These questions concern concern issues with substitution and Skolemization.
1. Given the premise ${\forall\,x\;\;} {\exists\,y\;\;} P(x,y)$, it is not valid to conclude that ${\exists\,q\;\;} P(q,q)$. Give an example of a predicate $P$ where the first is true but the second is false.
2. Suppose that an inference engine is incorrectly written with the occurs check omitted, so that it allows a literal like $P(x,F(x))$ to be unified with $P(q,q)$. (As mentioned, most standard implementations of Prolog actually do allow this.) Show that such an inference engine will allow the conclusion ${\exists\,y\;\;} P(q,q)$ to be inferred from the premise ${\forall\,x\;\;} {\exists\,y\;\;} P(x,y)$.
3. Suppose that a procedure that converts first-order logic to clausal form incorrectly Skolemizes ${\forall\,x\;\;} {\exists\,y\;\;} P(x,y)$ to $P(x,Sk0)$—that is, it replaces $y$ by a Skolem constant rather than by a Skolem function of $x$. Show that an inference engine that uses such a procedure will likewise allow ${\exists\,q\;\;} P(q,q)$ to be inferred from the premise ${\forall\,x\;\;} {\exists\,y\;\;} P(x,y)$.
4. A common error among students is to suppose that, in unification, one is allowed to substitute a term for a Skolem constant instead of for a variable. For instance, they will say that the formulas $P(Sk1)$ and $P(A)$ can be unified under the substitution $\{ Sk1/A \}$. Give an example where this leads to an invalid inference.