Why is "$P$ only if $Q$" equivalent to "if $P$ then $Q$"?

Question

Why is the statement "$P$ only if $Q$" equivelant to "if $P$ then $Q$"? Is there a paper explaining this in detail.

For instance, I know statement "$P$ is sufficient for $Q$" or "$Q$ is necessary for $P$" are equivalent to $P\Rightarrow Q$, since sufficient means "just enough" and $P$ need not be true for $P\Rightarrow Q$ to be true; however, necessary means "absolutely essential" and $Q$ has to be true for $P\Rightarrow Q$ to be true. Therefore, both statements translate to "if $P$ then $Q$".

In the case of "$P$ only if $Q$", this following states that "$P$ only if $Q$" translates to "if not $Q$ then not $P$" and "if not $Q$ then not $P$" translates into "if $P$ then $Q$". The problem I don't understand is:

Why does "$P$ only if $Q$" translate to "if not $Q$ then not $P$"?

Is there an intuitive explanation?

@TankutBeygu Yes, this answers the question. You can close as a duplicate. — Arbuja, Jun 25 '23 at 20:45
It is a good practice to use the search facility of the site with the relevant tag added before posting a question. — Tankut Beygu, Jun 25 '23 at 20:49
I would say I tried looking this up on the internet but it didn't show me the original question. — Arbuja, Jun 25 '23 at 20:49
Example: It is raining ($R$) only if it is cloudy ($C$). This rules out that it is raining and not cloudy, i.e. we have $\neg (R\land \neg C)$ . Using a truth table, we can prove $\neg (R\land \neg C)\iff (R \implies C)$. — Dan Christensen, Jun 26 '23 at 19:02

score 1 · Accepted Answer · answered Jun 25 '23 at 20:33

1

$P$ can happen only if $Q$ happens.

So if $Q$ doesn't happen then $P$ can't happen.

"May I go to the movies?"

"Only if you have done your homework."

That is:

"If you don't do your homework then you may not go to the movies."

answered Jun 25 '23 at 20:33

Prime Mover

5,005

Why is "$P$ only if $Q$" equivalent to "if $P$ then $Q$"?

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