I hereby introduce an unimportant new logical connective: if or only if!
It's like if and only if, but with a disjunction of material conditionals instead of a conjunction. Since we already have iff, I suggest calling it "foof", for iF Or Only iF. So "p foof q" is true if p→q or q→p.
Those of you who are quick with logic will notice the thing that makes foof so unimportant. No matter what the truth-values of p and q are, p foof q is true! If either p or q is false, at least one material conditional is trivially true because of a false antecedent. If neither is false, both conditionals are true.
Since foof claims always come out true, I don't think anything significant depends on them. But maybe someone can show me otherwise, or make a cool point about foof with kinds of logic I'm not good at. Then I'll be happy to give up my claim about the unimportance of foof. Maybe foof is useful for conditionals other than the material conditional, since some of them won't guarantee trivial truth.
Even with the material conditional, I suppose you might like foof if you have a weird and intense love of truth. Whenever you see a claim of the form "p foof q", you know it's true! The other connectives you learned in introductory logic could never guarantee you that.
