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.

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 $A \Rightarrow B$
