How is it that 'if A then B' can be equivalent to 'A only if B'? I've seen this stated multiple times that the two are equal. However, I think it's actually logically impossible for the two terms to be the same, as either one of the statements seems to preclude the other.
I know that's a bold statement, but consider:
If A then B, means that A happens first, B happens second, consequently.
So, 'A only if B', should thus be the same as 'only if B, then A. 
This means 'only if B then A' = 'if A then B'. However, A and B cannot happen as a consequent of each other, as that's literally impossible (well... I guess there's that whole chicken or the egg thing, but still).
 A: I think you might misunderstand what "A only if B" means.  We consider "only if" as a phrase on its own, independent of "if."
What "A only if B" means is that the only way that A can possibly hold is if B holds.  That is clearly the same as saying "If B does not hold, there is no way that A can hold."  That is the same as saying "If A holds, then B holds."  The last that I mentioned is the principle of contraposition (https://en.wikipedia.org/wiki/Contraposition).
A: Both expressions mean that $B$ is necessary condition (with respect to $A$), or equivalently that $A$ is a sufficient condition (with respect to $B$). They are both denoted as 
$$A\Rightarrow B$$
and are both equivalent to "$A$ implies  $B$".
The expression "if $A$ then $B$" stresses the sufficiency of $A$ while the expression "$A$ only if $B$" stresses the necessity of $B$, in the following sense: It says that $A$ holds true only if $B$ does. This becomes evident, since $A$ implies $B$, so when $A$ holds true then it is an automatic consequence that $B$ also does. If $B$ weren't true then neither would $A$ be. 
A: The only way to refute "if A then B" is to find a case where A is true, yet B is false.
Which is exactly what "A onlyif B" says -- A is supposed to hold only if B does also, which holds except if A is true yet B is false.
Note that neither "if A then B" nor "A onlyif B" say anything about a sequence of events. Classical logic has no concept of "before" and "after". It doesn't even really have the concept of a "cause". In fact, "if A then B" (in classical logic) is usually defined as being an abbreviation for "either B or not A". Which, again, says that for "if A then B" to hold, in all cases either B must be true, or if not, then A must be false as well.
This definition of "if A then B" is responsible for quite a few of the counter-intuitive properties of classical logic. It is, for example, responsible for "ex falso quodlibet", meaning if A is never true, then "if A then B" is always true. Therefore, if you can prove that "earth is not a disc", then in classical logic, you can also prove "if earth is a disc, santa exists". Such sentences are calles vacuous truths, because while logically true, they don't provide any actual information (in this case about the existance of santa)
