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I can never figure out (because the English language is imprecise) which part of "if and only if" means which implication.

($A$ if and only if $B$) = $(A \iff B)$, but is the following correct:

($A$ only if $B$) = $(A \implies B)$

($A$ if $B$) = $(A \impliedby B)$

The trouble is, one never comes into contact with "$A$ if $B$" or "$A$ only if $B$" using those constructions in everyday common speech.

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    I'm not sure if this is a genuine question or not, but in case it is: A (must hold) if B (does). A (can hold) only if B (does). This makes it pretty unambiguous.2010-08-01
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    The number of people who got this wrong when I marked introductory logic exams was unbelievable...2010-09-08
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    Unambiguous, I think. But surely, confusing!2011-09-03
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    @MattE What leads you to not be sure if this a genuine question?2018-09-26

3 Answers 3

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It's easier to work out if you have a specific example:

Let A:I am a parent B:I have a child

I am a parent if and only if I have a child has two parts:

I am a parent if I have a child can be rephrased: If I have a child, then I am a parent. B => A

I am a parent only if I have a child can be understood to mean: if I do not have a child, then I am not a parent: ~B -> ~A But this is logically equivalent to if I am a parent, then I have a child: A=> B

So the "if and only if" locution implicitly involves some grammatical transformations. The meaning may not be immediately obvious, but it can be worked out.

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This example may be more clear, because apples ⊂ fruits is more obvious:

"This is an apple if it is a fruit" is false.
"This is an apple only if it is a fruit" is true.
"This is a fruit if it is an apple" is true.
"This is a fruit only if it is an apple" is false.

A is an apple => A is a fruit

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    '"This is an apple only if it is a fruit" is true.' well being a fruit isn't the only thing that makes an apple an apple.2012-08-09
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    @user10389 You're confusing "only if" with "if and only if". The former implies the possibility of further neccessary requirements, while the latter is considered an equivalence.2012-08-11
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I wouldn't say that I never come into contact with those phrasings--they are certainly rare in technical use, but perhaps more common in plain language. Below is a table of equivalent phrasings of p=>q, from UCSMP Precalculus and Discrete Mathematics, 3rd ed., © 2010 Wright Group/McGraw Hill (Lesson 1-5).

table http://www.imgftw.net/img/171340741.png

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    Certainly the last box of the right column is not true! I intend to accept this answer when it becomes possible to do so, however.2010-07-29
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    The last box of the right column is logically equivalent to the rest of the statements in the column--the relationship between necessary/sufficient conditions and if/only-if can be difficult to get a handle on.2010-07-29
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    Ah, I was replacing "condition" with "motivation" in the two bottom right boxes.2010-07-29
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    Some of those statements are not a way that anyone would speak or would expected to understand. To make it (a bit) more natural, I would have written the following. For "p only if q", I would use: "You have applied early only if you have increased your chance of receiving aid". For q is a necessary condition for p, I would write: "Having increased your chance of receiving aid is a necessary condition for having applied early" or "is a necessary consequence of having applied early".2010-07-29
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    @Casebash: While it's less of an issue in the table, in the context of the lesson as a whole, the change in tense would have been problematic. (Also, given that this is in Chapter 1 of a high school book, applying early and financial aid are both in the future for the student.)2010-07-29
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    @Isaac: Yeh, but as is the sample doesn't clarify, only confuse2010-07-29
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    @Casebash: Perhaps this is an issue of usage in the U.S. versus Australia (or elsewhere)--all but the necessary/sufficient samples are well within normal speech patterns for U.S. english.2010-07-29
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    @Isaac: Regarding tense, we can always use the "future looking back tense" - "You will have applied early only if your chance of receiving aid has increased"2010-07-29
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    I dislike the conflating of "If A then B" with "A implies B". The second suggests some kind of logical dependence that need not exist with the material conditional. (Because a false antecedent makes the conditional true, no matter what the consequent is).2010-09-08