Q19 - At a gathering at which bankers

[nanagyanewa]

Could someone please help me understand this question? Here's what I got from the question:
A-> B
-B->-A
L->-B->-A
Isn't that what answer E says? I could not differentiate between B and C. any help will be greatly appreciated

[jm12345]

i got it as.

B -> A
L -> ~B
-----
B -> ~L

so you get

B -> A
and
B -> ~L

so some athletes (A) are not lawyers (~L)

[ManhattanPrepLSAT2]

Though there are conditional statements in this very tough q, they are not the only things being tested, and focusing on just the conditionals can get you in some trouble here.

Having said that, of course it's important to understand the conditional relationships correctly, and I'm happy to walk through the diagramming --

There are two conditional statements in this argument:

"All of the bankers are athletes."

"None of the lawyers are bankers."

One process you can use to figure out which way the arrow should go is to think about what conditional relationship MUST BE TRUE.

Let's take the first statement:

"All of the bankers are athletes."

We know our elements are "banker" and "athlete."

Based on this statement, if someone is a banker, MUST IT BE TRUE that that person is an athlete? Yes, and so we know

B --> A is a valid condition. (We also automatically know that its contrapositive, - A -> - B, is valid as well).

Based on this statement, is someone is an athlete, MUST IT BE TRUE that that person is a banker? No. Someone could be lawyer who is an athlete, and that wouldn't violate the constraint.

Therefore, we know A --> B is NOT a valid condition.

Now, let's look at the second statement: "None of the lawyers are bankers."

What do we know MUST BE TRUE if someone is a lawyer? That they can't be a banker:

L --> - B.

What do we know MUST BE TRUE if someone is a banker? They can't be a lawyer."

B --> - L.

In this case, the must be true test happened to yield the contrapositive, and that's fine. Again, the key is to understand both the orientation of the arrow, and the +/- of the element, correctly, and if a process like the one mentioned above helps you do that, it's worth investing the time.

So, now we've got 4 conditional relationships:

B - > A
- A - > - B
B -> - L
L -> - B

When I solve these problems, I do not predict an answer -- typically there are just too many possibilities -- instead, I'll just look for answers I know I can't prove and eliminate them.

I can't prove (A). Note that (A) means A -> B, and we don't have that relationship anywhere.

I can't prove (B). This is not a pure conditional statement, so I have to think about it a bit differently -- could it be possible that all of the lawyers are athletes? Don't see why not -- we can eliminate (B).

(C) is also a "some" statement, and needs to considered more carefully. Must it be true that some athletes are not bankers?

Yes it does.

Since all bankers are athletes, it holds that at least some athletes are bankers. We know all bankers are not lawyers, so at least some athletes are not lawyers.

Another way you could think about (C) -- could it be possible that all of the athletes are lawyers? No -- it's pretty easy to see that this is not possible because some bankers are athletes.

I know for sure (D) is wrong -- I know all bankers are not lawyers.

(E) is another conditional -- L -> - A, and we can't prove that using what we've been given.

Tough q! Hope that was helpful!

[tlahood]

Is this how this problem could be approached?

Since in the information, we are given in the information that at the gathering, there are bankers, athletes, and lawyers.

Meaning there is at least one of each.

Since all bankers are athletes, and we know for sure that there is a banker present, we can assume that there is at least one athlete who is also banker (not meaning that all athletes are bankers, which would be a mistaken reversal if applied universally), since for a banker to be at this meeting (which there is) he MUST be an athlete.

From this we follow the contrapositive that all bankers are not lawyers, that there is at least one athlete who is a banker, who is not a lawyer (with "at least one" being equivalent to "some"

Is this basically what you were saying?

[LSAT-Chang]

Hey Mike,
I have a quick question on this one.
So we have:

All B --> All A
All B --> All NOT L

since we have two all statements, we can deduce a "some" statement, right? So some As NOT Ls -- but can it not go the other way around? Like some NOT Ls NOT As (basically going from bottom up if we are to look at the 2 conditional statements above). Does it always go top-down?

A --> B
A --> C

B some C

NOT

C some B????

Would (B) have been correct if it said: "Some who are NOT lawyers are athletes" since that is the same thing as "Some who are athletes are NOT lawyers"??

[timmydoeslsat]

Yes, that version would be correct as it is simply switching the entities in the some statement like this:

A some B

B some A

Same statement logically.

I would diagram this stimulus like this:

We know we have B, A, L.

