↔ ⇔ ≡
Iff means, in logic and mathematics, the phrase if and only if. When used in a sentence, for example A occurs iff B occurs, it actually states two conditions: that A occurs if B occurs, and B occurs if A occurs. These two conditions together imply that A and B always occur together.
Wikipedia has a page on if and only if. On it, we find: ‘In that it is biconditional, the connective can be likened to the standard material conditional (“if”) combined with its reverse (“only if”); hence the name.’
Note that only if is mentioned as the reverse of if. Meaning, whereas A if B conveys B implies A, A only if B would convey A implies B.
I have a problem with that.
When I say that I write poetry if it rains, it means the following: rain is a sufficient but not necessary condition for writing poetry. It’s not necessary that I write poetry only when it rains. Therefore, rain would imply that I am writing poetry, but my writing poetry does not imply that it’s raining. I could also write poetry on dry days. Hence, the times when it rains is a subset of the times when I write poetry.
Now, if only if is to be the reverse of if, it must convey the reverse condition when used in a sentence, meaning necessity instead of sufficiency. For example, if I say that I write poetry only if it rains, according to Wikipedia and also according to mathematicians and logicians, it’s supposed to mean that it may rain whenever it wants to, but it should be raining whenever I am writing poetry. So, the times when I write poetry should be a subset of the times when it rains. But does I write poetry only if it rains mean that? No. In normal usage it means something different. Consider the following everyday conversation:
Do you, like, have any gay habits?
Yeah, I write poetry.
Cool. That’s so gay.
But I have a condition for writing poetry. A very gay-ish condition.
I write poetry only if it rains.
Okay, through this dialogue, the gay guy conveys the following:
Whenever I am writing poetry, it’s raining.
Whenever it rains, I write poetry.
Thus, only if single-handedly works as a bi-conditional. It says that the times when it’s raining and the times when the poetry gets written are the same set, and these two events always occur together.
Thus, instead of if and only if, mathematicians could use only only if. Considering all their fuss about compact (and consequently unreadable) language, I think it’s a little shameful how this redundant phrase survived so long in their literature. Seriously, they should have thought twice about labelling only if as the reverse of if. Wikipedia even tries to work some unconvincing logic involving some pudding and a Madison character to justify this.
Well, I wouldnt’ve written this if I were I huge fan of all the abstractions of maths that I had to gulp down today. That I am not. All you pure mathematicians reading this may breathe a little mathematical curse at me now.
As iff I care.