I already know, and so ask NOT about, the proof of: $A$ only if $B$ = $A \Longrightarrow B$.
Because I ask only for intuition, please do NOT prove this or use truth tables.
My problem: I try to avoid memorisation. So whenever I see this statement, I always need to pause for 5 minutes to remember my linked post above, in order to determine the meaning.
This pause reveals deficiency in my knowledge that stifles me. So please help me dig deeper. How can this statement be naturalised?
On
There have been some comments about this requiring memorization or it being different from the way the word is used in normal conversation, but this simply isn't true. It's English.
You can walk the dog only if you have a leash. Therefore, if you can walk the dog, it follows logically that you must have a leash. Can walk the dog implies have a leash.
You can legally drive a car only if you have a license. Therefore, if you can legally drive a car, it follows that you must have a license. Can legally drive a car implies have a license.
Etc.
I understand "A only if B" as "A can be true only in those possible worlds where B is true also".