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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?

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    I have to pause often. I don't see it as a gap in my knowledge or my understanding. Mathematics can be complicated, and it requires precision and careful attention to details. Sometimes you just have to accept things and then get used to them. – Asaf Karagila Apr 22 '15 at 19:38
  • @turkeyhundt Thanks, but how does that help? Sorry; I don't see the deeper meaning. –  Apr 22 '15 at 19:39
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    I would add that part of the reason this confuses you is likely that it is not how we use the word "if" in every day conversation. – Sam Clearman Apr 22 '15 at 19:39
  • This isn't really an explanation, but a mnemonic. i think of it as the opposite direct of "$A$ if $B$," which is an alternate way of saying "If $B$ then $A$." So "$A$ only if $B$" is the oppositve implication. – Thomas Andrews Apr 22 '15 at 19:40
  • It might help to consider the "if" as meaning "when", such as "$A$ only when $B$", thus leading to "$A$ means that we must have $B$". – abiessu Apr 22 '15 at 19:41
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    "I try to avoid memorisation.": Math is going to be hard... You had to "memorize" what the word "if" means, you will have to memorize what "only if" means too. – Najib Idrissi Apr 22 '15 at 19:42
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    Here's a way of thinking about it: $A$ only if $B$ means $A$ can only happen if $B$ "happens first". That is, if $A$ (happened), then $B$ (must have happened beforehand). – Ben Grossmann Apr 22 '15 at 19:42
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    @Omnomnomnom No, no, no. It has nothing to do with order or time. – Thomas Andrews Apr 22 '15 at 19:43
  • @ThomasAndrews well, the sequence of events doesn't change the logic. I'm just throwing it in here to make the language clearer. – Ben Grossmann Apr 22 '15 at 19:44
  • But it obscures the meaning, and gives the wrong intuition of implication in mathematics. It is such a common error that you should avoid perpetuating it. @Omnomnomnom – Thomas Andrews Apr 22 '15 at 19:45
  • @ThomasAndrews I don't see how it obscures the meaning. Little in math has any "chronological" order, but the everyday example of physical causality is a useful mnemonic for implication. Event $A$ causes event $B$, so if $A$ occurs, then $B$ occurs later. I think that this is a fine mnemonic for someone who has seen implication enough to know that it isn't tied to a sequence of events. – Ben Grossmann Apr 22 '15 at 19:51
  • @Omnomnomnom Quite a few beginners have a hard time with the fact that implication doesn't mean what the standard "if" means. Causality is a very dangerous intuition for "if $A$ then $B$" in general. – Thomas Andrews Apr 22 '15 at 19:56

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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.

Matt Samuel
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I understand "A only if B" as "A can be true only in those possible worlds where B is true also".

hmakholm left over Monica
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