Question

Using rules of inference, which step correctly proves (∃x.P(x))⇒Q implies (∀x.P(x))⇒Q?

Answer 1

To prove (∃x.P(x))⇒Q implies (∀x.P(x))⇒Q using rules of inference, we can assume (∃x.P(x))⇒Q is true and (∀x.P(x)) is false. From this, we can use disjunctive syllogism to conclude that Q is false. Hence, (∀x.P(x))⇒Q is true.

Step-by-step explanation:

1. Assume (∃x.P(x))⇒Q is true.
2. Assume (∀x.P(x)) is false.
3. From assumption 2, we can derive ∃x.P(x) is false.
4. Using disjunctive syllogism, we can conclude Q is false.
5. Therefore, (∀x.P(x))⇒Q is true.