Given the number of comments I read on people being confused between contrapositive and negation, merits an individual video lesson on it.

@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.

@ConqueringLSAT

Contrapositive:

If cat, then mammal

C->M

/M->/C

If not mammal, then not cat

This is the exact same sentence as: If cat, then mammal. Contrapositive is another way of saying the same things but with nots & nones.

Negation:

If cat, then not mammal

You destroy the subset superset relationship by taking out the subset out of the superset i.e. take out the cat from being a mammal.

Original statement: In order for it to be a cat, it must be a mammal.

Negated statement: In order for it to be a cat, it does not require to be a mammal. It is a cat and not a mammal.

C and /M

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

@krzhou_1 The difference between many and some is that it makes an existence claim, not a percentage claim unlike the difference between many vs. most. If X is many, that means that X exists (at least one of X exists)= X is some

@KindlyQualifiedChampionship

some means at least one can be all or not, but has to be at least one. So, we need to attack what we know must be 100% true for negation. To negate (opposite) this you need to at none.

Some pets are dogs.

Negation: No pets are dogs.

Contrapositives keeps the meaning same by slapping negation to both sides of the sufficient necessary clauses, but keeps the meaning same. Lets try it with this example:

Some pets, then dogs.

P←s→D

Contrapositive: /P←s→/D

Some non-pets are not dogs. Does this mean the same thing as some pets are dogs? NO, it completely changes its meaning.

I was confused at first but I think I know why some can include all:

Some students pass the LSAT only if they study everyday.

In order for the necessary condition to be true it has to include at least 1 student, but can also include all students, that is, even if we say all students have to study for passing LSAT does not fail the necessary. So it can include all!

Essentially we need to look for anything can cause the necessary condition to fail and exclude that interpretation.

Can someone help me with this pls?? #help

Aren't we suppose to assume that the disjunction is inclusive, that is, an "or" is always an inclusive-or if not stated otherwise? So, when we interpret a disjunction, don't we also have to consider the possibility of the inherent "and" unless we are considering an exclusive-or? For example:

If I am adopted, then either Bill or Mary is adopted (but could be both). I don’t think that this is inclusive or? If both are adopted, doesn't it become the same as interpreting it as an "and"? If that is the case how would the negation of this be any different from the negation of using an "and"?

@lilywong

Using set math:

Statement A: If Bill (B) and Mary (M) are adopted, then I (A) am adopted.

If B and M, then A

= B and M->A

= (B∩M)->A

= /A->/B ∪ /M

=/A->/B or /M or both not (inclusive or)

If I am not adopted, then either Bill or Mary is not adopted, or both are not adopted.

Statement B: If I (A) am adopted, then both Bill (B) and Mary (M) are adopted.

If I, then B and M

= A-> B and M

=A->(B∩M)

= /B ∪ /M or both not (inclusive or)-> /A

If either Bill or Mary is not adopted, or both are not adopted, then I am not adopted

You can use the same method to negate for sentences using "or" as the conjunction. As set theory OR (A or B, or both AB) is an inclusive or like the LSAT. Exclusive or is either A or B, no union represented by XOR.

I literally read it as some "trade support foreign policies" instead of seeing support as the main verb lol

@AlvinB Exactly! that's what's been on my mind throughout these lessons. I'm glad that someone else is seeing that too.

We wouldn't have time in the exam to do all this logical analysis, so we need to find which clause sounds like an absolute fact statement (conclusion) and which clause supports that clause (premise). For example,

It will not rain unless Zeus is mad.

=If not Zeus is mad, it will not rain.

=If it rains, then Zuest must be mad.

/Z->/R

R->Z

It is being claimed that for it to rain something specific/absolute happens, which is, Zeus needs to be mad i.e. it is required that Zeus is mad.

When you put it through the "why, because" test for premise-conclusion, you'll see that:

Zeus must be mad, why?-> Conclusion(receiving support)/Necessary

Because it will rain.->Premise (throwing support)/Sufficient

@SusanLeifker Seeing it like this helped me:

Birds that cannot flap their wings cannot fly.

If birds cannot flap their wings, then birds cannot fly.

/FW->/F

F->FW

Why do we have to flip to negate/or remove a negate? because that's how math is, if you want to keep the direction of the arrow consistent.

-5<-4

or, 4<5

@amygao It does. I found that the more I do these the less I need to rely on explicitly writing down these logical structures.

My first question is what has to happen for something to happen? That becomes my necessary condition.

