The problem asks for the correct justification of line (6) in the proof, which asserts $\forall x \, q(x)$.

Here’s a step-by-step breakdown of the proof and how it leads to line (6):

1. Line (1): $\forall x \, (p(x) \to q(x))$
   This is the universal statement given as part of the premises. It states that for all $x$, if $p(x)$ holds, then $q(x)$ holds as well.

2. Line (2): $p(c) \to q(c)$
   This is derived from (1) by Universal Instantiation. Since $c$ is an arbitrary element of the universe, you apply the universal statement to $c$, which gives the implication $p(c) \to q(c)$.

3. Line (3): $\forall x \, p(x)$
   This is another premise, stating that $p(x)$ holds for all $x$.

4. Line (4): $p(c)$
   This is derived from (3) by Universal Instantiation, applying the universal statement $\forall x \, p(x)$ to the arbitrary element $c$.

5. Line (5): $q(c)$
   This follows by Modus Ponens from lines (2) and (4). Since $p(c)$ is true and $p(c) \to q(c)$, we can conclude that $q(c)$ must be true.

6. Line (6): $\forall x \, q(x)$
   This is the final step of the proof. Since $c$ was arbitrary and we derived that $q(c)$ holds, we can generalize the result to say that $q(x)$ holds for all $x$. This step is justified by the Rule of Universal Generalization, which allows us to generalize from an arbitrary element to the entire domain.

Correct Answer:

• The correct justification for line (6) is (5) and the Rule of Universal Generalization.

Would you like a more detailed explanation or any further clarifications? Here are 5 questions to consider for deeper understanding:

1. What is the Rule of Universal Generalization, and when is it valid to use?
2. How does Universal Instantiation work in formal logic proofs?
3. Why is it necessary to prove for an arbitrary element $c$ before generalizing to all $x$?
4. How does Modus Ponens operate in logical arguments?
5. What are the other rules of inference, and how do they compare to Universal Generalization?

Tip: When dealing with universal quantifiers in logical proofs, always remember that after deriving a statement for an arbitrary element, you can apply Universal Generalization to extend the result to all elements.