# Is there a name for a biconditional that is itself conditional on something else?

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.

Thanks!

• "Second-order conditional" isn't a good descriptor for a conditional (hypothetical?) equivalence, as it suggests second-order logic. Nov 30 '21 at 16:49
• I would like to apply this to a context where the word "equivalence" would seem odd. Like the example I gave: "hot" and "August" imply or strongly indicate each other in the northern hemisphere, but "hot" and "August" are not "equivalent", they are different things. Nov 30 '21 at 17:03
• Then you want the word "suggest" instead, since in logic and mathematics, 'imply' $(\Rightarrow)$ has an even stronger mean than 'strongly indicate' ('suggest' typically also has a slightly stronger connotation than 'propose/hint/evoke'). In any case, consider supplying, in your Question, the full context that you mention; this increases the chance of getting an Answer. Nov 30 '21 at 17:24