#19 - At a gathering at which bankers, athletes, and lawyers

[Francis O'Rourke]

Hi Khodi,

because it's essentially saying B is A

This might be a source of your difficulties. There is a difference between saying B is A, and All B's are A's.

The interpretation of the former statement, B is A, is rather strong. For example in the statement "Blue is my favorite color" the two ideas are identical: Blue is my favorite color and my favorite color is blue. The word "is" functions as biconditional in this example.

Compare that statement to the following: "All blue paintings are beautiful." From this rule, whenever you find a painting that is blue, you know that it is beautiful. However, we don't know anything about a painting if we know that it is beautiful; it is possible that red paintings are also beautiful.

In this way, given the rule All B's are A's, it is possible for some A's to not be B's. You essentially committed the mistaken negation by assuming B is A.

Let me know if this helps!

[Khodi7531]

Hey Francis I know what you mean but that isn't where my struggles are

I was just typing fast and just got lazy to get into a if B then A statement since i'm trying to address the larger point


Which is that in this diagram for the question, there is a some statement. So if anyone can, please show me EVER SINGLE POSSIBLE WAY that the statement can be read IN A SOME way. Considering the negated version, contrapositives if you can, and everything.

[Jonathan Evans]

Hey Khodi,

Good questions. Let's break this down and evaluate what we know and don't know:

1. There are A, B, and L (There are Athletes, Bankers, and Lawyers).
2. B A (All Bankers are Athletes; if you're a Banker, then you are also an Athlete)
3. L B (No Lawyers are Bankers; if you're a Lawyer, then you are not a Banker)

Let's look at the contrapositives from statements (2) and (3) above:

• A B (All non-Athletes are not Bankers; if you're not an Athlete, then you are not a Banker)
• B L (No Bankers are Lawyers; if you're a Banker, then you are not a Lawyer)

"Some" statements follow from (2) and (3):

• A B (since all Bankers are Athletes, and since there are Bankers present, we may conclude that some Athletes are Bankers)
• B L (some non-Bankers are Lawyers; this doesn't tell us much, since we already know that there are Lawyers!)

We therefore may infer:

• A B L (Some of the Athletes are not Lawyers)

However, we may not infer that there are Lawyers who are not Athletes. "Some" is indeed bi-directional, but there's no connection between L and A there. All we have is a connection between A and L. We may conclude that the Athletes who are Bankers are not Lawyers. Therefore, we may conclude that there exists an Athlete who is not a Lawyer.

However, there could be other Athletes present! All the Lawyers, who are not Bankers, could possibly be among these Athletes. We have no information either way. Therefore, answer choice (B) is not a valid inference. It could be true. It might not be true.

I hope this helps!

[Khodi7531]

Finally!!!!


Ok there you go, now I see what you're saying and I see what my "issue was".

All B's are A's never got me to think that "some A's are B's". This I didn't even know was something people considering - although I see it as being redundant since some A's being B's is essentially stated in All B's are A.


Also, question about your diagram of the biconditional between A and B and the negated L. So for some statements, you're saying that you CAN NOT say some L is not A ? One of my questions was about all the possible ways to read a some statement. Can you read a some statement in a contrapositive?

[Jonathan Evans]

Hi, Khodi,

"Some" statements are not biconditionals. Neither condition is necessary nor sufficient for the other. All you know is that "certain A are B." You do not know which A are B. In other words, there could be lots of athletes who are not bankers.

Basically, here's a quick rule of thumb: you cannot follow a logic chain through a some statement.

Let me illustrate with a different example:

• Some vegetables are actually fruits. Some vegetables are leafy greens.
  
  F V LG

We cannot infer that some fruits are leafy greens or vice versa.

There is one exception to this situation, it is when you're dealing with "most":

• Most residents of Smallville buy groceries on Tuesday. Most residents of Smallville eat at a restaurant on Tuesday.
  
  SV G
  SV ER

From these two statements, we may infer that there is at least one resident of Smallville who both buys groceries on Tuesday and eats at a restaurant on Tuesday.

There may be other more esoteric formal constructions, but this is roughly as deep as it gets on the LSAT.

I hope this helps clear this up for you.

[Khodi7531]

Yeah I know that, I was just saying biconditional to say that some statements can be read in both ways - wasn't saying that one is necessary and one is sufficient.


And yeah I understood those rules of some statements already but my question is; Can a some statement be controposed? If some A is not L .... can I say some L is not A?

[Jonathan Evans]

No, there is no contrapositive of a "some" statement. Valid contrapositives only apply to to conditional statements.

[Khodi7531]

Awesome, thanks.

[phoenixflame]

I tried to solve this problem using formal logic, but I got the question wrong. I wanted to show you how I got the answer, and I hope you may explain to me how I got the question wrong.

All of the bankers are athletes:

A (bankers) B (athletes)

This is the contrapositive:

B (athletes) A (Bankers)

None of the lawyers are bankers:

L (lawyers) B (bankers)

In order to turn a "some" statement into an "all" statement, you must negate the necessary condition:

L (lawyers) B (Bankers)

The following is the contrapositive of the "all" statement

B L

To use formal logic on both the "all" statements:

B A
B L

A L

[James Finch]

Hi Phoenix,

Honestly, you lost me when you used "A" for bankers and "B" for athletes. Here's how I broke down the conditional reasoning:

Bankers (B) Athlete (A), so A B

Lawyers (L) B , so B:arrow: L

Meaning B A + L

What does this mean logically? That all bankers are athletes and not lawyers, so we know that at least some athletes must not be lawyers (those that are bankers).

Answer choice (C) says exactly this, making it the correct answer.

Hope this helps!