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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.

Adriano
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4 Answers4

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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

Larry Wang
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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. – user10389 Aug 09 '12 at 22:00
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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. – Arthur Aug 11 '12 at 01:00
12

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 $A \Rightarrow B$
Nima
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    Thank you for being the only person to literally answer the question. I came here to quickly check which direction is which. Your answer was helpful. – 1Teaches2Learn Oct 21 '21 at 08:15
5

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.

Confutus
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-1

IF-AND-ONLY IF statements have a double arrow symbol ($\Leftrightarrow$) indicating both direction of the statement is true no matter which direction you start from.

Eg: Former statement $\Leftrightarrow$ Latter statement.

IF is the Backward direction ($\Leftarrow$). That is, assume the Latter statement is true and prove the Former statement (often the easier part)

IF-AND-ONLY is the Forward direction ($\Rightarrow$). That is, assume the Former statement is true and prove the Latter statement (often the difficult part)

holala
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