@LiviaLSAT "but only well-trained Pokémon are capable of evolving"

That means if you have evolved you must have been well trained. For you to evolve it is necessary for you to have been well-trained. You having evolved is sufficient to say that you must have been well-trained.

Evolve (sufficient)->Well-trained (necessary)

@malop91 I think when you introduce "can" its easier to imagine a subset superset relationship. Lift (superset) and pure of heart (subset). You can/must lift if you have a pure heart. Similarly to what Kevin said, you can/must be wealthy, if you are an athlete. That's how I see it

@SarahSmile I understand you now! tnx

@GrahamSC You can look into this with two different lenses:

Lens 1: when you don't have time in the exam to do all this logical analysis, you need to find which clause sounds like a absolute fact statement clause (almost like a support/premise) and which clause is a claim dependent on that clause (conclusion)

for this sentence, it is being claimed that it will not rain unless something specific/absolute happens, which is, Zeus being mad.

It will not rain unless Zeus is mad.

=In order for it to rain Zeus must (necessity indicator) be mad

=It is necessary for Zeus to be mad in order for it to rain.

R->ZM.

Again,

Juxtapose it into the cat-mammal example:

It is necessary for it to be a mammal for it to be a cat.

C->M

Lens 2: Substitute it with the If not

It will not rain unless Zeus is mad.

=It will not rain if not Zeus is mad

=If Zeus is not mad, then it will not rain

/ZM->/R

R->ZM

@TaylarGottsegen Yes Group 1 sufficient indicators are:

a - all, any (+ as long as—rare indicator, but still used)

e - every

i - if

o - the only

double u (W) - when/whenever, where

@LamontNarcisse It goes like,

If tree, then perennial plant with elongated stem.

Think of it in a subset-superset relationship: Trees are a class (subset) of perennial plant with elongated stem (superset). There can be other types of perennial plant with elongated stem, but trees is just one example in that class.

If tree, then perennial plant with elongated stem.

Sufficient ->Necessary

If it is a tree->then it must be a perennial plant with elongated stem. In order for it to be a tree, it must be a perennial plant with elongated stem.

It being a tree is suffiecient to say that it is a perennial plant with elongated stem.

***Important note: If the sentence was "Trees are perennial plants with an elongated stem" it will also have the same visual translation in subset-superset relationship.

@danielamordonez Absolutely agree. There are LR questions which would have chain of causation (i.e. negations and contrapositives) and would conveniently skip a chain which we would later need to identify as a necessary assumption.

@SarahSmile That is a good way of looking at it when there is a subset and superset relationship. However, it becomes a problem in a sufficiency and necessity causal relationship. For example:

A brain death is going to get you killed (body will stop) surely i.e. is sufficient for you to get killed. But that is not necessary for you to get killed. You can get killed in all sorts of ways kidney failure, cancer etc. Death does not require brain death.

This is an example of sufficient but not necessary. That is, sufficiency is not guaranteeing neccessity because maybe there are no necessary ways to get killed.

@yanasok398 there have been questions on the previous LSATs that test on these concepts along with percentages. A simple format for example:

1. 60% of law students take Constitutional Law.

2. 40% of law students participate in Moot court.

3. 25% of law students both take Constitutional Law and participate in Moot court.

∣C∪M∣=∣C∣+∣M∣−∣C∩M∣ =60+40−25=75%

A question might test what we can infer from this, that is 75% of the student only does exclusively one or the other.

The questions get really tricky when they start introducing abstract numeral words like all, more, most, some, instead of a solid number. Like for example:

1. Most (more than 50% -> almost all) law students take Constitutional Law.

2. Some (at least one person) law students participate in moot court.

3. All (M subset of C ) law students who participate in moot court also take Constitutional Law.

And now you are trying to draw Venn Diagrams and inferences based on these sets of data. For example:

Inference 1: Some students who take Constitutional Law participate in moot court.

Inference 2: Most law students do not participate in moot court.

@prestonbigley759 Such a good explanation, thank you!

@gracegabriellejimenez810 I had thought the same too. So confusing

The reason why Tigers argument is flawed:

1. the "suitability" is not clearly defined in the conclusion. It can very well be subjective i.e. what is suitable to me might not be suitable to you. "Suitable" here is a vague term, pointing to nothing definite.

2. "Aggressive and can cause serious injuries to people" is just a fact and does not support suitability and unsuitability in the conclusion. For instance, I might have a cage at home or I can bring a timid breed of tigers that only eats things that I give.

3. What can make the argument fool proof would be the necessary assumption that ties these to missing links: "Aggressive and can cause serious injuries to people" is not suitable. Then the argument will be fool proof.