@CharlizeMorrison The thing on the left of the arrow is always the sufficient, and the thing on the right is always necessary.
A --> B
A is sufficient for B
/B --> /A
Here, not having B is sufficient for not having A. So when looking at the contrapositive, "/B" is sufficient and "/A" is necessary. This is just another way of describing the relationship that "A" is sufficient for "B".
I'm not totally clear on your Lawgic/syntax here, but remember, you have to FLIP and NEGATE to do a contrapositive correctly. Using my dumb example about tall people hitting their heads on door frames, you CANT say:
(((((the following is incorrect btw to show what you CANT do given a statement)))))
/Hit head on door frame ---> /Tall
(or: Tall ---> Hit head on door frame)
"so therefore...."
Hit head on door frame ---> tall
it DOESNT work that way, that's confusing necessary with sufficient. And also not doing contrapositives correctly.
In my scenario, there are so many other ways you can hit your head on a doorframe, you could be wearing heels, it could be a short door, you could be wearing stilts. Being Tall is not a requirement. I was just saying, in my other comment, in that scenario, if you are tall, you WILL hit your head on door frames. Hitting your head is a requirement, IF you are tall. But if you hit your head, who's to say why that is happening, being Tall is not necessary.
But if somehow you are NOT hitting your head, by golly you are definitely NOT tall. That's the contrapositive. Because one of these variables was NECESSARY to happen if the other one happened.
If I am tall, then I will hit my head on door frames.
T ----> DF
If I do not hit my head on door frames, then I am not tall.
/DF -----> /T
Being Tall is Sufficient, as in it is enough, to then hit my head on door frames. If I am tall, it is necessary that I hit my head on door frames.
If I do not hit my head on door frames, it is Sufficient, as in it is enough to then say, I am not tall. Because it is necessary to NOT be tall if I am NOT hitting my head.
If one thing is necessary for a Sufficient thing, then not having that necessary thing means I cant possibly have the sufficient thing.
Contrapositive is basically saying, the Necessary item is NOT HAPPENING, so the Sufficient one cannot be true.
It took me embarrassingly long to understand this but I want to post it for anybody who is having trouble understanding the "Arrow question"-meaning what happens to the sufficient and necessary titles:
I need to preface that you need to start off with the right premise because i didn't. I didn't understand Conditions. Conditions are not this is sufficient and this is necessary-it's more like this condition causes the other one to happen or be true.
A condition: A situation or state that causes another to have to be true. That's what the LSAT means by conditional logic.
Not Mammal becomes sufficient condition because it's ENOUGH to guarantee Not Cat-BUT it is sufficent because you can be Not Cat in other ways like being a dog. It's a condition because Whenever not mammal happens not cat will happen-thats the condition.
Not Cat is necessary because we know that when Not Mammal is true then Not Cat MUST happen/Be true. That is why it's necessary. When not mammal happens not cat must happen to/has to follow.
If x then Y actually means Xis the trigger (sufficient) - Y is guaranteed (Necessary).
Not mammal -> Not cat ---- means : whenever the condition of not mammal happens, the condition of not cat must also happen
To me the left means subset and the right means superset so this was confusing because I was like wait so now mammals is the subset and cats is the superset?
@gracegairani I think the left-right subset-superset rule would only apply if the conditions weren't negated. When we take the contrapositive of /M = /C, mammals is still the superset. If mammals were a subset and cats were the superset this would mean that something that wasn't a mammal could still be a cat as there could be a type of cat in the cat superset that wasn't a mammal, but this isn't the case.
I am a little bit confused on one thing. When we negate both claims, the original necessary condition now becomes sufficient and the sufficient becomes necessary. Now what I don't understand is for this example, not being a cat is the new necessary condition so we are saying, 'not being a cat is necessary for not being a mammal' how does that make sense? it could be another animal like a dog for example.
Always read it left to right, and remember that this arrow: "→" always means "sufficient"
I'm gonna try to correct a couple parts of your comment.
