# 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?

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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
@BrianM.Scott If a shop keeper said to you: "I will give you this cake only if you give me a dollar", I doubt you would be thinking there were other conditions you may need to fulfill such as having to hand over another dollar. – John McVirgo Aug 14 '13 at 16:03

Please note that this annex/partial answer responds predominantly to the OP's concern that "B is the only condition that needs to be true for A to be true," by exemplifying Professor Scott's inestimable comment.

My summary of Professor Scott's comment:

$A$ only if $B$
= $A$ is the case/can only happen only if $B$ is the case/has happened.
= $B$ is a necessary (pre)condition for $A$.
= $A \Longrightarrow B$.

I had been confused why $[A$ only if $B] \neq [B \Longrightarrow A]$ until I devised the following aquatic example where I define:
$F$ := There is freshwater fish.
$W$ := There is water.

$\color{blue}{\text{Based upon "natural" knowledge, it is "natural" to assume and be given that:$F$only if$W$.}}$

Notice that I purposely did not specify the kind of water defined in $W$. Thus, $W$ is NOT a sufficient condition, because $W$ says nothing about the salinity of the water and other conditions necessary for freshwater fish. I wittingly pretermitted these details to make my point here.

Thus, because we know nothing about the water in $W$ and because $F$ may require other conditions, we don't know that $W \overset{?}{{\Longrightarrow}} F$. $\color{blue}{\text{All we know is from the given in blue:$F \implies W$.}}$

Remark: It is possible that $W$ could refer to FRESH water. Then $W \Longrightarrow F$ would be true. But again, we are not given this information so we cannot know this expressly and univocally.

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+1 for the explanation. Though I think your answer exaggerates the way in which the formal logic captures the truth of the statements. If you don't provide a rule which lets one deduce $F\Rightarrow W$, then I don't think it's part of "All we know". What if the fish is dead, etc. – NikolajK Aug 9 '13 at 7:28
Thanks for the effort, but this doesn't answer my question which is looking for its history of use. Kcd's accepted answer in the second link in my question makes clear that mathematicians have defined their own version of "only if" which isn't the same as how it's used in common English. I think it would be better if you moved your answer as a reply to either of the questions in the links given in the question. – user10389 Aug 9 '13 at 12:33