Logic – Why ‘A is Sufficient for B’ Implies ‘B is Necessary for A’

logic

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 $B$?

Best Answer

  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.

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