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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".

Anton
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    Usually, "multiple" refers only to integers , making the first option valid. And "iff" is much better than "=" – Peter Feb 03 '21 at 18:16
  • Well in naming even numbers you are defining a set of numbers. Does your proposed definition define the right set? Is it a construction which uses the structural machinery of your underlying set theory? The set you define is a subset of a pre-existing set - so the definition might depend on what that set is. In general I would say the foundational questions here are deeper than is necessary for functional understanding - if you want back to basics, Russell and Whitehead's observations on $1+1=2$ (https://en.wikipedia.org/wiki/Principia_Mathematica) provide a classic discouragement. – Mark Bennet Feb 03 '21 at 18:50
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    "=" in a mathematical theory is a relation between two objects in that theory. "Even number" and "a natural number that is a multiple of $2$" are relations of the theory of numbers, not objects in it. So within the theory, your second definition is gibberish. However, it does make sense in the metatheory describing the original theory. But like physics is preferable to metaphysics, mathematics is preferable to metamathematics. ;-) – Paul Sinclair Feb 04 '21 at 02:00
  • See also this post – Mauro ALLEGRANZA Feb 04 '21 at 08:54
  • The equal sign $=$ is inappropriate, too informal. –  Feb 04 '21 at 09:11
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    "$=$" is used between "names" (terms). "iff" is used between sentences. When we define what an even number is, we define the predicate "to be even" and the form of the definition is "n is Even iff ..." – Mauro ALLEGRANZA Feb 04 '21 at 11:40

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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?)

Karl
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