Precise definition of "weaker" and "stronger"?
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".
If $A\Rightarrow B$, then for every $C$, if $B\Rightarrow C$ we have that $A\Rightarrow C$. Therefore $A$ implies at least the same propositions that $B$ implies.
We have two options from here:
- $B\Rightarrow A$, in which case $A$ is equivalent to $B$, and they imply the same things.
- $B\nRightarrow A)$, that is $B$ does not imply $A$. We have if so that $A$ is stronger than $B$ since $A\implies A$, but $B$ does not.
In essence "$A$ is stronger than $B$" is when $\{C\mid B\Rightarrow C\}\subsetneq\{C\mid A\Rightarrow C\}$, and equivalent is when the sets are equal.
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 $$(B \implies X) \implies (A\implies X).$$