1. **State the problem:** We want to identify the error in the argument that tries to prove: $$\forall x (P(x) \lor Q(x)) \implies \forall x P(x) \lor \forall x Q(x)$$ 2. **Recall the logical equivalence and rules:** - The premise is $$\forall x (P(x) \lor Q(x))$$ meaning for every element $$x$$, either $$P(x)$$ or $$Q(x)$$ is true. - The conclusion claims $$\forall x P(x) \lor \forall x Q(x)$$, meaning either $$P(x)$$ is true for all $$x$$ or $$Q(x)$$ is true for all $$x$$. - Important: Universal instantiation allows us to infer $$P(c) \lor Q(c)$$ for an arbitrary element $$c$$. - Universal generalization requires that the statement holds for all elements, not just one. 3. **Analyze each step:** - Step 1: $$\forall x (P(x) \lor Q(x))$$ (Premise) — correct. - Step 2: $$P(c) \lor Q(c)$$ (Universal instantiation) — correct for arbitrary $$c$$. - Step 3: $$P(c)$$ (Simplification from (2)) — **incorrect**. From $$P(c) \lor Q(c)$$, you cannot infer $$P(c)$$ alone without additional information. - Step 4: $$\forall x P(x)$$ (Universal generalization from (3)) — invalid because step 3 is invalid. - Step 5: $$Q(c)$$ (Simplification from (2)) — also **incorrect** for the same reason. - Step 6: $$\forall x Q(x)$$ (Universal generalization from (5)) — invalid. - Step 7: $$\forall x (P(x)) \lor \forall x (Q(x))$$ (Conjunction from (4) and (6)) — invalid conclusion based on invalid previous steps. 4. **Summary of the error:** The key mistake is in step 3 and 5, where the argument incorrectly applies simplification to a disjunction $$P(c) \lor Q(c)$$ to conclude either $$P(c)$$ or $$Q(c)$$ individually. This is a logical fallacy because from $$A \lor B$$, you cannot infer $$A$$ or $$B$$ alone without further proof. 5. **Final conclusion:** The argument is invalid because it incorrectly simplifies a disjunction and then generalizes from an invalid premise. **Answer:** The error is the incorrect simplification of $$P(c) \lor Q(c)$$ into $$P(c)$$ or $$Q(c)$$ in steps 3 and 5, which is not logically valid. \n\n