Impossibility of making an "If and only if statement" stronger?

Question

Is it impossible to prove a stronger version of an "if an only if" result?

It seems to me that the answer would be "no" because to strengthen "$A$ iff $B$" I would need to either weaken $A$ or $B$. Suppose I weaken $A$ (so $A$ no longer holds but some condition implied by $A$ holds). Then $B$ cannot hold since $A$ is necessary for $B$ (i.e. $B\implies A$, so $\text{not } A\implies\text{not }B$).

So basically, if there is an "if and only if" result, in order to improve upon it would both $A$ and $B$ need to be weakened?

Thanks

Answer 1

If you weaken both $A$ and $B$ you simply get something that is neither stronger nor weaker than the original. For example,

I take a red umbrella to work if and only if it rains on a Tuesday

neither implies nor is implied by

I take an umbrella to work if and only if it rains on a weekday

(both A and B have been weakened).

The only way I can think of to make a stronger statement than $A\Longleftrightarrow B$ is to add something to it, for example finding a third condition $C$ such that $A$, $B$ and $C$ are all equivalent.

Answer 2

You're right: just working with the left or right side cannot result in a stronger statement. However, something like:

$$(A \leftrightarrow B) \land C$$

is a stronger claim than

$$A \leftrightarrow B$$

Answer 3

In practice, the way to "strenghten" an equivalence is often the following: Suppose we proved $$ A \iff B. $$ Now it's often the case that this equivalence is a 'corollary' of a more general result that we later discover. I.e. there are statements $A^-, B^-$ for which we can prove $$ A^- \iff B^- $$ and for which we (usually for trivial reasons) have $$ A \wedge B^- \implies B \text{ and } B \wedge A^- \implies A. $$

In this case the proof of $A^- \iff B^-$ can be viewed as a strengthening of the previous result that $A \iff B$.

The way this usually goes is that $A,B$ are 'global' statements and $A^-, B^-$ are 'localized' version of this. (This came up recently in a discussion about strong cardinals but I don't think that going into details will be very helpful to this thread.)