Which one of the following predicate formulae is NOT logically valid? Note that $W$ is a predicate formula without any free occurrence of $x$.

We can evaluate the logical validity of each equivalence using standard quantifier transformation rules, since $x$ does not occur free in $W$:

1. Option A: $\forall x(p(x) \lor W) \equiv \forall x\ p(x) \lor W$. Since $W$ does not depend on $x$, the universal quantifier distributes over disjunction with a constant formula. This is logically valid.

2. Option B: $\exists x(p(x) \land W) \equiv \exists x\ p(x) \land W$. Similarly, existential quantification distributes over conjunction with a constant formula. This is logically valid.

3. Option C: Using the equivalence $A \rightarrow B \equiv

eg A \lor B$:$$\forall x(p(x) \rightarrow W) \equiv \forall x( eg p(x) \lor W) \equiv \forall x( eg p(x)) \lor W \equiv eg(\exists x\ p(x)) \lor W \equiv \exists x\ p(x) \rightarrow W$$However, Option C equates this to$\forall x\ p(x) \rightarrow W$, which is generally not equivalent. Therefore, C is NOT logically valid.

4. Option D: $$\exists x(p(x) \rightarrow W) \equiv \exists x(

eg p(x) \lor W) \equiv \exists x( eg p(x)) \lor W \equiv eg(\forall x\ p(x)) \lor W \equiv \forall x\ p(x) \rightarrow W$$ This is logically valid.

Thus, C is the correct answer.