Answer
The error in this argument is in step 7:
Work Step by Step
The argument attempts to conclude that ∀x(P(x) ∨ ∀xQ(x)) is true, but this is not a valid inference. The argument incorrectly combines the universally quantified statement ∀x(P(x)) with the statement ∀xQ(x), treating it as if it's a universally quantified statement, which is not justified.
In other words, the argument mistakenly combines a universal quantification with an existential quantification (∀x(P(x)) with ∀xQ(x)), which is not a valid operation in predicate logic. The correct conclusion should be:
∀x(P(x) ∨ Q(x))
The correct conclusion is that for all values of x, either P(x) or Q(x) is true, but it doesn't necessarily mean that for all x, P(x) is true or for all x, Q(x) is true separately. The error lies in treating ∀xQ(x) as a valid universally quantified statement when it should not be combined in this way.
