'if' and 'if and only if' equivalence in mathematical definition

Question

I am always puzzled by the difference between 'if' and 'if and only if'. I know that an 'if' statement is one-way while an 'if and only if' is two-way. However, I also know that it is a convention to use 'if' statement in a mathematical definition. For example, 'a rectangle is a square if its two adjacent sides are equal in length'. It is puzzling to me why use this convention in the first place when this is clearly an if and only if statement. This convention feels logically inconsistent to me because it uses a one-way statement to describe something that is inherently two-way. To see the issue, imagine you are not told this example is in fact a definition. How are you going to judge if it is just an 'if' statement or an 'if and only if' statement in disguise?

EDIT: since there are many answers to this question, let me zoom the question into a more specific one: Given an if statement without being told whether the statement is a definition or not, how to judge if it is just an 'if' statement or an 'if and only if' statement in disguise?

EDIT2: as someone has pointed out, the example I gave on the square is actually a theorem, not a definition. So consider the following definition instead: a square is a regular quadrilateral. (an equivalent statement is (an object is a square if it is regular quadrilateral). I guess my confusion can be rephrased as how to tell apart a definition and a theorem, because if I know this example is a theorem, then the 'if' statement cannot mean 'if and only if'. In example it might be trivial to tell it is in fact a definition, but I suspect in general without being explicitly told, it can be difficult to tell them apart.

Answer 1

I am always puzzled by the difference between 'if' and 'if and only if'. I know that an 'if' statement is one-way while an 'if and only if' is two-way. However, I also know that it is a convention to use 'if' statement in a mathematical definition.

So, you are not puzzled, but in fact do fully understand the logic of, as well as the culture surrounding, 'if' versus 'iff'.

How are you going to judge if it is just an 'if' statement or an 'if and only if' statement in disguise?

Outside of definitions, ‘if’, ‘only if’ and ‘iff’ convey distinct ideas; none precludes any of the others. Your actual question is rather this: how to judge whether an 'if' statement is a definition or not a definition?

Definitions in mathematical texts are usually well-signposted. It also helps to pay attention to the statement's context and purpose (by carefully reading its surrounding text, including noticing whether the statement is introducing some object or phrase).

For example, 'a rectangle is a square if its two adjacent sides are equal in length'. To see the issue, imagine you are not told this example is in fact a definition.

No, this example is a theorem, not a definition.

It is puzzling to me why use this convention in the first place when this is clearly an if and only if statement. This convention feels logically inconsistent to me because it uses a one-way statement to describe something that is inherently two-way.

Omitting the converse direction of a bidirectional statement is neither illogical nor inconsistent.

You say that a definition is "clearly" an iff statement, and is "inherently" bidirectional. This sounds to me like supporting reasons for the convention—i.e., agreement— that a definition's converse direction is tacit.

Moreover, observe that unlike a standard-issue proposition, a definition is not a usual equivalence. Because its left side has no initial truth value, a definition is not falsifiable, and the ‘if’ or ‘iff’ or $\text‘{\overset{\text{def}}\iff}\text’$ within it isn't your run-of-the-mill inferential implication. As a metalogical biconditional, a definition is regarded differently from a regular statement.