"When we negate both claims, the original necessary condition now becomes sufficient..."
- Correction: When we negate both claims, the negation of the original necessary condition now becomes sufficient.
Try not to get hung up on the actual words and just follow the form. Nowhere are we saying that "not being a cat is necessary for not being a mammal"
- If cat, then mammal.
- If not mammal, then not cat.
Mammal is more restrictive than cat. If something is not a mammal, it definitely is not a cat. He explains this well in the second half of the video, using earth as an example of a non-mammal and therefore a non-cat.
So in contrapositives, if one switches the sufficient and necessary conditions, does the negated sufficient condition somehow logically become the necessary condition and vice versa?
Like in the cats and mammals example. We go:
C -> M to /M -> /C
Is /C the new sufficient condition? If one is not a mammal, then one is not a cat. So if one is not a cat, would that be necessary or sufficient to say one is not a mammal? Or do they retain their respective statuses as sufficient ent and necessary after they are switched and negated?
Realistically, I can answer my own question. It seems like if one not being a mammal is still necessary for one not being a cat, because if one was not a cat, they could still be a dog and be a mammal. Im just wondering if I'm missing something, or if this plays out differently in any other examples?
I would love an answer to the same question. If we keep the same conventions, the left side of the arrow is the sufficient condition, and the right side is the necessary one. This does not make sense to me for the contapositive statement. Not being a cat is sufficient but not necessary for not being
Okay actually ignore what I just said! I think the convention still stands with the left side of the arrow being the sufficient condition and the right side being the necessary condition.
Just like cats are always mammals in C --> M, not-mammals are always not-cats in /M --> /C. So, /M is sufficient in determining that an organism is /C.
Just like mammals can be cats (but aren't always), not-cats can be not-mammals (but aren't always). So, /C is necessary in determining that an organism is /M (in other words, determining that an organism in not a cat is necessary but not sufficient to determine that it is not a mammal).
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52 comments
If there is sun outside, then it must be a great day.
SO > GD
If it not a great day, then there is no sun outside
/GD > /SO
If one eats buffalo chicken cheese fries, then they must be my good friend Kileki.
BCCF>K
If one is NOT my good friend Kileki, then they must've NOT eaten buffalo chicken cheese fries.
/K>/BCCF
Valid contrapositive argument, false claim lol. We love forms!
If I am at my desk studying for the LSAT, my cats will be sitting on my desk.
D -> C
If my cats are not sitting on my desk, then I am not at my desk studying for the LSAT.
/C -> /D
In a contrapositive, does the "if" clause become the necessary condition and the "then" clause sufficient? since it's flipped around?
@CharlizeMorrison The thing on the left of the arrow is always the sufficient, and the thing on the right is always necessary.
A --> B
A is sufficient for B
/B --> /A
Here, not having B is sufficient for not having A. So when looking at the contrapositive, "/B" is sufficient and "/A" is necessary. This is just another way of describing the relationship that "A" is sufficient for "B".
I ran across an argument that stated using Simon’s argument /DF~/T DF~T What is going on with this argument?
@ShamellBenson
I'm not totally clear on your Lawgic/syntax here, but remember, you have to FLIP and NEGATE to do a contrapositive correctly. Using my dumb example about tall people hitting their heads on door frames, you CANT say:
(((((the following is incorrect btw to show what you CANT do given a statement)))))
/Hit head on door frame ---> /Tall
(or: Tall ---> Hit head on door frame)
"so therefore...."
Hit head on door frame ---> tall
it DOESNT work that way, that's confusing necessary with sufficient. And also not doing contrapositives correctly.
In my scenario, there are so many other ways you can hit your head on a doorframe, you could be wearing heels, it could be a short door, you could be wearing stilts. Being Tall is not a requirement. I was just saying, in my other comment, in that scenario, if you are tall, you WILL hit your head on door frames. Hitting your head is a requirement, IF you are tall. But if you hit your head, who's to say why that is happening, being Tall is not necessary.
