Another basic Logic Question

Question

Translate this statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people. a) ∀x(C(x) → F(x))

The answer given in the book is:"Every comedian is funny."

The problem I am getting is that the proposition ∀x(C(x) → F(x)) according to me will also be true if the premise is false.So if x is some person who is not a comedian then the proposition will be true for that x too.So should't the answer be somewhat more specific.

Answer 1

If C(x) is false, then the statement C(x)→F(x) is true regardless of whether F(x) is true or false. That is, if someone is not a comedian, although the statement is true, we learn no information about whether or not they are funny from this statement. Since this statement gives us no information about people who are not comedians, there's no real reason to include anything about non-comedians in our English translation.

Another way to think about it: Say we do as you suggest, and write this proposition as "Every comedian is funny, and every non-comedian is either funny or not funny" to cover all cases that satisfy the statement C(x)→F(x). Since the second half of this sentence is tautological, logically it is equivalent to "Every comedian is funny". Neither sentence is technically an incorrect interpretation, but the latter is more concise/does not include redundant information.

Answer 2

$A\to B$ means "whenever $A$ is true, then $B$ is true" but makes no claim at all as to what $B$ happens to be when $A$ is false.   Thus either $A$ is false (and $B$ is anything) or $A$ is true (wherefore so is $B$).   That is: "$A$ is false or $B$ is true".

Hence $\;A\to B\;$ is logically equivalent to $\;\neg A \vee B\;$.

In this case $\forall x\,\big(C(x)\to F(x)\big)$, "everyone, if they are a comedian, then they are funny," is equivalent to $\forall x\,\big(\neg C(x)\vee F(x)\big)$, "everyone either is not a comedian or is funny".

However the former expression has a more intuitive reading; that we are only making the consequent claim ("is funny") whenever the antecedent ("is a comedian") is satisfied.   Thus we abbreviate the English sentence to "Every comedian is funny."