This exercise is confusing me. Let $S(x,y,z):= $ $z$ is the child of $x$ and $y$, where $x$ is the mother and $y$ is the father. Express the following sentence in predicate logic using the predicate $S(x,y,z)$:
"There exist a being thats is a father or a mother of another being" $(1)$
My first thought was to write: \begin{equation} \exists x\exists y\exists z \quad S(x,y,z)\quad(2) \end{equation} But then I realized that if there exists a being that only has a father and no being has a mother(I know it sounds stupid), then $(1)$ is true but not $(2)$.
Now I was thinking, what if we interpret the domain as $x\in B, y\in \emptyset, z\in B$, where $B$ is the set of beings and then write
$$\exists x\exists y\exists z \quad S(x,y,z)\lor S(y,x,z)$$
Is this correct? Or am I confusing the meaning of logical interpretation?
$$∃x~∃y~∃z~~S(x,y,z)∨S(y,x,z)$$
Yes, you will clearly need a witness in either parental position, a witness in the child position, and an implicit witness in the remaining position.
Of course, you can simplify this to just: $$\exists x~\exists y~\exists z~S(x,y,z)$$
They are equivalent.
It is not stupid, you simply cannot express it with the given predicate because it requires three terms.