But if somehow you are NOT hitting your head, by golly you are definitely NOT tall. That's the contrapositive. Because one of these variables was NECESSARY to happen if the other one happened.
If I am tall, then I will hit my head on door frames.
T ----> DF
If I do not hit my head on door frames, then I am not tall.
/DF -----> /T
Being Tall is Sufficient, as in it is enough, to then hit my head on door frames. If I am tall, it is necessary that I hit my head on door frames.
If I do not hit my head on door frames, it is Sufficient, as in it is enough to then say, I am not tall. Because it is necessary to NOT be tall if I am NOT hitting my head.
If one thing is necessary for a Sufficient thing, then not having that necessary thing means I cant possibly have the sufficient thing.
Contrapositive is basically saying, the Necessary item is NOT HAPPENING, so the Sufficient one cannot be true.
@SimonArmendariz Ser Dunk, is that you? lol
from A knight of the seven kingdoms show
if something is icy, then it's cold.
I --> C
/C --> /I
if something isn't cold, then it's not icy.
i think the way i made sense of it is: if it something large doesn't fit in this one box, it won't fit in this smaller box inside either.
If one is a king, then one is a man.
K --> M
Condition: A situation or state that causes another to have to be true. That's what the LSAT means by conditional logic.
If one is not a man, then one is not a king.
/M --> /K
If i am studying for the LSAT, I have a coffee
SL --> C
If I am not having a coffee, I am not studying for the LSAT
/C --> /SL
If I am studying for the LSAT, I'm in my office
SL->O
If I am not in my office, I am not studying for the LSAT
/O->/SL
It took me embarrassingly long to understand this but I want to post it for anybody who is having trouble understanding the "Arrow question"-meaning what happens to the sufficient and necessary titles:
I need to preface that you need to start off with the right premise because i didn't. I didn't understand Conditions. Conditions are not this is sufficient and this is necessary-it's more like this condition causes the other one to happen or be true.
A condition: A situation or state that causes another to have to be true. That's what the LSAT means by conditional logic.
Not Mammal becomes sufficient condition because it's ENOUGH to guarantee Not Cat-BUT it is sufficent because you can be Not Cat in other ways like being a dog. It's a condition because Whenever not mammal happens not cat will happen-thats the condition.
Not Cat is necessary because we know that when Not Mammal is true then Not Cat MUST happen/Be true. That is why it's necessary. When not mammal happens not cat must happen to/has to follow.
If x then Y actually means Xis the trigger (sufficient) - Y is guaranteed (Necessary).
Not mammal -> Not cat ---- means : whenever the condition of not mammal happens, the condition of not cat must also happen
If I am pale, then I am sick.
P -> S
If I am not Sick, then I am not pale.
/S -> /P
If I am drinking coffee, then I am awake.
If C--> A
If I am not awake, then I am not drinking coffee
If /A --> /C
To me the left means subset and the right means superset so this was confusing because I was like wait so now mammals is the subset and cats is the superset?
@gracegairani I think the left-right subset-superset rule would only apply if the conditions weren't negated. When we take the contrapositive of /M = /C, mammals is still the superset. If mammals were a subset and cats were the superset this would mean that something that wasn't a mammal could still be a cat as there could be a type of cat in the cat superset that wasn't a mammal, but this isn't the case.
Conditional Claim: If one drinks coffee in the morning, then one has energy for the day.
Contrapositive: If one does not have energy for the day, then one did not drink coffee in the morning.
-drinking coffee in the morning is sufficient for having energy for the day.
-having energy for the day is necessary for drinking coffee in the morning.
example...?
"if I leave for work after 8, I will be late.
I will not be late if I leave for work at or before 8."
If you're looking for the contrapositive to your argument. It's basically taking an "if then" statement and flipping it, adding "not".
In your first it's if I leave for work after 8, then I will be late.
