I'm trying to find whether there is a technical name for something like this:
$(A \iff B)$ given $C$
So $A$ and $B$ imply each other only if $C$ applies. For example, "August" and "hot" (fuzzily?) imply each other given that you are in the northern hemisphere.
Would it be reasonable to call this a second-order conditional? Since it is a conditional relationship that is itself conditional on something else.