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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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    $\begingroup$ 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. $\endgroup$ – Matt E Aug 1 '10 at 4:36
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    $\begingroup$ The number of people who got this wrong when I marked introductory logic exams was unbelievable... $\endgroup$ – Seamus Sep 8 '10 at 12:21
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    $\begingroup$ Unambiguous, I think. But surely, confusing! $\endgroup$ – Srivatsan Sep 3 '11 at 0:49
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    $\begingroup$ @MattE What leads you to not be sure if this a genuine question? $\endgroup$ – Al Jebr Sep 26 '18 at 16:13
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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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  • $\begingroup$ 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. $\endgroup$ – user126 Jul 29 '10 at 11:01
  • $\begingroup$ 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. $\endgroup$ – Isaac Jul 29 '10 at 11:07
  • $\begingroup$ Ah, I was replacing "condition" with "motivation" in the two bottom right boxes. $\endgroup$ – user126 Jul 29 '10 at 11:18
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    $\begingroup$ 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". $\endgroup$ – Casebash Jul 29 '10 at 13:54
  • $\begingroup$ @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.) $\endgroup$ – Isaac Jul 29 '10 at 14:04
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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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  • $\begingroup$ '"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. $\endgroup$ – user10389 Aug 9 '12 at 22:00
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    $\begingroup$ @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. $\endgroup$ – Arthur Aug 11 '12 at 1:00
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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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The explanation in this link clearly and briefly differentiates the meanings and the inference direction of "if" and "only if". In summary, $A \text{ if and only if } B$ is mathematically interpreted as follows:

  • '$A \text{ if } B$' : '$A \Leftarrow B$'
  • '$A \text{ only if } B$' : '$\neg A \Leftarrow \neg B$' which is the contrapositive (hence, logical equivalent) of $A \Rightarrow B$
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