3
$\begingroup$

Suppose there are two theorems $A \Rightarrow B$ and $C \Rightarrow A$. Then we have $C \Rightarrow B$.

Now comparing $A \Rightarrow B$ and $C \Rightarrow B$, we know that $C \Rightarrow A$ means C is a stronger condition than A. Is it to say $A \Rightarrow B$ is stronger or weaker than $C \Rightarrow B$? I personally think $A \Rightarrow B$ is a stronger result/theorem than $C \Rightarrow B$, but I also saw $A \Rightarrow B$ is said to be weaker than $C \Rightarrow B$.

Thanks!

$\endgroup$
  • 2
    $\begingroup$ Where did you see "$A\Rightarrow B$ is said to be weaker than $C \Rightarrow B$"? $\endgroup$ – Ben Millwood Jul 5 '12 at 12:57
  • 1
    $\begingroup$ I looked back, but forgot where I saw it. $\endgroup$ – Tim Jul 5 '12 at 13:02
  • $\begingroup$ Surely $\mathcal{P}$ is only stronger (more general) than $\mathcal{Q}$ if $\mathcal{P}\Rightarrow \mathcal{Q}$. In your example, $A\Rightarrow B$ does not imply $C\Rightarrow B$, and $C\Rightarrow B$ does not imply $A\Rightarrow B$. So...I'd say they were incomparable. Neither is stronger than the other. $\endgroup$ – user1729 Jul 5 '12 at 13:05
  • $\begingroup$ @user1729: in this case, we already know $C\Rightarrow A$, so one of those does imply the other. $\endgroup$ – Ben Millwood Jul 5 '12 at 13:08
  • $\begingroup$ @BenMillwood: Is the OP not asking about the raw implications? If we throw away our knowledge that $C\Rightarrow A$ then which is stronger? I mean, ($A\Rightarrow B$ and $C\Rightarrow A$) is stronger than $C\Rightarrow B$, but we need the fact that $C\Rightarrow A$ to make this so... $\endgroup$ – user1729 Jul 5 '12 at 13:13
2
$\begingroup$

Suppose we later find that $D\Rightarrow A$. Then we can use $A\Rightarrow B$ to show $D\Rightarrow B$, but we can't use $C\Rightarrow B$ to do that.

Conversely, if we find that $E\Rightarrow C$, then we can use either $A\Rightarrow B$ or $C\Rightarrow B$ to prove $E\Rightarrow B$, since the latter can be recovered from the former and $C\Rightarrow A$.

So, in the presence of $C\Rightarrow A$, the statement $A\Rightarrow B$ is stronger than $C\Rightarrow B$.

$\endgroup$
  • $\begingroup$ I would be intrigued to hear the downvoters reason for the downvote? $\endgroup$ – user1729 Jul 5 '12 at 13:31
0
$\begingroup$

You are correct, $A \Rightarrow B$ is stronger than $C \Rightarrow B$ as it is the more direct path.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.