8 --> L where 8 symbolizes "If I leave for work after 8" and the arrow symbolizes "then" and L symbolizes "I will be late"
/L --> /8 is the contrapositive.
If I'm not late, then I did not leave after 8. It could be true that you left at or before 8.
I'm confused by the use of 'left side of the arrow is sufficient condition' and 'right side of the arrow is a necessary condition.
I can have it memorized in my head: if A then B
If not B then not
but where does sufficient and necessary come into this?
It is sufficient to be a cat → to be necessary to be a mammal
It is not sufficient to be a mammal → to be necessary to be cat
Is that how this works?
hey! i think it is like: being a cat is sufficient for being a mammal; being a mammal is necessary for being a cat.
@ep2660 Bingo
Would this be logically equivalent:
If one is not a cat then they are a reptile
(Lawgic: /C —> R)
Contra:
If one is not a reptile then they are a cat.
(Lawgic: /R —> C)
Yes thats correct
I am a little bit confused on one thing. When we negate both claims, the original necessary condition now becomes sufficient and the sufficient becomes necessary. Now what I don't understand is for this example, not being a cat is the new necessary condition so we are saying, 'not being a cat is necessary for not being a mammal' how does that make sense? it could be another animal like a dog for example.
Always read it left to right, and remember that this arrow: "→" always means "sufficient"
I'm gonna try to correct a couple parts of your comment.
"When we negate both claims, the original necessary condition now becomes sufficient..."
- Correction: When we negate both claims, the negation of the original necessary condition now becomes sufficient.
Try not to get hung up on the actual words and just follow the form. Nowhere are we saying that "not being a cat is necessary for not being a mammal"
- If cat, then mammal.
- If not mammal, then not cat.
Mammal is more restrictive than cat. If something is not a mammal, it definitely is not a cat. He explains this well in the second half of the video, using earth as an example of a non-mammal and therefore a non-cat.
If one is pulling out their hair, then one is studying for the LSAT.
PTH —> LSAT
If one is not studying for the LSAT, then one is not pulling out their hair
/LSAT —> /PTH
so the contrapositive is "If not a mammal, then one is not a cat"
The "See you in the next lesson" caught me off guard lol
If I am in NYC, then I am in the USA
NYC --> USA
Being in the USA is necessary to be in NYC
Being in NYC is sufficient to be in the USA
/USA --> /NYC
Not being in the USA means I am not in NYC
So in contrapositives, if one switches the sufficient and necessary conditions, does the negated sufficient condition somehow logically become the necessary condition and vice versa?
Like in the cats and mammals example. We go:
C -> M to /M -> /C
Is /C the new sufficient condition? If one is not a mammal, then one is not a cat. So if one is not a cat, would that be necessary or sufficient to say one is not a mammal? Or do they retain their respective statuses as sufficient ent and necessary after they are switched and negated?
Realistically, I can answer my own question. It seems like if one not being a mammal is still necessary for one not being a cat, because if one was not a cat, they could still be a dog and be a mammal. Im just wondering if I'm missing something, or if this plays out differently in any other examples?
Any help is appreciated!
I would love an answer to the same question. If we keep the same conventions, the left side of the arrow is the sufficient condition, and the right side is the necessary one. This does not make sense to me for the contapositive statement. Not being a cat is sufficient but not necessary for not being
a mammal.
Okay actually ignore what I just said! I think the convention still stands with the left side of the arrow being the sufficient condition and the right side being the necessary condition.
Just like cats are always mammals in C --> M, not-mammals are always not-cats in /M --> /C. So, /M is sufficient in determining that an organism is /C.
Just like mammals can be cats (but aren't always), not-cats can be not-mammals (but aren't always). So, /C is necessary in determining that an organism is /M (in other words, determining that an organism in not a cat is necessary but not sufficient to determine that it is not a mammal).
To understand this better I would watch lessons on YouTube of DeMorgan's Law, its very helpful!
link: https://www.youtube.com/watch?v=93CxSLi89Ok