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?
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.
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?
With notes and re-reading to make sense of, these lessons take much longer than 2 or 3 minutes. This causes my study plan to be entirely out of reach. #feedback
Maybe this is me overthinking it, but I am trying to find ways that I might confuse your advice.
If we have a conditional statement and there is an indefinite binding word present, I believe that makes said statement ineligible for a contrapositive.
Ex: Some Presidents are Democrats
it logically follows as well that "Some Democrats are Presidents". But it does not follow that "if one is not a President, then one is not a Democrat" or "if one is not a Democrat, then one is not a President".
This is even the case with other indefinites, such as: most, a few, almost all, etc.
Here we have bilateral sufficiency, with no possible contrapositive.
This kind of makes me realize as well that the conditional argument from before is different in that it has unilateral necessity. If one is a cat, then one MUST be a mammal. Because of the binary strength of necessity, we can create a contrapositive. Indefinites, however, are gradient.
"flip it and reverse it" as missy elliott always said
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39 comments
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?
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."
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?
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)
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.
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!
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
I'm so glad I took logic classes in college lol great vid! I think this is really important to know for the test
what are contrapositives good for though?
this video helped me understand contrapositives !!
https://youtu.be/iZYk5PlzEDM?si=4pPdz5Kr_R4eI1fC
what is the difference between formally equivalent and logically equivalent?
With notes and re-reading to make sense of, these lessons take much longer than 2 or 3 minutes. This causes my study plan to be entirely out of reach. #feedback
Maybe this is me overthinking it, but I am trying to find ways that I might confuse your advice.
If we have a conditional statement and there is an indefinite binding word present, I believe that makes said statement ineligible for a contrapositive.
Ex: Some Presidents are Democrats
it logically follows as well that "Some Democrats are Presidents". But it does not follow that "if one is not a President, then one is not a Democrat" or "if one is not a Democrat, then one is not a President".
This is even the case with other indefinites, such as: most, a few, almost all, etc.
Here we have bilateral sufficiency, with no possible contrapositive.
This kind of makes me realize as well that the conditional argument from before is different in that it has unilateral necessity. If one is a cat, then one MUST be a mammal. Because of the binary strength of necessity, we can create a contrapositive. Indefinites, however, are gradient.
"flip it and reverse it" as missy elliott always said