If and only if, which direction is which?

Solution 1:

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

Solution 2:

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 $A \Rightarrow B$

Solution 3:

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.