Why every definition is an "iff"-type statement?

Question

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 ?

Answer 1

You are wrongly separating the propositions in your example, it should be: "a geometric object has three sides" $\iff$ "a geometric object is a triangle".

Answer 2

If you deny that

$X$ is a triangle $\Large\iff$ $X$ has three sides

but believe

$X$ is a triangle $\Large\impliedby$ $X$ has three sides

then there is some object $X$ that does not have three sides that you'd still call a triangle.

Answer 3

Here's a reason to include the "iff" rather than "if". We could say

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

This is a true statement, since every triangle is a polygon. This, however, fails to give us the definition of a polygon, since not every polygon has three sides.

Answer 4

A definition that wasn't bidirectional would be confusing and not very helpful. Say object $X$ is called $Y$ if (but not only if) it has properties $P$, then perhaps it is possible for $X$ to be $Y$ even if $X$ does not have properties $P$, but under what conditions? If we want to prove that $Z$ is also $Y$, what do we have to prove? Surely proving that $Z$ has properties $P$ is sufficient, but maybe it is not necessary. Leaving out the other direction has similar problems.

Answer 5

Maybe consider this example:

Definition. An integer $n$ is called odd if there exists an integer $k$ such that $n=2k+1$.

Theorem 1. $17$ is odd.

Proof. We have $17=2\cdot 8+1$, hence the definition of odd applies. $_\square$

Theorem 2. Suppose $m$ is an odd integer. Then $m^2$ is an odd integer.

Proof. Let $m$ be an odd integer. Then there exists $k$ with $m=2k+1$. We compute $m^2=(2k+1)^2=4k^2+4k+1=2\cdot(2k^2+2k)+1$, so $m^2$ is odd because $2k^2+2k$ is an integer. $_\square$

Read the proofs carefully. Do you see how we use the "if" part of the definition in the first proof? And also in the last step of the second proof? However, in "Then there exists $k$ with $m=2k+1$" we also use the (implied) only-if part of the definition. Without it the proof would not be possible. In fact, without the only-if it would be consistent that $2$ is odd (and $4$ still not odd) and the theorem false.

Answer 6

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

This doesn't exclude the possibility that a geometric object with four sides could also be a triangle. All it says is that every goemetric object with three sides is a triangle, while leaving open the possibility that other things could be triangles, too. Replacing "if" with "if and only if" is necessary to rule out the possibility that other things canould be triangles. It's equivalent to saying, "A geometric object is said to be a triangle if it has three sides. Nothing else is a triangle."

Answer 7

Another useful way of thinking about it might be to take the contrapositive of the 'only if' statement. The statement 'an object is a triangle if it has three sides' dictates a set of objects that are triangles, and then the contrapositive of the converse statement 'if an object is a triangle, then it has three sides' — namely, the statement 'an object is not a triangle if it does not have three sides' — dictates a set of objects that are not triangles. Since these two sets are complementary, we have a complete rule for deciding whether an object is a triangle or not (in computer-sciencey terms, you could say that the predicate of triangality is a total function rather than just a partial function; we've defined it for all possible inputs).

Answer 8

Here is another view on the matter:

We want definitions to be "if and only if" statements because the definition should be an abbreviation (not necessarily shorter) or an alias for some statement. This is because, we have a "meta-theorem" that basically says:

If $\alpha \Leftrightarrow \beta$ and $\alpha$ is a subformula of some formula $\phi(\alpha)$, let $\phi(\beta)$ be the formula obtained by replacing all occurences of $\alpha$ with $\beta$. Then: $\phi(\alpha) \Leftrightarrow \phi(\beta)$

(Here formula means a logical formula.)

or put differently:

$\phi(\alpha)$ and $\phi(\beta)$ have the same meaning, if $\alpha$ and $\beta$ have the same meaning

With this, we can always replace "is a triangle" with "has three sides" and vice versa without changing the content of a formula.

Answer 9

What makes something a definition, rather than an axiom, or a substantial statement whose truth is open to question?

Informally, the idea is that a definition is mere verbal shorthand. Thus, a definition should not introduce any genuinely new subject-matter, nor should it let us prove new results about subject-matter already under consideration.

Let $t$ be a term introduced by definition. Then

1. $t$ should be eliminable. That is, for any statement $\phi$, the definition supplies another statement $\phi'$ such that $\phi'$ doesn't contain $t$, and using the definition it's easily provable that $\phi$ and $\phi'$ are equivalent.

2. the definition of $t$ should be a "conservative" extension of the underlying theory. That is, if a statement not containing $t$ can be proved at all, this should be possible without using the definition of $t$.

Requiring a definition have the form "iff" rather than just "if" or "only if" is just an easy way to ensure fulfilment of the eliminability criterion.

For example, consider a statement of the form

D. "$Fx$ iff $\psi(x)$, for all $x$"

where $\psi$ does not contain $F$. Then, any further statement which contains $F$ is provably equivalent, using only D and pure logic, to a statement not containing $F$.

This is not in general the case if "iff" in D is replaced with "if" or "only if".

Answer 10

In order to have a definition of X, you need to know two things.

1. Under what conditions can I say that something is an X?
2. Under what conditions can I say that something is not an X?

If I only have 1, I don't really have a definition. For example, if all I know about triangles is that if something is a three sided closed plane figure, then it is a triangle, I have no idea whether a four-sided figure can also be a triangle.

So, something that is phrased in the form of "if" is not a definition. "If" provides only the information in 1, and you need "only if" to supply the additional information from 2 to have a complete definition.

Answer 11

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.