Dyadic Predicate Formalisation of Sentences

Question

Using predicates $Px$ for "x is a person" and $Rx,y$ for "x is the father of y" and secondly taking reference to people as implicit and using only the two-place predicate Rx,y.

1. No one has no father.
2. No one has fathered no one.
3. Some people are the fathers of someone’s father.
4. Anyone who has fathered someone’s father has fathered someone.

---

My ideas:

1. $\neg\exists x(Px \land \forall y(Py\rightarrow\neg Ry,x) \equiv \neg\exists x\forall y(\neg Ry,x)$

   ---

2. $\neg\exists x(Px \land \forall y(Py\land\neg Rx,y) \equiv \neg\exists x\forall y(\neg Rx,y)$

   ---

3. $\exists x(\exists y(Ry,x) \land \exists z(Rx,z))$

   ---

4. $\forall y\exists z\exists x(Ry,z \land Rz,x \land Ry,x)$

---

Any help would be kindly appreciated, I'm not too sure what the solutions are to these formalisations. They are quite tricky! Thank you very much in advanced.

Answer 1

1. $\forall x (Px \rightarrow \exists y (Py \land Ry,x))$ or $\neg \exists x (Px \land \forall y (Py \land \neg Ry,x))$ (without $Px$: $\forall x \exists y Ry,x$ or $\neg \exists x \forall y \neg Ry,x$)

2. $\neg \exists x (Px \land \forall y (Py \rightarrow \neg Rx,y))$ or $\forall x (Px \rightarrow \exists y (Py \land Rx,y))$ (without $Px$: $\neg \exists x \forall y \neg Rx,y$ or $\forall x \exists y Rx,y$)

3. $\exists x \exists y \exists z (Rx,y \land Ry,z)$

4. $\forall x (\exists y \exists z (Rx,y \land Ry,z) \rightarrow \exists w Rx,w)$

Answer 2

1. is fine, though you could also do $\forall x (P(x) \rightarrow \exists y ( P(y) \land R(y,x)))$ and $\forall x \exists y R(y,x)$ (everyone has a father)

2. is like 1, but with the role of $x$ and $y$ reversed. So: $\neg \exists x (P(x) \land \forall y (P(y) \rightarrow \neg R(x,y)))$ and $\neg \exists x \forall y \neg R(x,y)$. Alternatively: $\forall x (P(x) \rightarrow \exists y (P(y) \land R(x,y)))$ and (as Mauro suggests) $\forall x \exists y R(x,y)$

3. Why do you have a negation in here?! Try again.

4. Same here: why the negation?! Also, you will need four quantifiers: one for the first 'anyone', one for the persn they fathered (who itself is the father of someone, so that's a third), and a final one for the last 'someone'. Again, try this one again.