B ---> A
B ---> ~L

I decided upon diagramming this stimulus that it would be a great idea to line up the same sufficient condition to where I can see what is going on here.

We know that when we have a B, we have an A and ~L.

So I would go to the answer choices at this point.

A) Does not appear to be a must be true at all.

B) I would test this by doing: All of the L's are A's to see if I can show that this can still work, essentially disproving this answer choice.

B ---> A
B ---> ~L

NEW PREMISE: L---> A

I don't see anything contradictory there.

C) A some ~L

Hmm, let me see if I can disprove that. What if all A's were L's.


B ---> A
B ---> ~L

NEW PREMISE: A---> L


CONNECT THE PREMISES: B ---> A ---> L

That would not work because it leads to a logical contradiction. This would have B leading to L and ~L at the same time.

I then know that it must be the case that some A's are ~L.

[ManhattanPrepLSAT1]

We are give the following claims

B ---> A
L ---> ~B

B - banker, A - athlete, L - lawyer

What happens when you have two conditionals that have the same sufficient condition?

A ---> B
A ---> C

Well we can infer that some B's are C's.

In this case, if we take the contrapositive of the second statement we have just that

B ---> A
B ---> ~L

So we can infer that some athletes are not lawyers - best expressed in answer choice (C).

Timmy is exactly right when he says that it's easiest to see when you have the similar terms on the left of both of the conditional statements. I've been in heated debates with instructors from other companies who couldn't see the inference when the similar term was lined up on the right side of the conditionals as could have easily been the case here if you took the contrapositive of the first statement and ended up with

~A ---> ~B
L ---> ~B

But the same inference would still be implied that some athletes are not lawyers.

The wrong answer choices simply manipulate known statements or inferences into claims that are not supported.

(A) is a reversal of the first statement
(B) gets the signs wrong on the inference which might have been tempting if the similar terms were lined up on the right of the conditionals.
(D) contradicts the second claim
(E) states the relationship between lawyers and athletes more strongly than can be inferred.

Nice work Timmy and So!

[iridium77]

B -> A
L -> ~B
-----
B -> ~L

I guess I just don't see how C) is the only possible correct answer...

If I connect-up my chain, I see that:
B -> A -> ~L

...and the contrapositive of the last half results in:
L -> ~A

isn't that a paraphrase of answer E) ?
None of the lawyers are athletes ?

it seems to prove B) as well:
Some of the lawyers are not athletes...

I'm trying to learn to solve these strictly using logic chains, so any help would be appreciated...

[iridium77]

B -> A
L -> ~B
-----
B -> ~L

I guess I just don't see how C) is the only possible correct answer...

If I connect-up my chain, I see that:
B -> A -> ~L

...and the contrapositive of the last half results in:
L -> ~A

isn't that a paraphrase of answer E) ?
None of the lawyers are athletes ?

it seems to prove B) as well:
Some of the lawyers are not athletes...

I'm trying to learn to solve these strictly using logic chains, so any help would be appreciated...

[sumukh09]

If I connect-up my chain, I see that:
B -> A -> ~L

This is not a valid linkage.

You have

1. Bankers ---> Athletes
2. Lawyers ---> ~Bankers

There's no way to link these two statements to derive a chain.

In order to make a linkage we have to have the same necessary condition in one of them and the same sufficient condition in another.

So if we had A--->B an B---> C we could infer A ---> C since the necessary condition in the former conditional is the same as the sufficient condition in the latter conditional. However we don't have such a configuration in this problem to be able to derive a chain.

[iridium77]

This is not a valid linkage.

You have

1. Bankers ---> Athletes
2. Lawyers ---> ~Bankers

So what is the contrapositive of #2 ?

[sumukh09]

This is not a valid linkage.

You have

1. Bankers ---> Athletes
2. Lawyers ---> ~Bankers

So what is the contrapositive of #2 ?

Bankers ---> ~ Lawyers

but what you want to do is B --> A ---> ~L but this can't be done.

Just because B is sufficient for A and ~L doesn't mean you can make a valid linkage.

The only way to make a linkage is if you had A ---> B
and B ---> C

to yield A--->B--->C

but in this case you have

B-->A
B-->~L

[iridium77]

Thanks, I understand it now...

[patrice.antoine]

What if we have two similar necessary statements?

In this case:

Not A --> Not B
L --> Not B


Can't we make an inference from this?

