Question

Let $\mathit{P}\left(x\right)\mathbf{,}\mathit{Q}\left(x\right)\mathbf{,}$and$\mathit{R}\left(x\right)$be the statements “xis a professor,” “xis ignorant,” and “xis vain,” respectively. Express each of these statements using quantifiers; logical connectives; and$\mathit{P}\left(x\right)\mathbf{,}\mathit{Q}\left(x\right)\mathbf{,}$and$\mathit{R}\left(x\right)$where the domain consists of all people.

a) No professors are ignorant.

b) All ignorant people are vain.

c) No professors are vain.

d) Does (c) follow from (a) and (b)?

Short answer

a) $\forall x\left(P\left(x\right)\to \neg Q\left(x\right)\right)$

b)$\forall x\left(Q\left(x\right)\to R\left(x\right)\right)$

c)$\forall x\left(P\left(x\right)\to \neg R\left(x\right)\right)$

d) No

Step-by-step solution

No professors are ignorant.

We can write this statement as “ All professors are not ignorant.”

As we know ALL is represented by $\forall x$, NOT is represented by $\neg$and AND is represented by $\wedge$.

All ignorant people are vain.

As we know ALL is represented by $\forall x$, if someone is ignorant then they also have to be vain.

No professors are vain.

We can write this statement as “ All professors are not vain.”.

As we know ALL is represented by $\forall x$, NOT is represented by$\neg$and AND is represented by $\wedge$.

Does (c) follow from (a) and (b).

Suppose (a) and (b) are true, then we cannot sure that there are vain people that are not ignorant and thus we can’t be sure that the ignorant professors are not vain.