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# "no A are B" equals "all A are ~B"?

Core Member
in General 177 karma

Help! I thought "no A are B" equals "~B are A"

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• #### Is {A some /B} equal to {/A some B} ?I thought it was, but I'm second-guessing myself. Can you help me better understand this concept?

• Alum Member
1032 karma

I'm not sure I understand how you're using the "~", given that you use "no" as the negative in the first statement. But, the contrapositive of "no A are B" (A -->/B) is "no B are A" (B-->/A)

Hope that helps!

• Core Member
177 karma

Hey Yes that's what I thought too. But I was just browsing through Manhattan Prep for PT 9 stuff, the mentor there wrote something like this:

When we convert that "No A are B" into conditional logic, it is the same as saying "All A are ~B".

And that was alarming, because I always thought NoA=B equals NoB=A.

Does someone know what this mentor was talking about?

• Alum Member
2798 karma

@lsatyayy said:
Hey Yes that's what I thought too. But I was just browsing through Manhattan Prep for PT 9 stuff, the mentor there wrote something like this:

When we convert that "No A are B" into conditional logic, it is the same as saying "All A are ~B".

And that was alarming, because I always thought NoA=B equals NoB=A.

Does someone know what this mentor was talking about?

No As are B means that an A can never be a B. In other words No thing that is an A is ever a B. It would look like this: A------->/B or B--------->/A. So if we know that something is a B we know that it cannot be an A since no As are ever Bs.

Interpreting it as /B--------> A would mean that everything in the world that is not a B must be an A or conversely everything in the world that is not an A is a B. This is much different than No As are Bs.

A real world example would be if I told you that No humans can fly. Correctly translated it would be Human-----> /Fly. Contra positively we know that if we come across something that can fly than it cannot be a human. This is because being able to fly denies a necessary condition for being a human.

The erroneous interpretation of No humans can fly would be /Fly-------> Humans. This would mean that everything that does not fly is a human and everything that is not a human can fly. We know that this is not true and different than No humans can fly.

The lessons in the core curriculum starting here (https://7sage.com/lesson/logic/) are really useful for practicing this stuff and making it second nature. Hopefully this helps!

• Alum Member
268 karma

What about Not all A's are B's, does that translate into A some B?

• Alum Member
2227 karma

Try simplifying the language the way you would during the test. No As are Bs would translate simply into "If A, then not B." Contrapositive is "If B, then not A." 7sage lawgic uses a / for negation.

• Alum Member
2227 karma

@Harvey_l said:
What about Not all A's are B's, does that translate into A some B?

It means that some As are not Bs. In lawgic, A some /B
"Some" means at least 1, but could mean all. All implies most, and most implies some. At a minimum, at least 1 A is not a B.

• Core Member
177 karma

Thanks Lucas. That surely helped. Thanks a ton!

• Core Member
edited January 2019 177 karma

Okay just to really finish my understanding, so when there is "no" in front of A, it's the B that will be negated while it is A that will be negated when there is "not' in front of it?

no A equals B----> A->/B
not A equals B----> /A->B

• Alum Member
2227 karma

@lsatyayy said:
Okay just to really finish my understanding, so when there is "no" in front of A, it's the B that will be negated while it is A that will be negated when there is "not' in front of it?

no A equals B----> A->/B
not A equals B----> /A->B

I like “implies” more than “equals.” Might be just my preference, and if it makes sense to you then go with it. No A’s are B’s translates to A implies not B. You will not see sentences literally saying Not A’s are B’s, but you might see a sentence like “Anything that is not an A is a B. You correctly mapped that statement as /A -> B. “If you are not an A, then you are a B.” “Not being an A implies B.”

• Core Member
177 karma

Perfect. Thank you Dr. Brown!

• Alum Member
2227 karma

@lsatyayy said:
Perfect. Thank you Dr. Brown!

Lol not a doctor just my initials good luck

• Core Member
177 karma

lol spit my water reading yo comment. Thanks.