@Luna Yes I believe so. "Its not the case that all dogs are friendly" denies that all dogs are friendly. This translates to "some dogs are not friendly".
To me it's confusing to include /(D -> F) because in algebra, you would distribute whatever is outside of the parenthesis and get /D -> /F. But in Lawgic, it really means to negate the necessary condition to create a condition that contradicts the original claim. And you can't contradict the original claim without having the non-negated sufficient condition. If you negate the sufficient condition, then you are talking about a different scenario than the one you're trying to negate. That's how I'm interpreting these lessons, let me know if I have anything wrong though
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
@Tannercho06897 Contrapositive and negation are different!
Contrapositive is an equivalent way of expressing the relationship. We typically create a contrapositive of a normal conditional relationship (ex. cats are mammals) by flipping sufficient and necessary conditions, and negating each condition (ex. if not a mammal, then not a cat). These are two ways of expressing the same relationship.
Negation of an All statement/relationship ("it is not the case that all ...") is creating the opposite of the relationship. For example, the negation of "all cats are mammals" would be "some cats are not mammals". They express totally different relationships between the sets.
To negate "All dogs are friendly," we'd be focusing on the word "all" and disproving that specific word by saying, "It is not true that all dogs are friendly." This negated statement logically states that, at the very least, there exists at least one dog that is not friendly; hence, the existence of at least one unfriendly dog means that it's impossible for every single dog in existence to be friendly.
Going back a few lessons, we also know that "some" entails a numerical baseline of at least one. Therefore, this negated statement could also be written as, "Some dogs are not friendly." That's why the negation of "All dogs are friendly," is "Some dogs are not friendly!"
Good luck in your studies! Trust your gut and intuition, and reward yourself for every little bit of progress you make!!
So the difference between the contrapositive and the negation of a claim is that contrapositive reverses and negates to be logically equal to original. Negation of a claim means creating a contradictory statement.
Example : if it rains the ground is wet
R--> G
contrapositive: if the ground is not wet, it did not rain
But if I say that not all dogs are friendly, can we infer, or can we deny, that no dogs are friendly? It doesn't half to be the case but its not outside the realm of possibility here.
All A are B. To negate this claim we must deny the relationship, not the existence of a set. The way we think about this relationship is the quality of being "all". To deny the relationship, we would say some As are not B.
All jackfruit is splendid. J->S
Negated: J <-s->/S meaning some jackfruits are not splendid.
@PranjalChaudhary Not most, but some. We don't know what percentage of the A group that is not B. Most means more than half. It could just be one A that is not B, and some encompasses that range.
@ConqueringLSAT I think it can be if you understand what all means intuitively. We're just explicitly saying 'it's not the case that all dogs are friendly' so you don't fall into the trap of thinking not all means none.
@PruettJulia You're thinking of the contrapositive, which is just another way to express the same idea. The negation of a statement is what it means for the statement to be false.
@KoviFried Let me use an example, hopefully this clears something up.
statement: some LSAT studiers are full-time students.
If I say that all LSAT students are full-time students, it still means that some are (because some is at least one).
statement: all LSAT studiers are full-time students.
If I say this, it can't mean SOME are full-time students, because it would have to mean that 100% of studiers are full-time students, not just one. The upper boundary for all is 100%, whereas some is a bit more nebulous.
If a hot-dog cart had 10 hot-dogs and you said you wanted all of them and they only gave you five, you'd be pretty annoyed, right? But if you said you wanted some hot dogs and they gave you all ten, your request would still technically be satisfied, as some is more than one (and 10 is more than one, which happens to be all in this case).
I hope that helped! If that made it more confusing, I can clear it up.
@ArthurWhite /(D->F) looks like I'm going to say non dogs are friendly. I would say not all translates to D -s-> /F . It doesn't negate or contrapositive like conditionals. Short hand is translated different here.
There's a typo in this. In the 'lets review' section it says "in this instance, an 'all' relationship, you are trying deny that relationship." There is a 'to' missing- it should read "trying to deny that relationship."
Subscribe to unlock everything that 7Sage has to offer.
Hold on there, stranger! You need a free account for that.
We love that you want to get going. Just create a free account below—it only takes a minute—and then you can continue!
Hold on there, stranger! You need a free account for that.
We love that you came here to read all the amazing posts from our 300,000+ members. They all have accounts too! Just create a free account below—it only takes a minute—and then you’re free to discuss anything!
Hold on there, stranger! You need a free account for that.
We love that you want to give us feedback! Just create a free account below—it only takes a minute—and then you’re free to vote on this!
Hold on there, you need to slow down.
We love that you want post in our discussion forum! Just come back in a bit to post again!
Sorry, you need a subscription for that.
Happens all the time: now that you've had a taste of the lessons, you just can't stop -- and you don't have to! Click the button.
117 comments
are the sentences "its not the case that all dogs are friendly" and "some dogs are not friendly" logically equivalent?
@Luna Yes I believe so. "Its not the case that all dogs are friendly" denies that all dogs are friendly. This translates to "some dogs are not friendly".
To me it's confusing to include /(D -> F) because in algebra, you would distribute whatever is outside of the parenthesis and get /D -> /F. But in Lawgic, it really means to negate the necessary condition to create a condition that contradicts the original claim. And you can't contradict the original claim without having the non-negated sufficient condition. If you negate the sufficient condition, then you are talking about a different scenario than the one you're trying to negate. That's how I'm interpreting these lessons, let me know if I have anything wrong though
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
Was the F--->F to /(D-->F) turns into D <-S-> /F not taught anywhere before this???
wasnt it said earlier that "some" cannot have a contrapositive? or is negation different
@Tannercho06897 Contrapositive and negation are different!
