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
- 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.