# Why is it that if "A is sufficient for B" then "B is necessary for A"?

I know this may sound like a basic question in mathematical logic, implications and/or conditionals. But I haven't been able to find a simple and clear explanation as to why do we automatically call $B$ necessary for $A$, whenever we are given the fact that $A$ is sufficient for $B$?

I am not asking about the meaning of the words 'sufficient' and 'necessary' or how the truth table looks; I am asking why does one of these words represent one direction of relating A and B, whenever the other word represents the opposite direction? Why did we come to see them as opposite directions in the logical flow between two such events/statements?

Update:

It appears that the statement that $A$ is sufficient for $B$ doesn't tell us anything about $B$ influencing $A$ to happen. Yes, $A$ will be enough to guarantee $B$, but why does that lead us to conclude that $B$ is one of the necessary conditions for $A$ to happen? Why wouldn't $A$ happen without care about $B$?

• The reason you ask is precisely the meaning of the words. Commented Sep 19, 2016 at 13:46
• Comments are not for extended discussion; this conversation has been moved to chat. Commented Sep 21, 2016 at 9:30

1. $A$ is sufficient for $B$ means whenever $A$ happens, $B$ happens.

Now, for $A$ to happen, $B$ should also happen. Because if $A$ happens without $B$ happening, then statement 1 above will become false. So $B$ should definitely happen for $A$ to happen. So $B$ becomes necessity for $A$.

$B$ is a bigger set, and $A$ is a subset of $B$

'I am a mammal' is sufficient for 'I am an animal'. I have to be an animal to be a mammal.

Hope this helps.

• Got it. Many thanks. Commented Sep 19, 2016 at 15:53

The way one could think about it this way:

"A is sufficient for B" means "A is enough for B" which means "If A happens, B happens" which means "Whenever A happens, B happens" which means "B necessarily happens if A does" which means "B is necessary for A."

In other words "B is necessary for A" does not imply that if B happens, then A does. It implies that if A happens, B does, which is equivalent to saying "If B doesn't happen, A doesn't happen either." Thus, B happening is neccessary for A happening, but may be not enough.

• Got it. Many thanks. Commented Sep 19, 2016 at 15:52

Think of "$A$ is sufficient for $B$" as "Knowing that $A$ is true is sufficient for knowing that $B$ is true" and

"$B$ is necessary for $A$" as "If $A$ is true then $B$ is necessarily also true"

For this situation you may want to avoid thinking about cause-and-effect, or before and after. Imagine instead that these relationships happen instantaneously, and in parallel.

In our discrete mathematics course instead of sufficient we used the word implies to communicate the same meaning.

A implies B means that: if A is true, then B must also be true. (A is sufficient for B)

Saying that just slightly differently: for A to be true, B must also be true. (B is neccessary for A)

• This is generally good advice: mathematics doesn't know time and causality, it just knows correlations between the truthiness of various properties, as it were. Commented Sep 20, 2016 at 9:54

Example:

-- Being president is sufficient for being a human.

Assuming that is true, you are saying we now know the following is also

true:

-- Being human is necessary for being president.

To generalize this, make the following substitutions:

A <-> being president

B <-> being human

is a subset of <-> is sufficient for

is a superset of <-> is necessary for

After substituting we get:

A is a subset of B.

means the same as

B is a superset of A.

Apparently we standardized on the words "necessary" and "sufficient" for the simple notion of "superset" and "subset".

You write: "It appears that the statement that $A$ is sufficient for $B$ doesn't tell us anything about $B$ influencing $A$ to happen. Yes, $A$ will be enough to guarantee $B$, but why does that lead us to conclude that $B$ is one of the necessary conditions for $A$ to happen? Why wouldn't $A$ happen without care about $B$?"

This is one way to look at why $a\rightarrow b$ means "B is necessary for A" is to remember that is equivalent $\lnot B \rightarrow \lnot A$.

So if the implication is true, then if B is false, A must be false as well.

I.e. Given $(A\rightarrow B)$ which is equivalent to $(\lnot B \rightarrow \lnot A),$ the truth of $B$ is necessary to ensure the truth of A (otherwise, $\lnot A$)

Outside of Mathematics, it could be that some other "C is also sufficient for B". Therefore, if you see B, it could be due to A or to C. This might be what is confusing.

But your question only mentions A, therefore we have to assume that this is the only thing known when analyzing the logic of the statement. Therefore if we see B, it must have been due to A.

I also prefer to use "implies" instead of "is sufficient for", as I find it less confusing. Using "implies" means you're definitely talking math and logic. Using "is sufficient for" seems more open to interpretation in my opinion.

• it's common courtesy to explain downvote... Commented Oct 12, 2016 at 20:01
• Actually, it's not, it's more like an invitation to get downvoted yourself. It's not the way it should be, just the way it is. If I had been the one to downvote you (I wasn't), I would take a hit in my own score. By explaining the downvote, you could look at my profile and see if I have taken the downvoting hit.
– Lisa
Commented Apr 4, 2017 at 22:24