What is the logic symbol to write: $D$ only when $P$

Question

The text I am reading gives this problem:

Express the following using logic symbols.

The cat is out of the bag only when the contestant is bald.

$D$ is: The cat is out of the bag, and $P$ is: The contestant is bald, thus $D$ only when $P$.

I thought of "only when" as similar to "only if" and answered $D \iff P$. The text gives the answer as $D \implies P$.

That then is:

If the cat is out of the bag, then the contestant is bald, and I can not see that "if-then" has the same meaning as "only when".

As an absolute amateur at this, and assuming that the text is not in error, I have to look here for guidance. The logic text I am reading makes it very clear that connections do not imply causality or sequence in time. "Only when" does not imply a sequence, but does seem to separate the times when the contestant is bald and when, another, contestant is not. Please excuse me if I am not making sense about this.

How do I interpret "only when"?

Answer 1

"$D$ only when $P$" means exactly that "not $D$ when not $P$".   That is $\neg P\to\neg D$.

If it is $D$ only when it is $P$, then it must be not $D$ when it is not $P$.

If it is not $D$ when it is not $P$, then it can be $D$ only when it is $P$.

"$D$ only when $P$" also means exactly that "$P$ when $D$".   That is $D\to P$.

If it is $D$ only when it is $P$, then it must be $P$ when it is $D$.

If it is $P$ when it is $D$, then it must be $D$ only when it is $P$.

Answer 2

We read $D\implies P$ as "if $D$ then $P$," but what the notation really means is "either $D$ is false or $P$ is true or both."

Given that $D \implies P,$ is it possible that $D$ is true when $P$ is false? No, because then neither of the two halves of the "either or" version of the statement is true: it is not true that $D$ is false and it is not true that $P$ is true.

So the statement $D\implies P$ tells us that the only circumstances under which $D$ can be true are when $P$ also is true. "$D$ true and $P$ false" is ruled out. Hence $D$ is true only when $P$ is true.

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Another way to think of this is to recall that we read $D \iff P$ as "$D$ if and only if $P.$"

Now, "$D$ if $P$" is the same as "if $P$ then $D$", that is, it can be written $P \implies D$ or $D \impliedby P$.

So the "if" half of the "if and only if" gives us the $D \impliedby P$ direction of the arrow in $D \iff P$. The other direction of the arrow, $D \implies P$, is given by the "only if". That is, "$D$ only if $P$" is a legitimate way to read $D\implies P$.