Contrapositive is an equivalent way of expressing the relationship. We typically create a contrapositive of a normal conditional relationship (ex. cats are mammals) by flipping sufficient and necessary conditions, and negating each condition (ex. if not a mammal, then not a cat). These are two ways of expressing the same relationship.
Negation of an All statement/relationship ("it is not the case that all ...") is creating the opposite of the relationship. For example, the negation of "all cats are mammals" would be "some cats are not mammals". They express totally different relationships between the sets.
I think not all is logically equivalent to some are not.
To negate "All dogs are friendly," we'd be focusing on the word "all" and disproving that specific word by saying, "It is not true that all dogs are friendly." This negated statement logically states that, at the very least, there exists at least one dog that is not friendly; hence, the existence of at least one unfriendly dog means that it's impossible for every single dog in existence to be friendly.
Going back a few lessons, we also know that "some" entails a numerical baseline of at least one. Therefore, this negated statement could also be written as, "Some dogs are not friendly." That's why the negation of "All dogs are friendly," is "Some dogs are not friendly!"
Good luck in your studies! Trust your gut and intuition, and reward yourself for every little bit of progress you make!!
So the difference between the contrapositive and the negation of a claim is that contrapositive reverses and negates to be logically equal to original. Negation of a claim means creating a contradictory statement.
Example : if it rains the ground is wet
R--> G
contrapositive: if the ground is not wet, it did not rain
/G-->/R
negation: it rains but the ground is not wet
R and /G
is this correct reasoning?
But if I say that not all dogs are friendly, can we infer, or can we deny, that no dogs are friendly? It doesn't half to be the case but its not outside the realm of possibility here.
@Garrett_dom It's possible that no dogs are friendly. We can't infer that it's true, but it hasn't been ruled out.
Where would we use this idea of negating "all"? Like what question type on the lsat?
1. SOME DOGS ARE FRIENDLY
At least one dog is friendly
Not all dogs are not friendly
Negate: No dog is friendly
All dogs are not friendly.
2. ALL DOGS ARE FRIENDLY: 100% dogs->F
Negate: It’s not the case that all dogs are friendly.
Not all dogs are friendly: >100% dogs->F
Some dogs are not friendly
Arbys
@JDMurphy get this man some arbys
Do not confuse this with the creation of contrapositives which are logically equivalent to the original statement.
All A are B. To negate this claim we must deny the relationship, not the existence of a set. The way we think about this relationship is the quality of being "all". To deny the relationship, we would say some As are not B.
All jackfruit is splendid. J->S
Negated: J <-s->/S meaning some jackfruits are not splendid.
All pens are black
P -> B
Negated: Some pen's are not black
P <-S-> /B
To negate all relationships we are saying "It' not the case that all pens are black".
What this does not mean is all pens -> /black. THIS IS A TRAP. All this negated statement means is that "Some pens are not black"
Can this be applicable to "any" and "every"?
Original: All A are B
Negated: Most A are not B ?
@PranjalChaudhary Not most, but some. We don't know what percentage of the A group that is not B. Most means more than half. It could just be one A that is not B, and some encompasses that range.
I'm confused on why it can't be "Not all A are B" instead of bringing some in it
@ConqueringLSAT I think it can be if you understand what all means intuitively. We're just explicitly saying 'it's not the case that all dogs are friendly' so you don't fall into the trap of thinking not all means none.
I am going crazy. Why isn't the negation "if you are not friendly, then you must not be a dog" ?
@PruettJulia You're thinking of the contrapositive, which is just another way to express the same idea. The negation of a statement is what it means for the statement to be false.
How can not all mean some if some can mean all?
@KoviFried Let me use an example, hopefully this clears something up.
statement: some LSAT studiers are full-time students.
If I say that all LSAT students are full-time students, it still means that some are (because some is at least one).
statement: all LSAT studiers are full-time students.
If I say this, it can't mean SOME are full-time students, because it would have to mean that 100% of studiers are full-time students, not just one. The upper boundary for all is 100%, whereas some is a bit more nebulous.
If a hot-dog cart had 10 hot-dogs and you said you wanted all of them and they only gave you five, you'd be pretty annoyed, right? But if you said you wanted some hot dogs and they gave you all ten, your request would still technically be satisfied, as some is more than one (and 10 is more than one, which happens to be all in this case).
I hope that helped! If that made it more confusing, I can clear it up.
@kyorofan20 that helped a lot, thank you
So looking it from point of view of sets and subsets we notice that
Original: D → F
implies that Dogs is a subset of Friendly.
Whereas negated...
Negated: /(D → F)
The Dogs set is intersecting with the Friendly set? and some dogs are outside the Friendly set intersection.
https://miro.com/app/board/uXjVJL9zNgA=/?share_link_id=597707785172
Would that be correct? If so how can we get more out of thinking in sets?
@ArthurWhite /(D->F) looks like I'm going to say non dogs are friendly. I would say not all translates to D -s-> /F . It doesn't negate or contrapositive like conditionals. Short hand is translated different here.
Is there a drill set or specific questions to practice with quantifiers? This is one area where I struggle a lot.
There's a typo in this. In the 'lets review' section it says "in this instance, an 'all' relationship, you are trying deny that relationship." There is a 'to' missing- it should read "trying to deny that relationship."
For a nessscary assumptipn is this similar?