[timsportschuetz]

I have spent way too much time on this particular question! However, I am happy to say that I have a DEFINITIVE method of solving this particular conditional logic question:

Premises:
Bankers --> Athletes
Lawyers --> [NOT] Bankers

Step 1: Rearrange the premises in order to have the only similar term (Bankers) on the same side of the conditional logic diagram. It is very important to ensure that the similar term (Bankers) is on the LEFT and/or SUFFICIENT side of the conditional diagrams. If you choose to place the similar term on the RIGHT and/or NECESSARY side of the conditional diagrams, you are UNABLE to make any inferences (I need to qualify the previous statement - you can make inferences, albeit with MUCH MORE and unnecessary effort):

Bankers --> Athletes
Bankers --> [NOT] Lawyers

Step 2: Connect the above rearrangements in a slightly different way. Keep in mind that this method is fully valid and does not break any of the conditional logic rules. It simply visualizes the arrows in a different direction:

[NOT] Lawyers <-- Bankers --> Athletes

From this logic chain, you can ALWAYS infer the following (in formulaic terms):

SOME [NOT] Lawyers are Athletes
-AND-
SOME Athletes are [NOT] Lawyers

This holds true for all scenarios with the same structure - IE:

Premises:
B --> C
B --> A

Therefore:
A <-- B --> C

Thus, you can infer the following:
SOME A's are C's
-AND-
SOME C's are A's

For the people that would like to understand WHY you can make these inferences, please review the following rule: Any conditional statement that has an ALL and/or MOST relationship can be reversed (ie: "go against the direction of the arrow"). However, reversing the direction of the conditional logic arrow REQUIRES the addition of a "SOME" relationship. For example:
A --> B ("All A's are B's")

Inference:
SOME B's are A's

If you have: MOST A --> B, you can infer: Some B's are A's. (This works with both, "ALL" and "MOST" Conditional Logic Statements).

Please let me know if this helped...

[logicfiend]

For anyone still confused about B: I didn't get what I was doing wrong (and the explanations above) until I visually wrote out what this problem is saying.

All bankers are athletes

A --> B

None of the lawyers are bankers.

L --> ~B

So I was linking up as:

~A --> ~B
L --> ~B

And trying to find a relationship between ~A and L.

Visually, this looks something like this:

~A~B
~A~B
~A ~B
L ~B
L~B

All ~A are ~B and all L are ~B, but there is no overlap between ~A and L. Like all the explanations said before, there is no inference that can be drawn. Therefore, it's entirely possible for all lawyers to be athletes.

Instead, this linkage clearly demonstrates why (C) is right:

B --> A
B --> ~L (as the contrapositive)

So visually, there is an overlap between A and ~L, because they share a common sufficient trigger.

B A ~L
B A ~L
B A ~L

All Bs are As and we also know, all Bs are ~A. There must be some As that are NOT Ls.

[Aquamarine]

I still don't get why E is wrong.
As far as I remember about conditional reasoning, if:

1. A-> B
2. C-> ~A

I can combine 1 and 2 (~B-> ~A: Contrapositive #1 and C-> ~A) and turn out C-> ~B.

So the diagram about the stimulus I thought was:

1: B-> A
2: L-> ~B

So contrapositive #1: ~A-> ~B
And I combined 1+2 and turned out L-> ~A, so that's why I chose E, but the answer is C.

So I have no idea why E is wrong. Am I missing something?
And why is C an answer?

Please someone explain me.
Thanks!

[smsotolongo]

Timmy is exactly right when he says that it's easiest to see when you have the similar terms on the left of both of the conditional statements. I've been in heated debates with instructors from other companies who couldn't see the inference when the similar term was lined up on the right side of the conditionals as could have easily been the case here if you took the contrapositive of the first statement and ended up with

~A ---> ~B
L ---> ~B

But the same inference would still be implied that some athletes are not lawyers.

In this case, why do I think that B is the right answer then? What am i missing?

[meiy169]

Hey Mike,
I have a quick question on this one.
So we have:

All B --> All A
All B --> All NOT L

since we have two all statements, we can deduce a "some" statement, right? So some As NOT Ls -- but can it not go the other way around? Like some NOT Ls NOT As (basically going from bottom up if we are to look at the 2 conditional statements above). Does it always go top-down?

A --> B
A --> C

B some C

NOT

C some B????

Would (B) have been correct if it said: "Some who are NOT lawyers are athletes" since that is the same thing as "Some who are athletes are NOT lawyers"??

Yes Chang, I have the same question with you


A --> B
A --> C

B some C

NOT

C some B????