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In a definition like the following:

An object $x$ is called $P$ if [and only if] it has properties $p$ and $q.$

should one use an implication (if) or an equivalence (if and only if)?

It makes sense to use an equivalence seen as this is a definition and not a set of necessary conditions. However, in many texts I see the use of just an implication. Which is correct in the formal or stylistic sense?

user1892304
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1 Answers1

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It is quite common to use if instead of iff in definitions because it reads better.

An alternative is to use when instead of if.

A definition like “$x$ is called a blip when $x$ satisfies $p$ and $q$” is metalanguage.

The mathematical statement “$x$ is a blip iff $x$ satisfies $p$ and $q$” is true, as consequence of the definition, but is not quite the same thing.

lhf
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