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When reading books I pay attention that some of them giving a definition to some notion use "if" but others "if and only if". For example,

  • A set is called empty if it has no elements
  • A set is called empty if and only if it has no elements

or

  • A set is a subset of another if all elements of the former are in the latter
  • A set is a subset of another if and only if all elements of the first one are in the second one

As these statements involve naming of something, I can't fully understand which one should be correct. Of course "if and only if" seems reasonable for me and formal, but what about using just "if"?

If antecedent is true, then by a statement we would name it as it is described in the statement. From other side, given we call it with the name that coincides with the statement (consequent part holds), can we say its properties (antecedent part) hold? From my experience of reading books it seems true, but doesn't fully make sense to me.

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From a logical point of view, “if and only if” is the correct way of doing it. However, since we are dealing with a definition here, it is implicit that if one uses “if”, then what that actually means is “if and only if”.

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  • $\begingroup$ But what kind of implicity is this? Is this topic described or at least noted somewhere? I mean is it convention maybe or smth? $\endgroup$ – Turkhan Badalov Apr 7 '18 at 9:32
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    $\begingroup$ @TurkhanBadalov It's a matter of language, not of logic. If I say “a natural number is even if it is equal to $2k$ for some $k\in\mathbb N$” it is implicit that only those numbers are called even numbers, because what I am saying is what “even number” means and therefore it would make no sense that there were other even numbers besides these ones. $\endgroup$ – José Carlos Santos Apr 7 '18 at 9:37
  • $\begingroup$ @JoséCarlosSantos The way I've described it is what we are implicitly expecting is that definitions are exhaustive. The idea is if something else was a possibility it would have been stated. $\endgroup$ – Derek Elkins left SE Apr 7 '18 at 12:33

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