Why is my translation of $\exists{x}\,(C(x) \rightarrow F(x))$ into an English sentence wrong? Let $\text{C(x): x is a comedian}$ and $\text{F(x): x is funny}$
Let $$\alpha:\quad\exists{x}\,(C(x) \rightarrow F(x))$$ and the domain consists of all people.
I needed to translate $\alpha$ into English so what I did was I looked at when $\beta:\; C(x) \rightarrow F(x)$ is true. There are 2 cases for that:


*

*$C(x)$ is false and $F(x)$ is either true or false

*$C(x)$ is true and $F(x)$ is true.


Using this, I translated it as follows:
There is a comedian that is funny $\textit{(referring to case 2)}$ or there is no comedian. $\textit{(case 1)}$
The solution in the book is:
There exists a person such that if they are a comedian then they are funny.
My professor told me that the way I translated it was wrong.
Why is my translation wrong?

The domain is of people, not commedians.
$\exists x~(\lnot C(x)\lor F(x))$ would read as "There is some person who is not a commedian or is funny."
Although equivalent, $\exists x~(C(x)\to F(x))$ more directly translates as "There is some person who would be a commedian only if they were funny."
