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I understand the technical and logical distinction between "if" and "only if" and "if and only if". But I have always been troubled by the phrase "only if" even though I am able to parse and interpret it. Also in my posts on this and other sites I have frequently had to make edits to migrate the term "only", sometimes across multiple structural boundaries of a sentence, which is empirical evidence to myself that I don't intuitively know the meaning of the word.

Is there any simple rule that I can use to determine whether or not it is appropriate to use this word in a particular context in order to achieve more clarity? In mathematical discourse, what are some other common lexical contexts, meaningful or not, in which appears the word "only"? Why do I often write "only" in the wrong place?

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This question might be relevant:… – AgCl Aug 25 '10 at 7:59
I think that in formal writing, statements of the form "$P(\text{only }x)$" for some predicate $P$ and noun phrase $x$ have the meaning "$\not\exists y\neq x. P(y)$", but I haven't thought through enough examples to be sure. It might be helpful (although maybe a little embarrassing) if you posted examples where you initially wrote "only" in the wrong place. At least, I find it easier to correct someone's intuition if I know where that intuition is going wrong. – Rahul Aug 25 '10 at 9:40
English is not my mother-tong, but I dare saying that "only if" is the same as "provided that". If that is not the meaning in association with logic quantifiers, then I am missing an important (perhaps subtle) concept. – Américo Tavares Aug 25 '10 at 11:29
@Américo Tavares: That is the other way around. “A only if B” (where A and B are clauses) means “B if A,” and equivalent to “B provided that A.” (I am not a native speaker of English, either.) – Tsuyoshi Ito Aug 25 '10 at 11:51
@Tsuyoshi: In Wiktionary "provided" and "only if" are synonyms. Example from there: You can go to the party provided you finish all your homework first. I will comment no further because it now seems to me that my first comment doesn't give a relevant clarification. – Américo Tavares Aug 25 '10 at 14:22
up vote 17 down vote accepted

The terms "only" and "only if" mean different things. The first is English and the second is mathematical.

1) "only" in English means "just this and not others", where "others" could be an object or an action, depending on what "only" is connected with. For example, "We only proved that $p$ is prime" means you showed $p$ is prime but not anything further, which I think is synonymous with "We proved only that $p$ is prime"; it is just a stylistic judgment as to which of those you use and I prefer the second version at the moment. However, "We proved that only $p$ is prime" means you proved $p$ is prime while other numbers in the argument (that may possibly have been prime) are not prime.

The classical example, which is not meant as a personal remark, is to insert "only" in front of each word of the sentence "I love you". Each version has a completely different meaning. This is discussed under the topic "Modifiers" on the page

2) In math, "only if" has the same meaning as "implies". (Logically, it's the direction that is not "if", which is how I first was able to remember its meaning.) In terms of Tom's example, "shoes are on ==> socks are on" has the same meaning as "your shoes are on only if your socks are on."

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I think analogies in plain English are the way to internalize this... so here's one:

Given that you want to wear socks with your shoes, put your shoes on only if you have already put your socks on.

The idea is that there is no other way to arrive at the state of having your socks and shoes on (aside from the ridiculous possibility of placing your socks over your shoes).

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