Help with expressing statements using quantifiers

Question

I was hoping for someone to check over my answers for a question I am not totally confident about. I'm also not sure if they would require nested quantifiers (which I didn't use). Thank you in advance.

Let $P(x), Q(x), R(x)$, and $S(x)$ be the statements:

• $P(x): x$ is a duck
• $Q(x): x$ is one of my poultry
• $R(x): x$ is an officer
• $S(x): x$ is willing to waltz

The Domain is the set of all poultry. Express each of the statements using quantifiers; logical connectives; and $P(x), Q(x), R(x),$ and $S(x).$

1. No ducks are willing to waltz.

   ∀x,(P(x) → ¬S(x))
   

2. No officers ever decline to waltz.

   ∀x,(R(x) → S(x))
   

3. All of my poultry are ducks

   ∀x,(Q(x) → P(x))
   

4. My poultry are not officers

   ∀x,(Q(x) → ¬R(x))
   

5. Is it possible that I have poultry that is willing to waltz? Explain briefly.

   ∃x,(Q(x) → S(x)) ...Not quite sure how to explain
   

Answer 1

Your answers for 1–4 look correct to me. There's no need for nested quantifiers here, since all the relations are unary (have one variable).

I think that question 5 is asking you to decide if the statement is true assuming that statements 1–4 are true.