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From a linguistic standpoint, "if and only if" kinda sounds like an angry mother overexaggerating the "if", but mathematically this is not the case. Of course, this is completely irrelevant, I'm just trying to be funny.

I was told that "only if" is different from "if and only if". Instead of opening a Mathematical Logic book and looking through the definitions and proofs, I decided to take the easy route and ask here. If $p$ holds "only if" $q$ holds, shouldn't that mean that if $p$ holds then $q$ also holds, thus making "only if" the same as "if and only if"? Are these the exact same thing but we just ended up using "if and only if" predominantly?

Ben Grossmann
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Zero Pancakes
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    $p$ only if $q$ is $p\implies q$; $p$ if and only if $q$ is $p\iff q$ ($p$ only if $q,$ and $q$ only if $p$) – J. W. Tanner Mar 09 '20 at 16:05
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    $p$ only if $q$ can be written as $p\implies q$, but not $q\implies p$. – Andrew Chin Mar 09 '20 at 16:05
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    @AndrewChin "but not /necessarily/", the word necessarily is missing. $p$ only if $q$ does not directly imply that $q\not!!\implies p$ – JMoravitz Mar 09 '20 at 16:08
  • Oh okay, I see now. Now "if and only if" makes so much more sense. – Zero Pancakes Mar 09 '20 at 16:11
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    @MichalisP. My post was all mixed up, so I deleted it. See the page linked at the top, though – Ben Grossmann Mar 09 '20 at 16:23
  • @Omnomnomnom Yeah, it probably was mixed up because something seemed off in that sentence. – Zero Pancakes Mar 09 '20 at 16:24
  • It is true that you can be a bachelor only if you are male. But clearly it is not true that you are a bachelor if and only if you are male. 'only if' expresses necessary conditions, but those necessary conditions are not always sufficient. The 'if' of course expresses sufficient conditions. – Bram28 Mar 09 '20 at 17:09

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You are correct that "$p$ only if $q$" is the same as "$p$ implies $q$". The missing piece from your discussion is that "$p$ if $q$" is the same as "$q$ implies $p$". So "$p$ if and only if $q$" means that both implications hold, whereas "$p$ only if $q$" means that only one of the implications holds.

Will Orrick
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