In many mathematical textbooks the definitions are given in the following form (considering even numbers as an example):
"A natural number n is called even iff it is a multiple of 2".
Now as we know a definition should serve only as an abbrevation. That is I can use the following form for defining even numbers.
"Even number = a natural number that is a multiple of 2".
Is there any difference between the above definitions? I was thinking that the first makes the definition more strict. For example, suppose that I state:
"$1.5$ is an even number". According to the first definition this statement is meaningless because the term "even" is defined for natural numbers. According to the second definition the statement should be false because $1.5$ isn't a natural number. Which of the above definitions is more appropriate and why?
Edit I have read the question with the proposed answer before I asked this question. But there is a difference because I am asking what notation should I use and if there is a difference between equality sign and "iff" whereas the other question is about the difference between "if" and "iff".
The first one is more appropriate, and "if" is more conventional than "iff" in definitions (i.e. sentences introducing terminology with "is called" / "we say that" / etc.).
Your concern is that if our definition defines "even" only for natural numbers, then writing "1.5 is even" is a sort of "type error". But English text isn't typechecked computer code. We understand "x is even" in context to mean either "x is a natural number and x is even" or "(since here we've established that x is a natural number) x is even". The risk of miscommunication is small.
It's misleading to apply formal symbols like "=" to natural-language phrases, because the language isn't mathematically formalized and doesn't reliably follow simple compositional rules. (Is a sentence just a string of symbols? If two strings of symbols are "equal", does that mean you can substitute one for the other anywhere? And how can you state the answers to these questions if they haven't been answered yet?)