If you like playing with First Order Logic, I cannot recommend enough this playground site:

https://incredible.pm/

You can even introduce your 10 year old to FOL with it

There is a technical difference in logic between if and iff.

And “P if Q” would be better written “if Q then P” for clarity’s sake.

So yeah if covers the same ground as iff, but iff includes the exclusion of everything else that isn’t Q leading to P.

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.

This reminds me of how for centuries mathematics wrote “is equal to” or similar.

Recorde’s “=“ took centuries to catch on widely, so I guess it’s no surprise we still see “if and only if” instead of “≡.”

Got it, I'll just mentally rewrite it as "P when Q" which has a bidirectional equivalence in my mind moreso than if.