Question

Let $\mathit{P}\left(x\right)\mathbf{,}\mathit{Q}\left(x\right)\mathbf{,}\mathit{R}\left(x\right)$and$\mathit{S}\left(x\right)$be the statements “xis a baby,” “xis logical,” “xis able to manage a crocodile,” and “xis despised,” respectively. Suppose that the domain consists of all people. Express each of these statements using quantifiers; logical connectives; and$\mathit{P}\left(x\right)\mathbf{,}\mathit{Q}\left(x\right)\mathbf{,}\mathit{R}\left(x\right)$and$\mathit{S}\left(x\right)$.

a) Babies are illogical.

b) Nobody is despised who can manage a crocodile.

c) Illogical persons are despised.

d) Babies cannot manage crocodiles.

e) Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?

Short answer

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

b)$\exists x\left(R\left(x\right)\wedge \neg S\left(x\right)\right)$

c)$\forall x\left(\neg Q\left(x\right)\to S\left(x\right)\right)$

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

e) Yes

Step-by-step solution

Babies are illogical.

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

If x is a baby, then x is not logical. This statement should be true for every x.

Nobody is despised who can manage a crocodile.

$\exists x\left(R\left(x\right)\wedge \neg S\left(x\right)\right)$

If x is able to manage a crocodile, then x can't be despised. This statement should be true for every x.

Illogical persons are despised.

$\forall x\left(\neg Q\left(x\right)\to S\left(x\right)\right)$

If x is not logical, then x is despised. This statement should be true for every x.

Babies cannot manage crocodiles.

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

If x is a baby, then x cannot manage a crocodile. This statement should be true for every x.

Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?

Yes,

Suppose x is a baby, then from a) we know that x is illogical. c) then gives us that x is despised.

Let x can manage a crocodile, according to b) x can then not be despised, but we knew that x is despised, thus x cannot manage a crocodile.

Hence, we have been able to conclude d) from a), b) and c).