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?
Best Answer
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 http://www.kaptest.com/GMAT/Learn-and-Discuss/Community/blogs/tag/verbal/
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."