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?
Best Answer
Consider a world with no people. Then there is no comedian, so your translation is correct. However, the statement is false because there doesn't exist anyone, in particular no one who satisfies $C(x) \rightarrow F(x)$.
To correct your answer, you could say that in case 1, there is someone who is not a comedian.