Logical Translation 5Dce90
1. The problem is to translate the logical statement $\forall x (C(x) \lor \exists y (C(y) \land F(x,y)))$ into plain English.
2. The symbol $\forall x$ means "for all $x$" or "for every $x$."
3. The symbol $\lor$ means "or."
4. The symbol $\exists y$ means "there exists some $y$" or "there is at least one $y$."
5. The symbol $\land$ means "and."
6. $C(x)$ and $C(y)$ are predicates that depend on $x$ and $y$ respectively, meaning some property $C$ holds for $x$ or $y$.
7. $F(x,y)$ is a predicate involving both $x$ and $y$, meaning some relation $F$ holds between $x$ and $y$.
8. Putting it all together, the statement says: "For every $x$, either $x$ has property $C$, or there exists some $y$ such that $y$ has property $C$ and $x$ and $y$ satisfy relation $F$."
9. In simpler terms: "For all $x$, $x$ either has property $C$, or is related by $F$ to some $y$ that has property $C$."