.
Lawgic: /(A <-s-> B) which is equivalent to A --> /B.
What is the point of negating on the LSAT
I don't think "negating" itself is the point of this, it's more that practicing how to negate and learning the meaning of lawgic negations is what we learn through this.
For example a most seriously weakens questions:
"According to statistics on drunk driving some drivers with an alcohol permillage 5 times the legal limit are not a danger to other drivers"
Lets say the following are the two answers you're oscillating between:
a) no drivers with an alcohol permillage 5 times the legal limit are not a danger to other drivers
b) some drivers with an alcohol permillage 5 times the legal limit are a danger to other drivers
Do you see the difference between the answers? a) completely undermines the statement above. b), however, sort of agrees with the statement above. It says that there are some drivers who are a danger, which also implies that there are some drivers who are not a danger. Even though b) may seem like a 'negation' of the statement. (I know here it might be fairly obvious which answer is correct)
So learning how to recognize the correct negation of the different types of statements will help you recognize the right answer quicker and more accurately. Saving you time to focus on question you find more challenging.
honestly, this doesn't need to be overthought. Just trust your intuition...
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
"It's not the case that some parrots are clever."
"The negated claim actually means something much stronger: No parrots are clever."
but could it not also mean ALL parrots are clever? we already said some could mean all, so why isn't it all parrots are clever rather than none at all?
@GDatria715 my understanding is that some can be up to all. If all parrots are clever, than some are clever. You could say, its not only the case that some parrots are clever.
@GDatria715 Remember that "some" has a lower bound of at least 1 and an upper bound of ALL. For it not to be true, it has to be 0.
Could someone explain, in the parrot example, the outcome of the negated statement was:
Parrot --> /Clever
Would it also work to say:
Clever --> /Parrot
@SamI_02 I do believe that is the contrapositive, so yes it would work
Question:
If what we are doing is denying the "some" relationship, couldn't the negation work both in the negative and in the positive senses?
Namely:
['Negative' negation]
Initial: Some parrots are clever.
1st step: It's not the case that some parrots are clever.
Final: No parrots are clever.
['Positive' negation]
Initial: Some parrots are clever.
1st step: It's not the case that some parrots are clever.
Final: All parrots are clever.
If any error(s) in logic were made above, would appreciate corrections and explanations. Many thanks!
@brendan.curran.paul I'm not sure what you mean by negating "some" in the positive sense. It looks to me like that example involves interpreting "Some" as if it means "Less than all". Under this interpretation, the negation of "less than all" becomes "all."
But the problem is "some" doesn't have to imply "less than all." It just means "at least one." So negating "some" must mean 0 (your first example).
Note that informally people often use "some" to mean "some, but not all". But "some" just means "at least one" (without expressing an opinion about whether the quantity could include all).
@Kevin_Lin Thank you for this clarification. However could I trouble you for some additional insights into this, as your comment just sparked something in my brain as to whether it could be possible to understand the above to be in a parallel relationship with negating the "All" quantifier, namely:
[Negating "Some"]
1) Initial: Some A are B (A <-s-> B)
2) Negates to: No A are B (A -> /B)
[Negating "All"]
3) Initial: All A are B (A -> B)
4) Negates to: Some A are not B (A <-s-> /B)
But notice that the 1) and 4) above, don't they amount to saying the same thing? Namely, saying "some A are B" effectively the same thing as "some A are not B" (A <-s-> B = A <-s-> /B?)
IF it is the case that the above statements 1) and 4) mean the same thing, shouldn't their contrapositives 2) and 3) also effectively mean the same thing? Ah but here I realized I walked into a contradiction, as there's no way "No A are B" would mean the same as "All A are B"...!
Thanks for indulging me as I thought aloud here--but I guess my qualm with 1) and 4) above still stands, that whether defining "Some" to mean "at least one" rather than "less than all" would be considered more a matter of linguistic convention than it is a matter of strict logic.
The motive for asking this question also stems partly from the fact that I do not come from an English-speaking country, and could not help comparing the meaning of "some" in English to the meaning of "some" in certain other languages.
Hoping to hear back - Many thanks!
@brendan.curran.paul Usually, in a real life context, the only reason you'd say "some A are B" is because you're in a situation in which people recognize that some/most A are NOT B. So you're trying to emphasize that some of them are.
And in a real life context, often the only reason you'd say "some A are NOT B" is because you're in a situation in which people recognize that some/most A are B, and you're trying to emphasize that this isn't true for all As.
But when you examine the language itself -- "some A are B" -- it really does just mean "at least one A is B" without expressing any opinion about whether any As are NOT B. And "some A are NOT B" really does just mean "at least one A is NOT B" without expressing any opinion about whether any As are B.
It's a good idea to just translate "some" to "at least one."
"Some tests are difficult" = At least one test is difficult
Are there any tests that aren't difficult? We don't know based on that statement.
"Some people are friendly" = At least one person is friendly
Are there any people that aren't friendly? We don't know based on that statement.
"Some police officers are not corrupt." = At least one police officer is not corrupt.
Are there any police officers that are corrupt? We don't know based on that statement.
@Kevin_Lin Thank you for clearing this up for me! Appreciate it a lot :)
Following this logic:
Original: Some A are B
Negated: No A are B
Original: A ←s→ B
Negated: A → /B
Why isn't the last part no /A→ B, because the negated above reads No A are B ?
@nelsonmartins "No A are B" = If A, then NOT B
"No vegetables taste good." = If vegetable --> NOT taste good
"No dogs can fly." = If dogs --> NOT fly
"No" at the beginning of a statement can be confusing; you might think it applies to the beginning of the sentence. That's why you think "No A are B" starts with "If NOT A..." But that's not the right understanding.
so for example some vegans are kind
Negated: No vegans are kind
we are denying the intersection of the relationship in its entirety. Because few= at least 1 we must go with the definitive no claim to establish that no vegans are in fact kind
What is the importance of this? I do not get when and how I would use this?
#feedback, can I accurately negate "some" with "all/no" without concerning myself with the preface is it not the case"? Essentially whenever/wherever I am negating "some" just treat it as "all/no."
Could it be the case that negating a "some" claim is simply denying that it is merely some rather than most or all? For example, could "It is not the case that some parrots are clever" imply that, actually, most parrots are clever, or all parrots are clever? #help
Got it, thank you!
thank you, this makes sense
Nope, because "Some parrots are clever" literally means "At least one parrot is clever." What's the negation of "at least one"? Zero. The negation couldn't be "most" or "all", because those are consistent with the idea of at least one parrot being clever. Those quantities wouldn't actually contradict the idea of "some."
so basically we just negate all of them down one? so all to most, most to some and some to none?
is it correct to say in order to negate an ALL, SOME, and Conditional relationships, you have to deny only the necessary condition?
I thought that you could not negate some and most claims
@TimCrawford thats contrapositive! you can negate them. eg., [some tom cruise movies are boring]. TC-mov <- s -> boring. that is the same as saying [some boring things are tom cruise movies]. but you cannot say [no boring things are tom cruise movies] because the original suggests that "at least one" (remember, some means at least one) movie that stars tom cruise is boring.
however you can negate it. to negate "at least one" you have to argue "that there is not even one." so the negation would be no thats not true. there is no intersection between boring things and tom cruise movies. so, [none of tom cruise's movies are boring OR no tom cruise movies are boring].
the two statements cannot both be true, and they cannot both be false.
i hope this helps?
no cats are boneless Cats -> /boneless