Mathematical Definitions: "If" versus "Iff" One might expect, perhaps naively, that mathematical definitions should be interpreted as "iff" statements by default, unless stated otherwise. This expectation arises from the facts that a definition expresses an equivalence, and that equivalences are biconditional. But the more I read mathematics textbooks, the more it seems that the distinction between "if" and "iff" is not respected. Consider for example the following definition from Elementary Linear Algebra, 5ed, by Larson, Edwards, and Falvo (Houghton Mifflin Co.). My question is: Why are those ifs not iffs instead? Are the authors just being sloppy?