g --> h

~h v k
g
k --> w

Prove: W

g

h, since g-->h
k, since ~h or k, and we have h
w, since k-->w

To prove that W is true, we can follow the steps provided in the question:

1. Start with the premise: g --> h
2. Use the rule of inference known as the Disjunctive Syllogism: ~(~h v k) and (g) together imply ~(~h). This is because ~(~h v k) can be simplified to ~~h ^ ~k, and since ~~h is equivalent to h, we have h ^ ~k. And combining this with the premise g gives us h ^ g.
3. Use the rule of inference known as Simplification: From h ^ g, we can derive h.
4. Use the premise: k --> w
5. Use the Rule of Modus Ponens: From h and k --> w, we can derive w.

Therefore, we have proven that W is true.

To prove W, we need to show that it logically follows from the given premises. Let's use the rules of inference and logical equivalences to derive W.

1. g (Premise)
2. ~h v k (Premise)
3. g -> h (Premise)
4. ~(~h v k) (Assumption for contradiction)
5. ~~h ^ ~k (De Morgan’s Law, 4)
6. h ^ ~k (Double Negation, 5)
7. h (Simplification, 6)
8. ~k (Simplification, 6)
9. ~~g (Double Negation, 1)
10. h (Modus Ponens, 3, 9)
11. ~k v k (Disjunctive Syllogism, 2, 10)
12. k (Tautology, 11)
13. k -> w (Premise)
14. w (Modus Ponens, 12, 13)

Therefore, we have derived W using logical reasoning.