Translate the following statements into English, where $C(x)$ means '$x$ is a comedian', $F(x)$ means '$x$ is funny' and the domain consists of all people:
a) ∀x(C(x) → F(x))
b) ∀x(C(x) ∧ F(x))
c) ∃x(C(x) → F(x))
d) ∃x(C(x) ∧ F(x))
this my Sol :
a- for all people If comedians , then are funny.
b- Every people are comedians and funny.
c- some person If a comedian , then is funny.
d- some person comedian is funny.
Correct ?
Best Answer
If the object is to translate the symbolic formulas into English sentences that a non-mathematician can easily understand, here are my recommendations:
(Note the inclusion of the word "actually." It doesn't affect the logic of the claim, it just calls attention to the possibility that a comedian might not be funny. Similarly, the use of "at least one" instead of just "a" makes clear that you're not trying to say there's only one funny comedian.)
The formula c) warrants a little extra discussion. It can be translated directly into this:
But I don't think most people would immediately grasp that this statement is satisfied by the existence of a single non-comedian. To phrase it in a way that makes sense, I think you have to go with something like this:
Symbolically this says $(\forall x C(x))\rightarrow (\exists xF(x))$ instead of $\exists x(C(x)\rightarrow F(x))$, but these two formulations are tautologically equivalent. Translating from one "language" to another sometimes calls for reformulation, if clarity of expression is the goal.