Dyadic Predicate Formalisation of Sentences 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.

  
*
  
*No one has no father.
  
*No one has fathered no one.
  
*Some people are the fathers of someone’s father.
  
*Anyone who has fathered someone’s father has fathered someone.
  


My ideas:


*

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


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


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


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

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.
 A: *

*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)

*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) $\forall x \exists y R(x,y)$

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

*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.
A: *

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

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

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

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