I need help proving these arguments using rules of inference, please.
(S>B) > (SvK)
(Kv~D) ~ (H>S)
~S.W / ~H
~A > (B>~C)
~D > (~C >A)
D v ~A
~D
To prove these arguments using rules of inference, we'll go step by step and explain which rules are used at each step.
Argument 1:
1. (S > B) > (S v K) (Premise)
2. (K v ~D) > (H > S) (Premise)
3. ~S & W (Premise)
4. Assumption: ~H (Assumption for indirect proof)
5. Assumption: S (Assumption for conditional proof)
6. ~H & S (Conjunction of assumptions 4 and 5)
7. ~S (Simplification on line 3)
8. Contradiction: ~H & S and ~S on line 6 and 7
9. Therefore, ~H (Indirect proof, discharging assumption 4)
10. ~H (Reiteration)
Argument 2:
1. ~A > (B > ~C) (Premise)
2. ~D > (~C > A) (Premise)
3. D v ~A (Premise)
4. ~D (Premise)
5. Assumption: D (Assumption for disjunctive syllogism)
6. D v ~A (Premise)
7. ~A (Disjunctive syllogism on lines 5 and 6)
8. Assumption: ~C (Assumption for conditional proof)
9. B > ~C (Modus ponens on line 1 and 7)
10. ~C (Modus ponens on line 8 and 9)
11. ~C > A (Modus ponens on line 2 and 4)
12. A (Modus ponens on line 10 and 11)
13. Contradiction: A and ~A on lines 7 and 12
14. Therefore, ~C (Indirect proof, discharging assumption 8)
15. ~C (Reiteration)
In summary, we have proven ~H in Argument 1 and ~C in Argument 2 using the provided premises and the rules of inference.