Not all As are Bs.All As are Cs.What can you properly infer?

Mehran November 5, 2018

Hi @Yiming-Jin, thanks for your post.

Yes. This statement "not B-some-A ==> C" can be further reduced to "not B-some-C" which is the same as "not all Bs are Cs" and "some Cs are not Bs."

Hope that helps to clarify. Please let us know if you have any additional questions.

Yiming-Jin November 5, 2018

Sorry, I'm still confused why the answer is B and not E then? if "not B-some-C," how is it "no Bs are Cs?"

Mehran November 11, 2018

Hi again, @Yiming-Jin.

The sentence "not all Cs are Bs" is the same as "some Cs are not Bs," which is the transitive derived from the following:
First premise: A-some-not B [which is how you diagram "Not all As are Bs," which is the same as saying "Some As are not Bs"]
"some" statements can be flipped, so you could also write not B-some-A

Second premise: All As are Cs
A ==> C
contrapositive: not C ==> not A

You can combine these two premises, as follows:
Take the flip of the first premise, and add it to the positive of the second premise:
not B-some-A ==>C

The transitive is not B-some-C
which can be flipped as C-some-not B

Which in plain English means "some Cs are not Bs" or "Not all Cs are Bs."

ekowa1c July 16, 2019

Hello I got the right answer but I wanted to make sure I was on the right track on how I achieved it. The transitive of not all A's are B's is not B~A's and the transitive of All A's are C's is not C's- Not A so then if it's not B->A -> C -> not B ?

Lucas July 4, 2020

Why does Not B- some-Cs have to be switched? I'm so confused why I got that wrong

Skylar July 5, 2020

@ekowa1c, happy to help!

"Not all As are Bs" is written as: A - some - not B
Some statements are reversible, so we can also say: not B - some - A

"All As are Cs" is written as: A -> C
We reverse and negate to find the contrapositive: not C -> not A

We can then combine the reverse of our first statement (not B - some - A) with our original statement (A -> C) to get: not B - some - A -> C
This simplifies to: not B - some - C
Some statements are reversible, so we can also say: C - some - not B
This is read as: "Not all Cs are Bs"

Does that make sense? Please let us know if you have any other questions!

Skylar July 5, 2020

@Lucas, happy to help!

Some statements are reversible, so once we get to the point where we have "not B - some - C," we should automatically write out its reverse, which is "C - some - not B." These two statements have the same meaning, so only one will be listed as an answer choice. Knowing this, we should be sure to write out both so as to be aware of all possible valid options and to avoid getting stuck. Moreover, I anticipated that the answer would be "C - some - not B" because this can be written out nicely as "not all Cs are Bs" or "some Cs are not Bs," which flows better than writing out "not B - some - C."

Does that make sense? Please let us know if you have any other questions!