# What is the history of “only if” in mathematics?

A quick search on the use of "only if" returns questions asking about its use and meaning in mathematics, such as here, here and here, revealing confusion in its interpretation and use for some people.

Personally on first coming across "A only if B", it meant that B is the only condition that needs to be true for A to be true - which is incorrect. Even more confusing I find, is when its meaning can be taken from either philosophy as necessary conditions, or predicate logic as implication.

So my question is what is the history of "only if" used in mathematics and in particular where was the phrase first introduced?

• If you interpret "only if" to be a sufficient condition, you clearly haven't been subjected to much bureaucracy. – Niel de Beaudrap Aug 10 '12 at 23:51
• I've always interpreted "only if" as implication, because I've practically only encountered it as the second half of "if and only if". – Arthur Aug 11 '12 at 0:55
• I don’t really understand at gut level why people have trouble with it: it’s pretty straightforward English. A only if B says, depending on context, that A is the case/can happen only if B is the case/has happened, so clearly A implies B/B is a necessary (pre)condition for A. There’s nothing essentially mathematical about this, so I doubt that a single point of origin exists. – Brian M. Scott Aug 11 '12 at 7:03
• @user10389: No, the only in only if does not mean one. It means exactly what I said in my other comment. Just if is a slightly sloppy shorhand for precisely in case, which does indeed mean if and only if; it does not mean only if. – Brian M. Scott Aug 12 '12 at 18:07
• The mathematician R.L. Moore used "only if" to mean "if and only if". It sounds weird to us because we are so used to "if and only if", but I can see what Moore was thinking. "A only if B" is sort of like "A if B", but with an extra piece of information thrown in: not just "A if B", but in fact "A only if B." I suspect Moore would have agreed with the shop keeper example. – littleO Mar 1 '15 at 6:29