Can mathematical definitions of form "P if Q" mean "P if and only if Q"? (2012)

This post is specifically about definitions. What you describe only applies to predicates (properties, theorems, conjectures), not to definitions.

See the top answer which explains this well:

> Absolutely. The definition will state that we say [something] is P if Q. Thus, every time that Q holds, P also holds. The definition would be useless if the other direction didn't hold, though. We want our terms to be consistent, so it is tacitly assumed that we will also say P only if Q. Many texts prefer to avoid leaving this as tacit, and simply state it as "if and only if" in their definitions.

Emphasis mine: in a definition, "if" somehow can't mean anything else than "if and only if", so we can leave out the "and only if", and we usually do (probably for conciseness - if and only if is quite verbose and having to write and read it when it's useless would be quite boring in the long run).

I remember a teacher starting a definition like "a X is called Y if and only if... ah no, we don't put the 'only if' for definitions", and she erased it from the blackboard. It made me wonder why at the time. It was a useful teacher's mistake.