1. The problem is to identify the error in the argument that tries to prove: if $$\forall x (P(x) \lor Q(x))$$ is true, then $$\forall x P(x) \lor \forall x Q(x)$$ is true. 2. The argument steps are: 1. $$\forall x (P(x) \lor Q(x))$$ Premise 2. $$P(c) \lor Q(c)$$ Universal instantiation from (1) 3. $$P(c)$$ Simplification from (2) (Error here) 4. $$\forall x P(x)$$ Universal generalization from (3) (Invalid) 5. $$Q(c)$$ Simplification from (2) (Error here) 6. $$\forall x Q(x)$$ Universal generalization from (5) (Invalid) 7. $$\forall x P(x) \lor \forall x Q(x)$$ Conjunction from (4) and (6) (Invalid conclusion) 3. Explanation of the error: - From $$P(c) \lor Q(c)$$, you cannot simplify to just $$P(c)$$ or just $$Q(c)$$ because the disjunction means at least one is true, but not necessarily which one. - Simplification applies only to conjunctions, not disjunctions. - Therefore, steps 3 and 5 are invalid. - Consequently, universal generalization in steps 4 and 6 based on invalid simplifications is also invalid. - The conclusion in step 7 is not logically justified. 4. Important logical rule: - $$\forall x (P(x) \lor Q(x)) \not\Rightarrow \forall x P(x) \lor \forall x Q(x)$$ - The universal quantifier distributes over conjunction but not over disjunction. 5. Summary: The error is the incorrect use of simplification on a disjunction and invalid universal generalization based on that. $$\text{The argument is invalid because simplification cannot be applied to } P(c) \lor Q(c).$$