TITLE: Precise definition of "weaker" and "stronger"?
QUESTION [26 upvotes]: If I say that $A$ is stronger than $B$, do I mean that $A \Rightarrow B$, or that $B \Rightarrow A$?  (Or something else?)
I feel like I have seen both usages in literature, which is confusing.
Thoughts based on intuition:
$A \Rightarrow B$ means $A$ is a special case of $B$ -- $B$ is more general. This would seem to imply that $B$ is "stronger". (Example: $n$ is an integer implies $n$ is a real number.)
$A \Rightarrow B$ also means that whenever $A$ holds, $B$ must hold. This would seem to imply that $A$ is "stronger".

REPLY [16 votes]: Let's make the simplifying assumption that $\lnot(B \implies A)$.
Then $A \implies B$ can be informally expressed as "$A$ is (strictly) stronger than $B$."
It is certainly possible that in this situation, at some time, someone has instead written "$B$ is stronger than $A$."  Stuff happens.  We all have written $x<y$ when we meant $y<x$.  And interchange of "necessary condition" and "sufficient condition" happens so (relatively) often that it may be best to avoid these terms. 
But "$A$ is stronger than $B$" has only one correct interpretation in terms of the direction of the implication (with disagreement, possibly, in the case of equivalence.)
However, suppose that we have proved theorem $X$, $\:$(a) under the assumption $A$, and $\:$(b) under the assumption $B$.  Then the result (b) is considered to be a stronger result than the result (a). That is perfectly consistent with the ordinary meaning of "stronger," since
$$(B \implies X) \implies (A\implies X).$$