TITLE: Why every definition is an "iff"-type statement?
QUESTION [6 upvotes]: Suppose that we are trying to define a mathematical object $M$. The statement of the definition generally takes the form (or some of its equivalent variant),

A mathematical object is said to be $M$ $\color{red}{\text{if}}$ it satisfies some property or properties, say $P$.

It is said that the inclusion of "$\color{red}{\text{if}}$" in the definitions can be easily replaced by "$\color{blue}{\text{iff}}$". In other words, the above definition is equivalent to saying,

A mathematical object is said to be $M$ $\color{blue}{\text{iff}}$ it satisfies some property or properties, say $P$.

In Fraleigh's book on Abstract Algebra it says,



But I think that a definition is only an $\color{red}{\text{if}}$-type statement. Take for example the definition of a (Euclidean) triangle 

A geometric object is said to be a triangle $\color{blue}{\text{iff}}$ it has three sides.

This definition includes two parts,

A geometric object is said to be a triangle if it has three sides.

and,

If a geometric object is a triangle then has three sides.

But the second part is clearly nonsense if we assume that we have exactly one definition of a triangle for in the second part we have to know what a triangle actually is and by our assumption the definition of a triangle is provided by first part.
So, my question is,

Why is every definition an "if and only if" type statement ?

REPLY [3 votes]: The useful thing here is that the language of definitions (whether in mathematics, law, business, science) has to be incredibly precise. When you define something you must leave no room for ambiguity. For example, even the definition of triangle proposed is a poor one:

A polygon is a triangle if and only if it has three sides.

Well, what about a square? That has three sides, and an extra one to boot! Is it still a triangle? Nope. We must further refine our definition.

A polygon is a triangle if and only if it has exactly three sides.

Now this all seems a bit silly. Clearly a square has 4 sides, not 3. Who in their right mind would think we actually meant that? Well, the truth is it doesn't matter who. In the world of disciplined learning, we must approach our definitions with extreme caution. A number of fields are currently plagued with pseudoscience as a result of redefining terms.
Now this "legalese" as it's known is very different from common speech. If someone asks me "Do you take your coffee with milk or cream?" and I say "Yes, thank you!" I'm officially a terrible human being. (Exception: dad jokes. Naturally.) You don't hold people to these mathematical or literal interpretations of what they said because of the concept of colloquialism and known intent. If someone says "can I borrow your iPad" and you tell them "it's an Android" you're just going to sound snobby. It's okay for common speech to be imprecise. But this imprecision cannot carry over in the world of disciplined learning.