If and only if, which direction is which? 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.
 A: 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 $A \Rightarrow B$
A: 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. 
A: 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
A: 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)
