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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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    @MattE What leads you to not be sure if this a genuine question?2018-09-26

4 Answers 4

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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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    @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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    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