What do you say to the following way of teaching "if" and "the following are equivalent"? Has somebody ever taught it like this?
An implication A -> B can be viewed as asserting that B is at least as true as A. Thus if A -> B and A -> B then A and B have the same truth value ("at least as true" in both directions). Also, under this interpretation it is easy to see that it suffices to prove a cyclic chain of implication A_1 -> A_2 -> ... -> A_n -> A_1 in order to show that A_1, ..., A_n are equivalent (that is, that they have the same truth value).