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

  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.


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.


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"

Tags:

Logic

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