Question

Translate these staements into English whereC(x) is “xis a comedian” andF(x)is “xis funny” and the domain consists of all animals.

(a)$\forall \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\to \mathrm{F}\left(\mathrm{x}\right)\right)$ (b)$\forall \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{F}\left(\mathrm{x}\right)\right)$

(c)$\exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\to \mathrm{F}\left(\mathrm{x}\right)\right)$ (d) $\exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{F}\left(\mathrm{x}\right)\right)$

Short answer

The required quantification in English can be obtained by the interpretation of given symbols. The symbol $\exists$indicates “There exists” whereas the symbol $\forall$represents “All”.

Step-by-step solution

Definition of quantification

Quantificationis the connection of quantity signals to the predicate or subject of a statement in logic.

To find the truth values of given propositions.

(a) $\forall \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\to \mathrm{F}\left(\mathrm{x}\right)\right)$

The symbol $\forall$indicates “There exists”.

The required statement is “All commedians are funny.”

(b) $\forall \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{F}\left(\mathrm{x}\right)\right)$

The symbol $\forall$indicates “All”.

The required statement is “Every person is a comedian and funny.”

(c) $\exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\to \mathrm{F}\left(\mathrm{x}\right)\right)$

The symbol $\exists$indicates “There exists”.

The statement is “There exists a person such that, if the person is a comedian, then the person is funny.”

(d) $\exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{F}\left(\mathrm{x}\right)\right)$

The symbol $\exists$indicates “There exists”.

The statement is “There exists a person that is a comedian and funny.”

Therefore, the quantification can be expressed in English by the interpretation of given symbols.