Question

Let $\mathit{P}\left(x\right)\mathbf{,}\mathit{Q}\left(x\right)\mathbf{,}\mathit{R}\left(x\right)\mathbf{,}$and$\mathit{S}\left(x\right)$ be the statements “xis a duck,” “xis one of my poultry,” “xis an officer,” and “xis willing to waltz,” respectively. Express each of these statements using quantifiers; logical connectives; and$\mathit{P}\left(x\right)\mathbf{,}\mathit{Q}\left(x\right)\mathbf{,}\mathit{R}\left(x\right)\mathbf{,}$and$\mathit{S}\left(x\right)$.

(a) No ducks are willing to waltz.

(b) No officers ever decline to waltz.

(c) All my poultry are ducks.

(d) My poultry are not officers.

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

Short answer

The quantifier $\forall$indicates “All” whereas $\exists$indicates “There exits”. The given statements can be expressed using quantifiers, logical connectives and $P\left(x\right),Q\left(x\right),R\left(x\right),$ and$S\left(x\right)$ in the following way. Also, part (d) follows from (a), (b), and (c).

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

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

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

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

Step-by-step solution

Definition of Quantifier    

Quantifiers are terms that correspond to quantities such as "some" or "all" and indicate the number of items for which a certain proposition is true.

Expression of statements using quantifier, logical connectives, and P(x),Q(x),R(x), and S(x).

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

This indicates that if x is a duck, then x is not willing to waltz. This statement is true for all x.

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

This indicates that if x is an officer, then x is willing to waltz. This statement is true for all x.

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

This indicates that if x is one of my poultry, then x is a duck. This statement is true for all x.

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

This indicates that if x is one of my poultry, then x is not an officer. This statement is true for all x.

(e) Consider x is one of my poultry, from (c), it concludes that x is a duck and from (a) x is not willing to waltz. Now, consider x is an officer, from (b), it concludes that x is then willing to waltz but x is not willing to waltz. Hence, x cannot be an officer.

Thus, (d) is concluded from (a) and (c).

Therefore, the given statements can be expressed using quantifiers, logical connectives, and$P\left(x\right),Q\left(x\right),R\left(x\right),$and$S\left(x\right)$.