I find these not too challenging, but this one was really tricky and made me laugh after the fact.

I mapped it correctly, and my deduction is that Walnut Street and Main Street intersect if Pat returned not more than 10 to WS and got the discount - a bit of an excessive extrapolation on my part in retrospect, but I actually expected this to be an answer choice.

I chose B and totally glossed over the possibility that Pat isn't a member of the club as implied by the rules

Well done LSAC, I'm actually impressed with this one

They definitely exist

Yeah I think this is a poor and misleading analogy by the creators of the course because of the usage of the set diagrams in both this lesson and the conditional logic lesson

Visually you are right, but remember in the conditional logic section that the subset → superset diagramming with the subsumed circles doesn't apply very easily to a lot of the more abstract concepts we will encounter, but rather I think the teacher used it to illustrate the concepts of sufficiency vs. necessity to help build our intuitive and fundamental understanding of these concepts

He's using it here to talk about what inferences we can draw

Consider the following chain of conditional relationships:

All roofers must bring a hammer to work (R → H)

All hammers must have a blue handle (H → bh)

Only those who have hammers with blue handles bring lunch to work (bh → L)

So we have the chain:

R → H

H → bh

bh → L

Contrapositives:

/H → /R

/bh → /H

/L → /bh

These chain together due to the directionality of the arrows which allow us to make valid inferences/conclusions:

R→H→bh→L

contrapositive: /L→/bh→/H→/R

So I can infer that if someone has a hammer with a blue handle, then they bring lunch to work. But I can't infer that if someone brings lunch to work, that they bring a blue-handled hammer.

Similarly with the quantifiers, if I say "all" cats are pets, that also means "most" cats are pets, and "some" cats are pets, because all triggers most and some. They aren't exclusionary.

Also, "most" triggers "some" because some falls under "most" and isn't exclusionary

"Some" doesn't trigger most or all because their minimum thresholds lie ABOVE the maximum range that some encompasses

So basically it has to do with directionality and which inferences can be properly drawn, which is why he invoked the subset → superset relationship

I'd just keep it as some since he said it's a useful falsehood but given that the fundamentals are the same then use many if it makes you feel better

Many birds migrate south each winter:

So two intersecting sets: BIRDS and THINGS THAT MIGRATE SOUTH EACH WINTER

It is B ←s→ MS for some

And B ←many→ MS for many

The way I think of this is that "many" isn't necessarily about proportion but rather raw quantity

Many birds is still many of the things that migrate south each winter, thus the logical equivalence with "some" as a quantifier

It is neither unnecessary nor disparaging because when you're actually taking the questions, especially under the gun of time constraints during the real exam or timed PTs, then it's pretty easy to glaze over and accidentally view majority/most and overwhelming majority as being the same, which can actually be a major trap.

Imagine you're given a question with a set of quantifier premises and you set up your lawgic diagrams correctly and then you find out that "most" is a valid premise

But you select an answer that says "overwhelming majority" as the valid premise

But this is extrapolation from most and therefore not valid!

So it means that don't imply that some "not A" are "not B"

Because these statements don't tell us anything alone aside from the elements they explicitly mention

I'm pretty sure it would have to be stated explicitly otherwise it is unidirectional.

All cats are pets (C → P) does not mean all pets are cats (P → C)

But if it says "all cats are pets AND all pets are cats" then it would be bidirectional, but I have yet to see any prompt on any LSAT question where this sort of relationship exists...

I agree with Hannah and I'm sure we will learn in a later lesson but I will add that I am not sure the concept of a contrapositive translates to quantifiers, because while there is overlap, it is hard for these words to have specific logical equivalencies due to the ranges they include.

Sufficient condition and necessary condition is different because the sufficient condition coming about is enough to guarantee the necessary condition, and vice versa, the necessary condition being absent is enough to guarantee the sufficient condition not coming about.

If I go to school, then I will bring my books (S → B)

If I don't bring my books, then I don't go to school (/B → /S)

Logically equivalent statements

I don't think so because while most is included under some, i.e., given the range of most fits within some, they can't be contrapositives, because it doesn't have the same truth.

Contrapositives need to be logically equivalent statements

Since some can include anything from at least 1 to up to half (and further to all), but most can include anything from at least more than a half to all, therefore, it isn't logically equivalent

From what I can tell usually these will be the premises on which a conclusion is drawn, or the intermediate conclusions.

So for example:

Some cats are pets (C ←s→ P)

Most pets are kept in homes (P‑m→H)

Therefore, most things kept in homes are cats (H‑m→C)

Based on the premises above, the conclusion (therefore, most things kept in homes are cats) is demonstrably false due to the conditions laid out (i.e., C←s→P and P‑m→H).

A valid conclusion drawn from these premises would be some cats are kept in homes (I think, could be wrong!) (C←s→H)